text: ### 3.499 $$\int \frac{(d+e x)^3}{a+c x^2} \, dx$$
  
  Optimal. Leaf size=90 $\frac{e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}$
  
  [Out]
  
  (3*d*e^2*x)/c + (e^3*x^2)/(2*c) + (d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(3*
  c*d^2 - a*e^2)*Log[a + c*x^2])/(2*c^2)
  
  ________________________________________________________________________________________
  
  Rubi [A]  time = 0.0745295, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.235, Rules used = {702, 635, 205, 260} $\frac{e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}$
  
  Antiderivative was successfully veriﬁed.
  
  [In]
  
  Int[(d + e*x)^3/(a + c*x^2),x]
  
  [Out]
  
  (3*d*e^2*x)/c + (e^3*x^2)/(2*c) + (d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(3*
  c*d^2 - a*e^2)*Log[a + c*x^2])/(2*c^2)
  
  Rule 702
  
  Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
  x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])
  
  Rule 635
  
  Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
  (a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]
  
  Rule 205
  
  Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
  && PosQ[a/b]
  
  Rule 260
  
  Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
  [{a, b, m, n}, x] && EqQ[m, n - 1]
  
  Rubi steps
  
  \begin{align*} \int \frac{(d+e x)^3}{a+c x^2} \, dx &=\int \left (\frac{3 d e^2}{c}+\frac{e^3 x}{c}+\frac{c d^3-3 a d e^2+e \left (3 c d^2-a e^2\right ) x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}+\frac{\int \frac{c d^3-3 a d e^2+e \left (3 c d^2-a e^2\right ) x}{a+c x^2} \, dx}{c}\\ &=\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}+\frac{\left (d \left (c d^2-3 a e^2\right )\right ) \int \frac{1}{a+c x^2} \, dx}{c}+\frac{\left (e \left (3 c d^2-a e^2\right )\right ) \int \frac{x}{a+c x^2} \, dx}{c}\\ &=\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}+\frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}\\ \end{align*}
  
  Mathematica [A]  time = 0.0565024, size = 80, normalized size = 0.89 $\frac{e \left (\left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )+c e x (6 d+e x)\right )}{2 c^2}+\frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}$
  
  Antiderivative was successfully veriﬁed.
  
  [In]
  
  Integrate[(d + e*x)^3/(a + c*x^2),x]
  
  [Out]
  
  (d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(c*e*x*(6*d + e*x) + (3*c*d^2 - a*e^2
  )*Log[a + c*x^2]))/(2*c^2)
  
  ________________________________________________________________________________________
  
  Maple [A]  time = 0.045, size = 99, normalized size = 1.1 \begin{align*}{\frac{{e}^{3}{x}^{2}}{2\,c}}+3\,{\frac{d{e}^{2}x}{c}}-{\frac{\ln \left ( c{x}^{2}+a \right ) a{e}^{3}}{2\,{c}^{2}}}+{\frac{3\,\ln \left ( c{x}^{2}+a \right ){d}^{2}e}{2\,c}}-3\,{\frac{ad{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{{d}^{3}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  int((e*x+d)^3/(c*x^2+a),x)
  
  [Out]
  
  1/2*e^3*x^2/c+3*d*e^2*x/c-1/2/c^2*ln(c*x^2+a)*a*e^3+3/2/c*ln(c*x^2+a)*d^2*e-3/c/(a*c)^(1/2)*arctan(x*c/(a*c)^(
  1/2))*a*d*e^2+1/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*d^3
  
  ________________________________________________________________________________________
  
  Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  integrate((e*x+d)^3/(c*x^2+a),x, algorithm="maxima")
  
  [Out]
  
  Exception raised: ValueError
  
  ________________________________________________________________________________________
  
  Fricas [A]  time = 1.77174, size = 414, normalized size = 4.6 \begin{align*} \left [\frac{a c e^{3} x^{2} + 6 \, a c d e^{2} x +{\left (c d^{3} - 3 \, a d e^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} + 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) +{\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}, \frac{a c e^{3} x^{2} + 6 \, a c d e^{2} x + 2 \,{\left (c d^{3} - 3 \, a d e^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}\right ] \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  integrate((e*x+d)^3/(c*x^2+a),x, algorithm="fricas")
  
  [Out]
  
  [1/2*(a*c*e^3*x^2 + 6*a*c*d*e^2*x + (c*d^3 - 3*a*d*e^2)*sqrt(-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a
  )) + (3*a*c*d^2*e - a^2*e^3)*log(c*x^2 + a))/(a*c^2), 1/2*(a*c*e^3*x^2 + 6*a*c*d*e^2*x + 2*(c*d^3 - 3*a*d*e^2)
  *sqrt(a*c)*arctan(sqrt(a*c)*x/a) + (3*a*c*d^2*e - a^2*e^3)*log(c*x^2 + a))/(a*c^2)]
  
  ________________________________________________________________________________________
  
  Sympy [B]  time = 0.969667, size = 308, normalized size = 3.42 \begin{align*} \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{- a^{2} e^{3} - 2 a c^{2} \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{- a^{2} e^{3} - 2 a c^{2} \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \frac{3 d e^{2} x}{c} + \frac{e^{3} x^{2}}{2 c} \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  integrate((e*x+d)**3/(c*x**2+a),x)
  
  [Out]
  
  (-e*(a*e**2 - 3*c*d**2)/(2*c**2) - d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4))*log(x + (-a**2*e**3 - 2*a*c
  **2*(-e*(a*e**2 - 3*c*d**2)/(2*c**2) - d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4)) + 3*a*c*d**2*e)/(3*a*c*
  d*e**2 - c**2*d**3)) + (-e*(a*e**2 - 3*c*d**2)/(2*c**2) + d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4))*log(
  x + (-a**2*e**3 - 2*a*c**2*(-e*(a*e**2 - 3*c*d**2)/(2*c**2) + d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4))
  + 3*a*c*d**2*e)/(3*a*c*d*e**2 - c**2*d**3)) + 3*d*e**2*x/c + e**3*x**2/(2*c)
  
  ________________________________________________________________________________________
  
  Giac [A]  time = 1.34387, size = 105, normalized size = 1.17 \begin{align*} \frac{{\left (c d^{3} - 3 \, a d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} + \frac{{\left (3 \, c d^{2} e - a e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{c x^{2} e^{3} + 6 \, c d x e^{2}}{2 \, c^{2}} \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  integrate((e*x+d)^3/(c*x^2+a),x, algorithm="giac")
  
  [Out]
  
  (c*d^3 - 3*a*d*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c) + 1/2*(3*c*d^2*e - a*e^3)*log(c*x^2 + a)/c^2 + 1/2*(c*
  x^2*e^3 + 6*c*d*x*e^2)/c^2
explanation: The provided text is a detailed solution to an integral problem. It contains a math question and its corresponding answer.
  
  The question is to find the integral of \(\frac{(d + e x)^3}{a + c x^2}\) with respect to \(x\).
  
  The answer is given in multiple forms, including the optimal form, the form provided by Rubi (a computer algebra system), and the forms provided by Mathematica and Maple. Each form is verified to be correct.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: ### 3.499 $$\int \frac{(d+e x)^3}{a+c x^2} \, dx$$
  
  Optimal. Leaf size=90 $\frac{e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}$
  
  [Out]
  
  (3*d*e^2*x)/c + (e^3*x^2)/(2*c) + (d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(3*
  c*d^2 - a*e^2)*Log[a + c*x^2])/(2*c^2)
  
  ________________________________________________________________________________________
  
  Rubi [A]  time = 0.0745295, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.235, Rules used = {702, 635, 205, 260} $\frac{e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}+\frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}$
  
  Antiderivative was successfully veriﬁed.
  
  [In]
  
  Int[(d + e*x)^3/(a + c*x^2),x]
  
  [Out]
  
  (3*d*e^2*x)/c + (e^3*x^2)/(2*c) + (d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(3*
  c*d^2 - a*e^2)*Log[a + c*x^2])/(2*c^2)
  
  Rule 702
  
  Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
  x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])
  
  Rule 635
  
  Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
  (a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]
  
  Rule 205
  
  Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
  && PosQ[a/b]
  
  Rule 260
  
  Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
  [{a, b, m, n}, x] && EqQ[m, n - 1]
  
  Rubi steps
  
  \begin{align*} \int \frac{(d+e x)^3}{a+c x^2} \, dx &=\int \left (\frac{3 d e^2}{c}+\frac{e^3 x}{c}+\frac{c d^3-3 a d e^2+e \left (3 c d^2-a e^2\right ) x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}+\frac{\int \frac{c d^3-3 a d e^2+e \left (3 c d^2-a e^2\right ) x}{a+c x^2} \, dx}{c}\\ &=\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}+\frac{\left (d \left (c d^2-3 a e^2\right )\right ) \int \frac{1}{a+c x^2} \, dx}{c}+\frac{\left (e \left (3 c d^2-a e^2\right )\right ) \int \frac{x}{a+c x^2} \, dx}{c}\\ &=\frac{3 d e^2 x}{c}+\frac{e^3 x^2}{2 c}+\frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 c^2}\\ \end{align*}
  
  Mathematica [A]  time = 0.0565024, size = 80, normalized size = 0.89 $\frac{e \left (\left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )+c e x (6 d+e x)\right )}{2 c^2}+\frac{d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}$
  
  Antiderivative was successfully veriﬁed.
  
  [In]
  
  Integrate[(d + e*x)^3/(a + c*x^2),x]
  
  [Out]
  
  (d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(3/2)) + (e*(c*e*x*(6*d + e*x) + (3*c*d^2 - a*e^2
  )*Log[a + c*x^2]))/(2*c^2)
  
  ________________________________________________________________________________________
  
  Maple [A]  time = 0.045, size = 99, normalized size = 1.1 \begin{align*}{\frac{{e}^{3}{x}^{2}}{2\,c}}+3\,{\frac{d{e}^{2}x}{c}}-{\frac{\ln \left ( c{x}^{2}+a \right ) a{e}^{3}}{2\,{c}^{2}}}+{\frac{3\,\ln \left ( c{x}^{2}+a \right ){d}^{2}e}{2\,c}}-3\,{\frac{ad{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{{d}^{3}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  int((e*x+d)^3/(c*x^2+a),x)
  
  [Out]
  
  1/2*e^3*x^2/c+3*d*e^2*x/c-1/2/c^2*ln(c*x^2+a)*a*e^3+3/2/c*ln(c*x^2+a)*d^2*e-3/c/(a*c)^(1/2)*arctan(x*c/(a*c)^(
  1/2))*a*d*e^2+1/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*d^3
  
  ________________________________________________________________________________________
  
  Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  integrate((e*x+d)^3/(c*x^2+a),x, algorithm="maxima")
  
  [Out]
  
  Exception raised: ValueError
  
  ________________________________________________________________________________________
  
  Fricas [A]  time = 1.77174, size = 414, normalized size = 4.6 \begin{align*} \left [\frac{a c e^{3} x^{2} + 6 \, a c d e^{2} x +{\left (c d^{3} - 3 \, a d e^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} + 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) +{\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}, \frac{a c e^{3} x^{2} + 6 \, a c d e^{2} x + 2 \,{\left (c d^{3} - 3 \, a d e^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (3 \, a c d^{2} e - a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, a c^{2}}\right ] \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  integrate((e*x+d)^3/(c*x^2+a),x, algorithm="fricas")
  
  [Out]
  
  [1/2*(a*c*e^3*x^2 + 6*a*c*d*e^2*x + (c*d^3 - 3*a*d*e^2)*sqrt(-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a
  )) + (3*a*c*d^2*e - a^2*e^3)*log(c*x^2 + a))/(a*c^2), 1/2*(a*c*e^3*x^2 + 6*a*c*d*e^2*x + 2*(c*d^3 - 3*a*d*e^2)
  *sqrt(a*c)*arctan(sqrt(a*c)*x/a) + (3*a*c*d^2*e - a^2*e^3)*log(c*x^2 + a))/(a*c^2)]
  
  ________________________________________________________________________________________
  
  Sympy [B]  time = 0.969667, size = 308, normalized size = 3.42 \begin{align*} \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{- a^{2} e^{3} - 2 a c^{2} \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} - \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) \log{\left (x + \frac{- a^{2} e^{3} - 2 a c^{2} \left (- \frac{e \left (a e^{2} - 3 c d^{2}\right )}{2 c^{2}} + \frac{d \sqrt{- a c^{5}} \left (3 a e^{2} - c d^{2}\right )}{2 a c^{4}}\right ) + 3 a c d^{2} e}{3 a c d e^{2} - c^{2} d^{3}} \right )} + \frac{3 d e^{2} x}{c} + \frac{e^{3} x^{2}}{2 c} \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  integrate((e*x+d)**3/(c*x**2+a),x)
  
  [Out]
  
  (-e*(a*e**2 - 3*c*d**2)/(2*c**2) - d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4))*log(x + (-a**2*e**3 - 2*a*c
  **2*(-e*(a*e**2 - 3*c*d**2)/(2*c**2) - d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4)) + 3*a*c*d**2*e)/(3*a*c*
  d*e**2 - c**2*d**3)) + (-e*(a*e**2 - 3*c*d**2)/(2*c**2) + d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4))*log(
  x + (-a**2*e**3 - 2*a*c**2*(-e*(a*e**2 - 3*c*d**2)/(2*c**2) + d*sqrt(-a*c**5)*(3*a*e**2 - c*d**2)/(2*a*c**4))
  + 3*a*c*d**2*e)/(3*a*c*d*e**2 - c**2*d**3)) + 3*d*e**2*x/c + e**3*x**2/(2*c)
  
  ________________________________________________________________________________________
  
  Giac [A]  time = 1.34387, size = 105, normalized size = 1.17 \begin{align*} \frac{{\left (c d^{3} - 3 \, a d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} + \frac{{\left (3 \, c d^{2} e - a e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{c x^{2} e^{3} + 6 \, c d x e^{2}}{2 \, c^{2}} \end{align*}
  
  Veriﬁcation of antiderivative is not currently implemented for this CAS.
  
  [In]
  
  integrate((e*x+d)^3/(c*x^2+a),x, algorithm="giac")
  
  [Out]
  
  (c*d^3 - 3*a*d*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c) + 1/2*(3*c*d^2*e - a*e^3)*log(c*x^2 + a)/c^2 + 1/2*(c*
  x^2*e^3 + 6*c*d*x*e^2)/c^2
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a detailed solution to an integral problem. It contains a math question and its corresponding answer.
  
  The question is to find the integral of \(\frac{(d + e x)^3}{a + c x^2}\) with respect to \(x\).
  
  The answer is given in multiple forms, including the optimal form, the form provided by Rubi (a computer algebra system), and the forms provided by Mathematica and Maple. Each form is verified to be correct.
  
  Label: question-and-answer