Optimal. Leaf size=108 \[ \frac{(d+e x)^6 \left (a B e^2-2 A c d e+3 B c d^2\right )}{6 e^4}-\frac{(d+e x)^5 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac{c (d+e x)^7 (3 B d-A e)}{7 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]
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Rubi [A] time = 0.137743, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {772} \[ \frac{(d+e x)^6 \left (a B e^2-2 A c d e+3 B c d^2\right )}{6 e^4}-\frac{(d+e x)^5 \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac{c (d+e x)^7 (3 B d-A e)}{7 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^4 \left (a+c x^2\right ) \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right ) (d+e x)^4}{e^3}+\frac{\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^5}{e^3}+\frac{c (-3 B d+A e) (d+e x)^6}{e^3}+\frac{B c (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^5}{5 e^4}+\frac{\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^6}{6 e^4}-\frac{c (3 B d-A e) (d+e x)^7}{7 e^4}+\frac{B c (d+e x)^8}{8 e^4}\\ \end{align*}
Mathematica [A] time = 0.0532217, size = 194, normalized size = 1.8 \[ \frac{1}{6} e^2 x^6 \left (a B e^2+4 A c d e+6 B c d^2\right )+\frac{1}{5} e x^5 \left (a A e^3+4 a B d e^2+6 A c d^2 e+4 B c d^3\right )+\frac{1}{4} d x^4 \left (4 a A e^3+6 a B d e^2+4 A c d^2 e+B c d^3\right )+\frac{1}{3} d^2 x^3 \left (6 a A e^2+4 a B d e+A c d^2\right )+\frac{1}{2} a d^3 x^2 (4 A e+B d)+a A d^4 x+\frac{1}{7} c e^3 x^7 (A e+4 B d)+\frac{1}{8} B c e^4 x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 199, normalized size = 1.8 \begin{align*}{\frac{B{e}^{4}c{x}^{8}}{8}}+{\frac{ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) c{x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) c+B{e}^{4}a \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) c+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) a \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) c+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) a \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{4}c+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) a{x}^{2}}{2}}+A{d}^{4}ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03321, size = 262, normalized size = 2.43 \begin{align*} \frac{1}{8} \, B c e^{4} x^{8} + \frac{1}{7} \,{\left (4 \, B c d e^{3} + A c e^{4}\right )} x^{7} + A a d^{4} x + \frac{1}{6} \,{\left (6 \, B c d^{2} e^{2} + 4 \, A c d e^{3} + B a e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, B c d^{3} e + 6 \, A c d^{2} e^{2} + 4 \, B a d e^{3} + A a e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{4} + 4 \, A c d^{3} e + 6 \, B a d^{2} e^{2} + 4 \, A a d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (A c d^{4} + 4 \, B a d^{3} e + 6 \, A a d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a d^{4} + 4 \, A a d^{3} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60382, size = 498, normalized size = 4.61 \begin{align*} \frac{1}{8} x^{8} e^{4} c B + \frac{4}{7} x^{7} e^{3} d c B + \frac{1}{7} x^{7} e^{4} c A + x^{6} e^{2} d^{2} c B + \frac{1}{6} x^{6} e^{4} a B + \frac{2}{3} x^{6} e^{3} d c A + \frac{4}{5} x^{5} e d^{3} c B + \frac{4}{5} x^{5} e^{3} d a B + \frac{6}{5} x^{5} e^{2} d^{2} c A + \frac{1}{5} x^{5} e^{4} a A + \frac{1}{4} x^{4} d^{4} c B + \frac{3}{2} x^{4} e^{2} d^{2} a B + x^{4} e d^{3} c A + x^{4} e^{3} d a A + \frac{4}{3} x^{3} e d^{3} a B + \frac{1}{3} x^{3} d^{4} c A + 2 x^{3} e^{2} d^{2} a A + \frac{1}{2} x^{2} d^{4} a B + 2 x^{2} e d^{3} a A + x d^{4} a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.169384, size = 226, normalized size = 2.09 \begin{align*} A a d^{4} x + \frac{B c e^{4} x^{8}}{8} + x^{7} \left (\frac{A c e^{4}}{7} + \frac{4 B c d e^{3}}{7}\right ) + x^{6} \left (\frac{2 A c d e^{3}}{3} + \frac{B a e^{4}}{6} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac{A a e^{4}}{5} + \frac{6 A c d^{2} e^{2}}{5} + \frac{4 B a d e^{3}}{5} + \frac{4 B c d^{3} e}{5}\right ) + x^{4} \left (A a d e^{3} + A c d^{3} e + \frac{3 B a d^{2} e^{2}}{2} + \frac{B c d^{4}}{4}\right ) + x^{3} \left (2 A a d^{2} e^{2} + \frac{A c d^{4}}{3} + \frac{4 B a d^{3} e}{3}\right ) + x^{2} \left (2 A a d^{3} e + \frac{B a d^{4}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14459, size = 279, normalized size = 2.58 \begin{align*} \frac{1}{8} \, B c x^{8} e^{4} + \frac{4}{7} \, B c d x^{7} e^{3} + B c d^{2} x^{6} e^{2} + \frac{4}{5} \, B c d^{3} x^{5} e + \frac{1}{4} \, B c d^{4} x^{4} + \frac{1}{7} \, A c x^{7} e^{4} + \frac{2}{3} \, A c d x^{6} e^{3} + \frac{6}{5} \, A c d^{2} x^{5} e^{2} + A c d^{3} x^{4} e + \frac{1}{3} \, A c d^{4} x^{3} + \frac{1}{6} \, B a x^{6} e^{4} + \frac{4}{5} \, B a d x^{5} e^{3} + \frac{3}{2} \, B a d^{2} x^{4} e^{2} + \frac{4}{3} \, B a d^{3} x^{3} e + \frac{1}{2} \, B a d^{4} x^{2} + \frac{1}{5} \, A a x^{5} e^{4} + A a d x^{4} e^{3} + 2 \, A a d^{2} x^{3} e^{2} + 2 \, A a d^{3} x^{2} e + A a d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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