Optimal. Leaf size=62 \[ \frac{1}{3} x^3 (a B e+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{4} c x^4 (A e+B d)+\frac{1}{5} B c e x^5 \]
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Rubi [A] time = 0.0584914, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {772} \[ \frac{1}{3} x^3 (a B e+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{4} c x^4 (A e+B d)+\frac{1}{5} B c e x^5 \]
Antiderivative was successfully verified.
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Rule 772
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (a+c x^2\right ) \, dx &=\int \left (a A d+a (B d+A e) x+(A c d+a B e) x^2+c (B d+A e) x^3+B c e x^4\right ) \, dx\\ &=a A d x+\frac{1}{2} a (B d+A e) x^2+\frac{1}{3} (A c d+a B e) x^3+\frac{1}{4} c (B d+A e) x^4+\frac{1}{5} B c e x^5\\ \end{align*}
Mathematica [A] time = 0.0211432, size = 62, normalized size = 1. \[ \frac{1}{3} x^3 (a B e+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{4} c x^4 (A e+B d)+\frac{1}{5} B c e x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 55, normalized size = 0.9 \begin{align*} aAdx+{\frac{a \left ( Ae+Bd \right ){x}^{2}}{2}}+{\frac{ \left ( Acd+aBe \right ){x}^{3}}{3}}+{\frac{c \left ( Ae+Bd \right ){x}^{4}}{4}}+{\frac{Bce{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986852, size = 76, normalized size = 1.23 \begin{align*} \frac{1}{5} \, B c e x^{5} + \frac{1}{4} \,{\left (B c d + A c e\right )} x^{4} + A a d x + \frac{1}{3} \,{\left (A c d + B a e\right )} x^{3} + \frac{1}{2} \,{\left (B a d + A a e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65312, size = 163, normalized size = 2.63 \begin{align*} \frac{1}{5} x^{5} e c B + \frac{1}{4} x^{4} d c B + \frac{1}{4} x^{4} e c A + \frac{1}{3} x^{3} e a B + \frac{1}{3} x^{3} d c A + \frac{1}{2} x^{2} d a B + \frac{1}{2} x^{2} e a A + x d a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.191982, size = 66, normalized size = 1.06 \begin{align*} A a d x + \frac{B c e x^{5}}{5} + x^{4} \left (\frac{A c e}{4} + \frac{B c d}{4}\right ) + x^{3} \left (\frac{A c d}{3} + \frac{B a e}{3}\right ) + x^{2} \left (\frac{A a e}{2} + \frac{B a d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22705, size = 89, normalized size = 1.44 \begin{align*} \frac{1}{5} \, B c x^{5} e + \frac{1}{4} \, B c d x^{4} + \frac{1}{4} \, A c x^{4} e + \frac{1}{3} \, A c d x^{3} + \frac{1}{3} \, B a x^{3} e + \frac{1}{2} \, B a d x^{2} + \frac{1}{2} \, A a x^{2} e + A a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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