Optimal. Leaf size=80 \[ \frac{1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac{1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac{1}{4} x^4 (A c e+b B e+B c d)+\frac{1}{5} B c e x^5 \]
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Rubi [A] time = 0.0814684, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {771} \[ \frac{1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac{1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac{1}{4} x^4 (A c e+b B e+B c d)+\frac{1}{5} B c e x^5 \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a A d+(A b d+a B d+a A e) x+(b B d+A c d+A b e+a B e) x^2+(B c d+b B e+A c e) x^3+B c e x^4\right ) \, dx\\ &=a A d x+\frac{1}{2} (A b d+a B d+a A e) x^2+\frac{1}{3} (b B d+A c d+A b e+a B e) x^3+\frac{1}{4} (B c d+b B e+A c e) x^4+\frac{1}{5} B c e x^5\\ \end{align*}
Mathematica [A] time = 0.0314289, size = 80, normalized size = 1. \[ \frac{1}{3} x^3 (a B e+A b e+A c d+b B d)+\frac{1}{2} x^2 (a A e+a B d+A b d)+a A d x+\frac{1}{4} x^4 (A c e+b B e+B c d)+\frac{1}{5} B c e x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 76, normalized size = 1. \begin{align*}{\frac{Bce{x}^{5}}{5}}+{\frac{ \left ( c \left ( Ae+Bd \right ) +bBe \right ){x}^{4}}{4}}+{\frac{ \left ( Acd+b \left ( Ae+Bd \right ) +aBe \right ){x}^{3}}{3}}+{\frac{ \left ( Abd+a \left ( Ae+Bd \right ) \right ){x}^{2}}{2}}+aAdx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38863, size = 103, normalized size = 1.29 \begin{align*} \frac{1}{5} \, B c e x^{5} + \frac{1}{4} \,{\left (B c d +{\left (B b + A c\right )} e\right )} x^{4} + A a d x + \frac{1}{3} \,{\left ({\left (B b + A c\right )} d +{\left (B a + A b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a e +{\left (B a + A b\right )} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16278, size = 250, normalized size = 3.12 \begin{align*} \frac{1}{5} x^{5} e c B + \frac{1}{4} x^{4} d c B + \frac{1}{4} x^{4} e b B + \frac{1}{4} x^{4} e c A + \frac{1}{3} x^{3} d b B + \frac{1}{3} x^{3} e a B + \frac{1}{3} x^{3} d c A + \frac{1}{3} x^{3} e b A + \frac{1}{2} x^{2} d a B + \frac{1}{2} x^{2} d b A + \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 = 0.070365, size = 94, normalized size = 1.18 \begin{align*} A a d x + \frac{B c e x^{5}}{5} + x^{4} \left (\frac{A c e}{4} + \frac{B b e}{4} + \frac{B c d}{4}\right ) + x^{3} \left (\frac{A b e}{3} + \frac{A c d}{3} + \frac{B a e}{3} + \frac{B b d}{3}\right ) + x^{2} \left (\frac{A a e}{2} + \frac{A b d}{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 = 1.12722, size = 135, normalized size = 1.69 \begin{align*} \frac{1}{5} \, B c x^{5} e + \frac{1}{4} \, B c d x^{4} + \frac{1}{4} \, B b x^{4} e + \frac{1}{4} \, A c x^{4} e + \frac{1}{3} \, B b d x^{3} + \frac{1}{3} \, A c d x^{3} + \frac{1}{3} \, B a x^{3} e + \frac{1}{3} \, A b x^{3} e + \frac{1}{2} \, B a d x^{2} + \frac{1}{2} \, A b 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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