Optimal. Leaf size=100 \[ \frac{1}{5} x^5 \left (2 b c (A e+B d)+A c^2 d+b^2 B e\right )+\frac{1}{3} A b^2 d x^3+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{4} b x^4 (A b e+2 A c d+b B d)+\frac{1}{7} B c^2 e x^7 \]
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Rubi [A] time = 0.111995, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {771} \[ \frac{1}{5} x^5 \left (2 b c (A e+B d)+A c^2 d+b^2 B e\right )+\frac{1}{3} A b^2 d x^3+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{4} b x^4 (A b e+2 A c d+b B d)+\frac{1}{7} B c^2 e x^7 \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 d x^2+b (b B d+2 A c d+A b e) x^3+\left (A c^2 d+b^2 B e+2 b c (B d+A e)\right ) x^4+c (B c d+2 b B e+A c e) x^5+B c^2 e x^6\right ) \, dx\\ &=\frac{1}{3} A b^2 d x^3+\frac{1}{4} b (b B d+2 A c d+A b e) x^4+\frac{1}{5} \left (A c^2 d+b^2 B e+2 b c (B d+A e)\right ) x^5+\frac{1}{6} c (B c d+2 b B e+A c e) x^6+\frac{1}{7} B c^2 e x^7\\ \end{align*}
Mathematica [A] time = 0.0307333, size = 101, normalized size = 1.01 \[ \frac{1}{5} x^5 \left (2 A b c e+A c^2 d+b^2 B e+2 b B c d\right )+\frac{1}{3} A b^2 d x^3+\frac{1}{6} c x^6 (A c e+2 b B e+B c d)+\frac{1}{4} b x^4 (A b e+2 A c d+b B d)+\frac{1}{7} B c^2 e x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 97, normalized size = 1. \begin{align*}{\frac{B{c}^{2}e{x}^{7}}{7}}+{\frac{ \left ( \left ( Ae+Bd \right ){c}^{2}+2\,Bebc \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}d+{b}^{2}Be+2\,bc \left ( Ae+Bd \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Abcd+{b}^{2} \left ( Ae+Bd \right ) \right ){x}^{4}}{4}}+{\frac{A{b}^{2}d{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988261, size = 139, normalized size = 1.39 \begin{align*} \frac{1}{7} \, B c^{2} e x^{7} + \frac{1}{3} \, A b^{2} d x^{3} + \frac{1}{6} \,{\left (B c^{2} d +{\left (2 \, B b c + A c^{2}\right )} e\right )} x^{6} + \frac{1}{5} \,{\left ({\left (2 \, B b c + A c^{2}\right )} d +{\left (B b^{2} + 2 \, A b c\right )} e\right )} x^{5} + \frac{1}{4} \,{\left (A b^{2} e +{\left (B b^{2} + 2 \, A b c\right )} d\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28208, size = 290, normalized size = 2.9 \begin{align*} \frac{1}{7} x^{7} e c^{2} B + \frac{1}{6} x^{6} d c^{2} B + \frac{1}{3} x^{6} e c b B + \frac{1}{6} x^{6} e c^{2} A + \frac{2}{5} x^{5} d c b B + \frac{1}{5} x^{5} e b^{2} B + \frac{1}{5} x^{5} d c^{2} A + \frac{2}{5} x^{5} e c b A + \frac{1}{4} x^{4} d b^{2} B + \frac{1}{2} x^{4} d c b A + \frac{1}{4} x^{4} e b^{2} A + \frac{1}{3} x^{3} d b^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.195064, size = 121, normalized size = 1.21 \begin{align*} \frac{A b^{2} d x^{3}}{3} + \frac{B c^{2} e x^{7}}{7} + x^{6} \left (\frac{A c^{2} e}{6} + \frac{B b c e}{3} + \frac{B c^{2} d}{6}\right ) + x^{5} \left (\frac{2 A b c e}{5} + \frac{A c^{2} d}{5} + \frac{B b^{2} e}{5} + \frac{2 B b c d}{5}\right ) + x^{4} \left (\frac{A b^{2} e}{4} + \frac{A b c d}{2} + \frac{B b^{2} d}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23086, size = 166, normalized size = 1.66 \begin{align*} \frac{1}{7} \, B c^{2} x^{7} e + \frac{1}{6} \, B c^{2} d x^{6} + \frac{1}{3} \, B b c x^{6} e + \frac{1}{6} \, A c^{2} x^{6} e + \frac{2}{5} \, B b c d x^{5} + \frac{1}{5} \, A c^{2} d x^{5} + \frac{1}{5} \, B b^{2} x^{5} e + \frac{2}{5} \, A b c x^{5} e + \frac{1}{4} \, B b^{2} d x^{4} + \frac{1}{2} \, A b c d x^{4} + \frac{1}{4} \, A b^{2} x^{4} e + \frac{1}{3} \, A b^{2} d x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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