Optimal. Leaf size=127 \[ \frac{1}{7} c x^7 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{6} b x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 b e+3 c d)+\frac{1}{4} b^3 d^2 x^4+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9 \]
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Rubi [A] time = 0.109911, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ \frac{1}{7} c x^7 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{6} b x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 b e+3 c d)+\frac{1}{4} b^3 d^2 x^4+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9 \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx &=\int \left (b^3 d^2 x^3+b^2 d (3 c d+2 b e) x^4+b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^5+c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^6+c^2 e (2 c d+3 b e) x^7+c^3 e^2 x^8\right ) \, dx\\ &=\frac{1}{4} b^3 d^2 x^4+\frac{1}{5} b^2 d (3 c d+2 b e) x^5+\frac{1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac{1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac{1}{8} c^2 e (2 c d+3 b e) x^8+\frac{1}{9} c^3 e^2 x^9\\ \end{align*}
Mathematica [A] time = 0.0239765, size = 127, normalized size = 1. \[ \frac{1}{7} c x^7 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac{1}{6} b x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac{1}{5} b^2 d x^5 (2 b e+3 c d)+\frac{1}{4} b^3 d^2 x^4+\frac{1}{8} c^2 e x^8 (3 b e+2 c d)+\frac{1}{9} c^3 e^2 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 128, normalized size = 1. \begin{align*}{\frac{{c}^{3}{e}^{2}{x}^{9}}{9}}+{\frac{ \left ( 3\,{e}^{2}b{c}^{2}+2\,de{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,{e}^{2}{b}^{2}c+6\,deb{c}^{2}+{d}^{2}{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ({b}^{3}{e}^{2}+6\,{b}^{2}cde+3\,{d}^{2}b{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,de{b}^{3}+3\,{d}^{2}{b}^{2}c \right ){x}^{5}}{5}}+{\frac{{b}^{3}{d}^{2}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13131, size = 171, normalized size = 1.35 \begin{align*} \frac{1}{9} \, c^{3} e^{2} x^{9} + \frac{1}{4} \, b^{3} d^{2} x^{4} + \frac{1}{8} \,{\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, b c^{2} d^{2} + 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, b^{2} c d^{2} + 2 \, b^{3} d e\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42058, size = 301, normalized size = 2.37 \begin{align*} \frac{1}{9} x^{9} e^{2} c^{3} + \frac{1}{4} x^{8} e d c^{3} + \frac{3}{8} x^{8} e^{2} c^{2} b + \frac{1}{7} x^{7} d^{2} c^{3} + \frac{6}{7} x^{7} e d c^{2} b + \frac{3}{7} x^{7} e^{2} c b^{2} + \frac{1}{2} x^{6} d^{2} c^{2} b + x^{6} e d c b^{2} + \frac{1}{6} x^{6} e^{2} b^{3} + \frac{3}{5} x^{5} d^{2} c b^{2} + \frac{2}{5} x^{5} e d b^{3} + \frac{1}{4} x^{4} d^{2} b^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.237614, size = 138, normalized size = 1.09 \begin{align*} \frac{b^{3} d^{2} x^{4}}{4} + \frac{c^{3} e^{2} x^{9}}{9} + x^{8} \left (\frac{3 b c^{2} e^{2}}{8} + \frac{c^{3} d e}{4}\right ) + x^{7} \left (\frac{3 b^{2} c e^{2}}{7} + \frac{6 b c^{2} d e}{7} + \frac{c^{3} d^{2}}{7}\right ) + x^{6} \left (\frac{b^{3} e^{2}}{6} + b^{2} c d e + \frac{b c^{2} d^{2}}{2}\right ) + x^{5} \left (\frac{2 b^{3} d e}{5} + \frac{3 b^{2} c d^{2}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31292, size = 181, normalized size = 1.43 \begin{align*} \frac{1}{9} \, c^{3} x^{9} e^{2} + \frac{1}{4} \, c^{3} d x^{8} e + \frac{1}{7} \, c^{3} d^{2} x^{7} + \frac{3}{8} \, b c^{2} x^{8} e^{2} + \frac{6}{7} \, b c^{2} d x^{7} e + \frac{1}{2} \, b c^{2} d^{2} x^{6} + \frac{3}{7} \, b^{2} c x^{7} e^{2} + b^{2} c d x^{6} e + \frac{3}{5} \, b^{2} c d^{2} x^{5} + \frac{1}{6} \, b^{3} x^{6} e^{2} + \frac{2}{5} \, b^{3} d x^{5} e + \frac{1}{4} \, b^{3} d^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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