Optimal. Leaf size=225 \[ \frac{1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac{1}{8} e x^8 \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac{1}{7} d x^7 \left (18 b^2 c d e^2+4 b^3 e^3+12 b c^2 d^2 e+c^3 d^3\right )+\frac{1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac{1}{4} b^3 d^4 x^4+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11} \]
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Rubi [A] time = 0.215499, antiderivative size = 225, 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} \[ \frac{1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac{1}{8} e x^8 \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac{1}{7} d x^7 \left (18 b^2 c d e^2+4 b^3 e^3+12 b c^2 d^2 e+c^3 d^3\right )+\frac{1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac{1}{4} b^3 d^4 x^4+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx &=\int \left (b^3 d^4 x^3+b^2 d^3 (3 c d+4 b e) x^4+3 b d^2 \left (c^2 d^2+4 b c d e+2 b^2 e^2\right ) x^5+d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right ) x^6+e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right ) x^7+3 c e^2 \left (2 c^2 d^2+4 b c d e+b^2 e^2\right ) x^8+c^2 e^3 (4 c d+3 b e) x^9+c^3 e^4 x^{10}\right ) \, dx\\ &=\frac{1}{4} b^3 d^4 x^4+\frac{1}{5} b^2 d^3 (3 c d+4 b e) x^5+\frac{1}{2} b d^2 \left (c^2 d^2+4 b c d e+2 b^2 e^2\right ) x^6+\frac{1}{7} d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right ) x^7+\frac{1}{8} e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right ) x^8+\frac{1}{3} c e^2 \left (2 c^2 d^2+4 b c d e+b^2 e^2\right ) x^9+\frac{1}{10} c^2 e^3 (4 c d+3 b e) x^{10}+\frac{1}{11} c^3 e^4 x^{11}\\ \end{align*}
Mathematica [A] time = 0.0347559, size = 225, normalized size = 1. \[ \frac{1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac{1}{8} e x^8 \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac{1}{7} d x^7 \left (18 b^2 c d e^2+4 b^3 e^3+12 b c^2 d^2 e+c^3 d^3\right )+\frac{1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac{1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac{1}{4} b^3 d^4 x^4+\frac{1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac{1}{11} c^3 e^4 x^{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 232, normalized size = 1. \begin{align*}{\frac{{c}^{3}{e}^{4}{x}^{11}}{11}}+{\frac{ \left ( 3\,{e}^{4}b{c}^{2}+4\,d{e}^{3}{c}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,{e}^{4}{b}^{2}c+12\,d{e}^{3}b{c}^{2}+6\,{d}^{2}{e}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ({e}^{4}{b}^{3}+12\,d{e}^{3}{b}^{2}c+18\,{d}^{2}{e}^{2}b{c}^{2}+4\,{d}^{3}e{c}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 4\,d{e}^{3}{b}^{3}+18\,{d}^{2}{e}^{2}{b}^{2}c+12\,{d}^{3}eb{c}^{2}+{d}^{4}{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{b}^{3}+12\,{d}^{3}e{b}^{2}c+3\,{d}^{4}b{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,{d}^{3}e{b}^{3}+3\,{d}^{4}{b}^{2}c \right ){x}^{5}}{5}}+{\frac{{b}^{3}{d}^{4}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18534, size = 309, normalized size = 1.37 \begin{align*} \frac{1}{11} \, c^{3} e^{4} x^{11} + \frac{1}{4} \, b^{3} d^{4} x^{4} + \frac{1}{10} \,{\left (4 \, c^{3} d e^{3} + 3 \, b c^{2} e^{4}\right )} x^{10} + \frac{1}{3} \,{\left (2 \, c^{3} d^{2} e^{2} + 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} x^{9} + \frac{1}{8} \,{\left (4 \, c^{3} d^{3} e + 18 \, b c^{2} d^{2} e^{2} + 12 \, b^{2} c d e^{3} + b^{3} e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (c^{3} d^{4} + 12 \, b c^{2} d^{3} e + 18 \, b^{2} c d^{2} e^{2} + 4 \, b^{3} d e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b c^{2} d^{4} + 4 \, b^{2} c d^{3} e + 2 \, b^{3} d^{2} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, b^{2} c d^{4} + 4 \, b^{3} d^{3} e\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46311, size = 556, normalized size = 2.47 \begin{align*} \frac{1}{11} x^{11} e^{4} c^{3} + \frac{2}{5} x^{10} e^{3} d c^{3} + \frac{3}{10} x^{10} e^{4} c^{2} b + \frac{2}{3} x^{9} e^{2} d^{2} c^{3} + \frac{4}{3} x^{9} e^{3} d c^{2} b + \frac{1}{3} x^{9} e^{4} c b^{2} + \frac{1}{2} x^{8} e d^{3} c^{3} + \frac{9}{4} x^{8} e^{2} d^{2} c^{2} b + \frac{3}{2} x^{8} e^{3} d c b^{2} + \frac{1}{8} x^{8} e^{4} b^{3} + \frac{1}{7} x^{7} d^{4} c^{3} + \frac{12}{7} x^{7} e d^{3} c^{2} b + \frac{18}{7} x^{7} e^{2} d^{2} c b^{2} + \frac{4}{7} x^{7} e^{3} d b^{3} + \frac{1}{2} x^{6} d^{4} c^{2} b + 2 x^{6} e d^{3} c b^{2} + x^{6} e^{2} d^{2} b^{3} + \frac{3}{5} x^{5} d^{4} c b^{2} + \frac{4}{5} x^{5} e d^{3} b^{3} + \frac{1}{4} x^{4} d^{4} b^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.218664, size = 257, normalized size = 1.14 \begin{align*} \frac{b^{3} d^{4} x^{4}}{4} + \frac{c^{3} e^{4} x^{11}}{11} + x^{10} \left (\frac{3 b c^{2} e^{4}}{10} + \frac{2 c^{3} d e^{3}}{5}\right ) + x^{9} \left (\frac{b^{2} c e^{4}}{3} + \frac{4 b c^{2} d e^{3}}{3} + \frac{2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac{b^{3} e^{4}}{8} + \frac{3 b^{2} c d e^{3}}{2} + \frac{9 b c^{2} d^{2} e^{2}}{4} + \frac{c^{3} d^{3} e}{2}\right ) + x^{7} \left (\frac{4 b^{3} d e^{3}}{7} + \frac{18 b^{2} c d^{2} e^{2}}{7} + \frac{12 b c^{2} d^{3} e}{7} + \frac{c^{3} d^{4}}{7}\right ) + x^{6} \left (b^{3} d^{2} e^{2} + 2 b^{2} c d^{3} e + \frac{b c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac{4 b^{3} d^{3} e}{5} + \frac{3 b^{2} c d^{4}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33182, size = 327, normalized size = 1.45 \begin{align*} \frac{1}{11} \, c^{3} x^{11} e^{4} + \frac{2}{5} \, c^{3} d x^{10} e^{3} + \frac{2}{3} \, c^{3} d^{2} x^{9} e^{2} + \frac{1}{2} \, c^{3} d^{3} x^{8} e + \frac{1}{7} \, c^{3} d^{4} x^{7} + \frac{3}{10} \, b c^{2} x^{10} e^{4} + \frac{4}{3} \, b c^{2} d x^{9} e^{3} + \frac{9}{4} \, b c^{2} d^{2} x^{8} e^{2} + \frac{12}{7} \, b c^{2} d^{3} x^{7} e + \frac{1}{2} \, b c^{2} d^{4} x^{6} + \frac{1}{3} \, b^{2} c x^{9} e^{4} + \frac{3}{2} \, b^{2} c d x^{8} e^{3} + \frac{18}{7} \, b^{2} c d^{2} x^{7} e^{2} + 2 \, b^{2} c d^{3} x^{6} e + \frac{3}{5} \, b^{2} c d^{4} x^{5} + \frac{1}{8} \, b^{3} x^{8} e^{4} + \frac{4}{7} \, b^{3} d x^{7} e^{3} + b^{3} d^{2} x^{6} e^{2} + \frac{4}{5} \, b^{3} d^{3} x^{5} e + \frac{1}{4} \, b^{3} d^{4} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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