Optimal. Leaf size=75 \[ \frac{1}{5} b^2 x^5 (b e+3 c d)+\frac{1}{4} b^3 d x^4+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{2} b c x^6 (b e+c d)+\frac{1}{8} c^3 e x^8 \]
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Rubi [A] time = 0.062641, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {631} \[ \frac{1}{5} b^2 x^5 (b e+3 c d)+\frac{1}{4} b^3 d x^4+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{2} b c x^6 (b e+c d)+\frac{1}{8} c^3 e x^8 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int (d+e x) \left (b x+c x^2\right )^3 \, dx &=\int \left (b^3 d x^3+b^2 (3 c d+b e) x^4+3 b c (c d+b e) x^5+c^2 (c d+3 b e) x^6+c^3 e x^7\right ) \, dx\\ &=\frac{1}{4} b^3 d x^4+\frac{1}{5} b^2 (3 c d+b e) x^5+\frac{1}{2} b c (c d+b e) x^6+\frac{1}{7} c^2 (c d+3 b e) x^7+\frac{1}{8} c^3 e x^8\\ \end{align*}
Mathematica [A] time = 0.0102053, size = 75, normalized size = 1. \[ \frac{1}{5} b^2 x^5 (b e+3 c d)+\frac{1}{4} b^3 d x^4+\frac{1}{7} c^2 x^7 (3 b e+c d)+\frac{1}{2} b c x^6 (b e+c d)+\frac{1}{8} c^3 e x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 76, normalized size = 1. \begin{align*}{\frac{{c}^{3}e{x}^{8}}{8}}+{\frac{ \left ( 3\,eb{c}^{2}+d{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,e{b}^{2}c+3\,db{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( e{b}^{3}+3\,d{b}^{2}c \right ){x}^{5}}{5}}+{\frac{{b}^{3}d{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11304, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{8} \, c^{3} e x^{8} + \frac{1}{4} \, b^{3} d x^{4} + \frac{1}{7} \,{\left (c^{3} d + 3 \, b c^{2} e\right )} x^{7} + \frac{1}{2} \,{\left (b c^{2} d + b^{2} c e\right )} x^{6} + \frac{1}{5} \,{\left (3 \, b^{2} c d + b^{3} e\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35286, size = 182, normalized size = 2.43 \begin{align*} \frac{1}{8} x^{8} e c^{3} + \frac{1}{7} x^{7} d c^{3} + \frac{3}{7} x^{7} e c^{2} b + \frac{1}{2} x^{6} d c^{2} b + \frac{1}{2} x^{6} e c b^{2} + \frac{3}{5} x^{5} d c b^{2} + \frac{1}{5} x^{5} e b^{3} + \frac{1}{4} x^{4} d b^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.22159, size = 80, normalized size = 1.07 \begin{align*} \frac{b^{3} d x^{4}}{4} + \frac{c^{3} e x^{8}}{8} + x^{7} \left (\frac{3 b c^{2} e}{7} + \frac{c^{3} d}{7}\right ) + x^{6} \left (\frac{b^{2} c e}{2} + \frac{b c^{2} d}{2}\right ) + x^{5} \left (\frac{b^{3} e}{5} + \frac{3 b^{2} c d}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22847, size = 109, normalized size = 1.45 \begin{align*} \frac{1}{8} \, c^{3} x^{8} e + \frac{1}{7} \, c^{3} d x^{7} + \frac{3}{7} \, b c^{2} x^{7} e + \frac{1}{2} \, b c^{2} d x^{6} + \frac{1}{2} \, b^{2} c x^{6} e + \frac{3}{5} \, b^{2} c d x^{5} + \frac{1}{5} \, b^{3} x^{5} e + \frac{1}{4} \, b^{3} d x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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