Optimal. Leaf size=75 \[ \frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{4} A b^3 x^4+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{2} b c x^6 (A c+b B)+\frac{1}{8} B c^3 x^8 \]
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Rubi [A] time = 0.0551682, 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} \[ \frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{4} A b^3 x^4+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{2} b c x^6 (A c+b B)+\frac{1}{8} B c^3 x^8 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^3+b^2 (b B+3 A c) x^4+3 b c (b B+A c) x^5+c^2 (3 b B+A c) x^6+B c^3 x^7\right ) \, dx\\ &=\frac{1}{4} A b^3 x^4+\frac{1}{5} b^2 (b B+3 A c) x^5+\frac{1}{2} b c (b B+A c) x^6+\frac{1}{7} c^2 (3 b B+A c) x^7+\frac{1}{8} B c^3 x^8\\ \end{align*}
Mathematica [A] time = 0.0119994, size = 75, normalized size = 1. \[ \frac{1}{5} b^2 x^5 (3 A c+b B)+\frac{1}{4} A b^3 x^4+\frac{1}{7} c^2 x^7 (A c+3 b B)+\frac{1}{2} b c x^6 (A c+b B)+\frac{1}{8} B c^3 x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 76, normalized size = 1. \begin{align*}{\frac{B{c}^{3}{x}^{8}}{8}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,Ab{c}^{2}+3\,B{b}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{b}^{2}c+{b}^{3}B \right ){x}^{5}}{5}}+{\frac{A{b}^{3}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01821, size = 99, normalized size = 1.32 \begin{align*} \frac{1}{8} \, B c^{3} x^{8} + \frac{1}{4} \, A b^{3} x^{4} + \frac{1}{7} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B b^{2} c + A b c^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63444, size = 182, normalized size = 2.43 \begin{align*} \frac{1}{8} x^{8} c^{3} B + \frac{3}{7} x^{7} c^{2} b B + \frac{1}{7} x^{7} c^{3} A + \frac{1}{2} x^{6} c b^{2} B + \frac{1}{2} x^{6} c^{2} b A + \frac{1}{5} x^{5} b^{3} B + \frac{3}{5} x^{5} c b^{2} A + \frac{1}{4} x^{4} b^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.076817, size = 80, normalized size = 1.07 \begin{align*} \frac{A b^{3} x^{4}}{4} + \frac{B c^{3} x^{8}}{8} + x^{7} \left (\frac{A c^{3}}{7} + \frac{3 B b c^{2}}{7}\right ) + x^{6} \left (\frac{A b c^{2}}{2} + \frac{B b^{2} c}{2}\right ) + x^{5} \left (\frac{3 A b^{2} c}{5} + \frac{B b^{3}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10901, size = 104, normalized size = 1.39 \begin{align*} \frac{1}{8} \, B c^{3} x^{8} + \frac{3}{7} \, B b c^{2} x^{7} + \frac{1}{7} \, A c^{3} x^{7} + \frac{1}{2} \, B b^{2} c x^{6} + \frac{1}{2} \, A b c^{2} x^{6} + \frac{1}{5} \, B b^{3} x^{5} + \frac{3}{5} \, A b^{2} c x^{5} + \frac{1}{4} \, A b^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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