Optimal. Leaf size=57 \[ -\frac{a^2 c^3 (a-b x)^4}{b}-\frac{c^3 (a-b x)^6}{6 b}+\frac{4 a c^3 (a-b x)^5}{5 b} \]
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Rubi [A] time = 0.0303072, antiderivative size = 57, 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 = {43} \[ -\frac{a^2 c^3 (a-b x)^4}{b}-\frac{c^3 (a-b x)^6}{6 b}+\frac{4 a c^3 (a-b x)^5}{5 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x)^2 (a c-b c x)^3 \, dx &=\int \left (4 a^2 (a c-b c x)^3-\frac{4 a (a c-b c x)^4}{c}+\frac{(a c-b c x)^5}{c^2}\right ) \, dx\\ &=-\frac{a^2 c^3 (a-b x)^4}{b}+\frac{4 a c^3 (a-b x)^5}{5 b}-\frac{c^3 (a-b x)^6}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0029511, size = 68, normalized size = 1.19 \[ c^3 \left (\frac{1}{2} a^2 b^3 x^4-\frac{2}{3} a^3 b^2 x^3-\frac{1}{2} a^4 b x^2+a^5 x+\frac{1}{5} a b^4 x^5-\frac{1}{6} b^5 x^6\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 73, normalized size = 1.3 \begin{align*} -{\frac{{b}^{5}{c}^{3}{x}^{6}}{6}}+{\frac{a{b}^{4}{c}^{3}{x}^{5}}{5}}+{\frac{{a}^{2}{b}^{3}{c}^{3}{x}^{4}}{2}}-{\frac{2\,{a}^{3}{c}^{3}{b}^{2}{x}^{3}}{3}}-{\frac{{a}^{4}{c}^{3}b{x}^{2}}{2}}+{a}^{5}{c}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99351, size = 97, normalized size = 1.7 \begin{align*} -\frac{1}{6} \, b^{5} c^{3} x^{6} + \frac{1}{5} \, a b^{4} c^{3} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{3} x^{4} - \frac{2}{3} \, a^{3} b^{2} c^{3} x^{3} - \frac{1}{2} \, a^{4} b c^{3} x^{2} + a^{5} c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34014, size = 154, normalized size = 2.7 \begin{align*} -\frac{1}{6} x^{6} c^{3} b^{5} + \frac{1}{5} x^{5} c^{3} b^{4} a + \frac{1}{2} x^{4} c^{3} b^{3} a^{2} - \frac{2}{3} x^{3} c^{3} b^{2} a^{3} - \frac{1}{2} x^{2} c^{3} b a^{4} + x c^{3} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.076897, size = 78, normalized size = 1.37 \begin{align*} a^{5} c^{3} x - \frac{a^{4} b c^{3} x^{2}}{2} - \frac{2 a^{3} b^{2} c^{3} x^{3}}{3} + \frac{a^{2} b^{3} c^{3} x^{4}}{2} + \frac{a b^{4} c^{3} x^{5}}{5} - \frac{b^{5} c^{3} x^{6}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05994, size = 97, normalized size = 1.7 \begin{align*} -\frac{1}{6} \, b^{5} c^{3} x^{6} + \frac{1}{5} \, a b^{4} c^{3} x^{5} + \frac{1}{2} \, a^{2} b^{3} c^{3} x^{4} - \frac{2}{3} \, a^{3} b^{2} c^{3} x^{3} - \frac{1}{2} \, a^{4} b c^{3} x^{2} + a^{5} c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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