Optimal. Leaf size=38 \[ \frac{c^3 (a-b x)^5}{5 b}-\frac{a c^3 (a-b x)^4}{2 b} \]
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Rubi [A] time = 0.013247, antiderivative size = 38, 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 = {43} \[ \frac{c^3 (a-b x)^5}{5 b}-\frac{a c^3 (a-b x)^4}{2 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x) (a c-b c x)^3 \, dx &=\int \left (2 a (a c-b c x)^3-\frac{(a c-b c x)^4}{c}\right ) \, dx\\ &=-\frac{a c^3 (a-b x)^4}{2 b}+\frac{c^3 (a-b x)^5}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0020905, size = 40, normalized size = 1.05 \[ c^3 \left (-a^3 b x^2+a^4 x+\frac{1}{2} a b^3 x^4-\frac{1}{5} b^4 x^5\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 45, normalized size = 1.2 \begin{align*} -{\frac{{b}^{4}{c}^{3}{x}^{5}}{5}}+{\frac{a{b}^{3}{c}^{3}{x}^{4}}{2}}-{a}^{3}{c}^{3}b{x}^{2}+{a}^{4}{c}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987381, size = 59, normalized size = 1.55 \begin{align*} -\frac{1}{5} \, b^{4} c^{3} x^{5} + \frac{1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72214, size = 89, normalized size = 2.34 \begin{align*} -\frac{1}{5} x^{5} c^{3} b^{4} + \frac{1}{2} x^{4} c^{3} b^{3} a - x^{2} c^{3} b a^{3} + x c^{3} a^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.071989, size = 44, normalized size = 1.16 \begin{align*} a^{4} c^{3} x - a^{3} b c^{3} x^{2} + \frac{a b^{3} c^{3} x^{4}}{2} - \frac{b^{4} c^{3} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12726, size = 59, normalized size = 1.55 \begin{align*} -\frac{1}{5} \, b^{4} c^{3} x^{5} + \frac{1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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