Optimal. Leaf size=54 \[ -\frac{8 a^3 c^3}{b (a+b x)}-\frac{12 a^2 c^3 \log (a+b x)}{b}+5 a c^3 x-\frac{1}{2} b c^3 x^2 \]
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Rubi [A] time = 0.0313632, antiderivative size = 54, 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} \[ -\frac{8 a^3 c^3}{b (a+b x)}-\frac{12 a^2 c^3 \log (a+b x)}{b}+5 a c^3 x-\frac{1}{2} b c^3 x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a c-b c x)^3}{(a+b x)^2} \, dx &=\int \left (5 a c^3-b c^3 x+\frac{8 a^3 c^3}{(a+b x)^2}-\frac{12 a^2 c^3}{a+b x}\right ) \, dx\\ &=5 a c^3 x-\frac{1}{2} b c^3 x^2-\frac{8 a^3 c^3}{b (a+b x)}-\frac{12 a^2 c^3 \log (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0188758, size = 46, normalized size = 0.85 \[ c^3 \left (-\frac{8 a^3}{b (a+b x)}-\frac{12 a^2 \log (a+b x)}{b}+5 a x-\frac{b x^2}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 53, normalized size = 1. \begin{align*} 5\,a{c}^{3}x-{\frac{b{c}^{3}{x}^{2}}{2}}-8\,{\frac{{a}^{3}{c}^{3}}{b \left ( bx+a \right ) }}-12\,{\frac{{a}^{2}{c}^{3}\ln \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01918, size = 72, normalized size = 1.33 \begin{align*} -\frac{1}{2} \, b c^{3} x^{2} - \frac{8 \, a^{3} c^{3}}{b^{2} x + a b} + 5 \, a c^{3} x - \frac{12 \, a^{2} c^{3} \log \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51771, size = 167, normalized size = 3.09 \begin{align*} -\frac{b^{3} c^{3} x^{3} - 9 \, a b^{2} c^{3} x^{2} - 10 \, a^{2} b c^{3} x + 16 \, a^{3} c^{3} + 24 \,{\left (a^{2} b c^{3} x + a^{3} c^{3}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{2} x + a b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.388527, size = 51, normalized size = 0.94 \begin{align*} - \frac{8 a^{3} c^{3}}{a b + b^{2} x} - \frac{12 a^{2} c^{3} \log{\left (a + b x \right )}}{b} + 5 a c^{3} x - \frac{b c^{3} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05334, size = 108, normalized size = 2. \begin{align*} \frac{12 \, a^{2} c^{3} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{8 \, a^{3} c^{3}}{{\left (b x + a\right )} b} + \frac{{\left (\frac{12 \, a c^{3}}{b x + a} - c^{3}\right )}{\left (b x + a\right )}^{2}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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