Optimal. Leaf size=44 \[ \frac{8 a^3}{b (a-b x)}+\frac{12 a^2 \log (a-b x)}{b}+5 a x+\frac{b x^2}{2} \]
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Rubi [A] time = 0.0309178, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {627, 43} \[ \frac{8 a^3}{b (a-b x)}+\frac{12 a^2 \log (a-b x)}{b}+5 a x+\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{(a+b x)^3}{(a-b x)^2} \, dx\\ &=\int \left (5 a+b x+\frac{8 a^3}{(a-b x)^2}-\frac{12 a^2}{a-b x}\right ) \, dx\\ &=5 a x+\frac{b x^2}{2}+\frac{8 a^3}{b (a-b x)}+\frac{12 a^2 \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0197676, size = 45, normalized size = 1.02 \[ -\frac{8 a^3}{b (b x-a)}+\frac{12 a^2 \log (a-b x)}{b}+5 a x+\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 45, normalized size = 1. \begin{align*}{\frac{b{x}^{2}}{2}}+5\,ax-8\,{\frac{{a}^{3}}{b \left ( bx-a \right ) }}+12\,{\frac{{a}^{2}\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 = 1.04831, size = 59, normalized size = 1.34 \begin{align*} \frac{1}{2} \, b x^{2} - \frac{8 \, a^{3}}{b^{2} x - a b} + 5 \, a x + \frac{12 \, a^{2} \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 = 1.73501, size = 134, normalized size = 3.05 \begin{align*} \frac{b^{3} x^{3} + 9 \, a b^{2} x^{2} - 10 \, a^{2} b x - 16 \, a^{3} + 24 \,{\left (a^{2} b x - a^{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 = 0.383747, size = 37, normalized size = 0.84 \begin{align*} - \frac{8 a^{3}}{- a b + b^{2} x} + \frac{12 a^{2} \log{\left (- a + b x \right )}}{b} + 5 a x + \frac{b x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22885, size = 74, normalized size = 1.68 \begin{align*} \frac{12 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{8 \, a^{3}}{{\left (b x - a\right )} b} + \frac{b^{5} x^{2} + 10 \, a b^{4} x}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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