Optimal. Leaf size=43 \[ \frac{2 a^2}{b (a-b x)^2}-\frac{4 a}{b (a-b x)}-\frac{\log (a-b x)}{b} \]
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Rubi [A] time = 0.0246914, antiderivative size = 43, 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} \[ \frac{2 a^2}{b (a-b x)^2}-\frac{4 a}{b (a-b x)}-\frac{\log (a-b x)}{b} \]
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 )^3} \, dx &=\int \frac{(a+b x)^2}{(a-b x)^3} \, dx\\ &=\int \left (\frac{4 a^2}{(a-b x)^3}-\frac{4 a}{(a-b x)^2}+\frac{1}{a-b x}\right ) \, dx\\ &=\frac{2 a^2}{b (a-b x)^2}-\frac{4 a}{b (a-b x)}-\frac{\log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0201887, size = 30, normalized size = 0.7 \[ -\frac{\frac{2 a (a-2 b x)}{(a-b x)^2}+\log (a-b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 47, normalized size = 1.1 \begin{align*} 2\,{\frac{{a}^{2}}{b \left ( bx-a \right ) ^{2}}}+4\,{\frac{a}{b \left ( bx-a \right ) }}-{\frac{\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 = 1.00333, size = 66, normalized size = 1.53 \begin{align*} \frac{2 \,{\left (2 \, a b x - a^{2}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} - \frac{\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 = 1.7725, size = 122, normalized size = 2.84 \begin{align*} \frac{4 \, a b x - 2 \, a^{2} -{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.420473, size = 39, normalized size = 0.91 \begin{align*} \frac{- 2 a^{2} + 4 a b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} - \frac{\log{\left (- a + b x \right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1591, size = 54, normalized size = 1.26 \begin{align*} -\frac{\log \left ({\left | b x - a \right |}\right )}{b} + \frac{2 \,{\left (2 \, a b x - a^{2}\right )}}{{\left (b x - a\right )}^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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