Optimal. Leaf size=26 \[ \frac{2 a}{b (a-b x)}+\frac{\log (a-b x)}{b} \]
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Rubi [A] time = 0.017393, antiderivative size = 26, 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} \[ \frac{2 a}{b (a-b x)}+\frac{\log (a-b x)}{b} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{a+b x}{(a-b x)^2} \, dx\\ &=\int \left (\frac{2 a}{(a-b x)^2}+\frac{1}{-a+b x}\right ) \, dx\\ &=\frac{2 a}{b (a-b x)}+\frac{\log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0077067, size = 23, normalized size = 0.88 \[ \frac{\frac{2 a}{a-b x}+\log (a-b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 29, normalized size = 1.1 \begin{align*} -2\,{\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 = 1.07968, size = 38, normalized size = 1.46 \begin{align*} -\frac{2 \, a}{b^{2} x - a 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 = 1.74987, size = 62, normalized size = 2.38 \begin{align*} \frac{{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{b^{2} x - a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.320725, size = 19, normalized size = 0.73 \begin{align*} - \frac{2 a}{- a b + b^{2} x} + \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 = 1.25697, size = 39, normalized size = 1.5 \begin{align*} \frac{\log \left ({\left | b x - a \right |}\right )}{b} - \frac{2 \, a}{{\left (b x - a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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