Optimal. Leaf size=31 \[ \frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x \]
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Rubi [A] time = 0.0214676, antiderivative size = 31, 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} \[ \frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^4}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{(a+b x)^2}{(a-b x)^2} \, dx\\ &=\int \left (1+\frac{4 a^2}{(a-b x)^2}-\frac{4 a}{a-b x}\right ) \, dx\\ &=x+\frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0195169, size = 32, normalized size = 1.03 \[ -\frac{4 a^2}{b (b x-a)}+\frac{4 a \log (a-b x)}{b}+x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 34, normalized size = 1.1 \begin{align*} x-4\,{\frac{{a}^{2}}{b \left ( bx-a \right ) }}+4\,{\frac{a\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 = 1.02119, size = 45, normalized size = 1.45 \begin{align*} -\frac{4 \, a^{2}}{b^{2} x - a b} + x + \frac{4 \, a \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 = 1.75862, size = 97, normalized size = 3.13 \begin{align*} \frac{b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \,{\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} x - a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.341498, size = 26, normalized size = 0.84 \begin{align*} - \frac{4 a^{2}}{- a b + b^{2} x} + \frac{4 a \log{\left (- a + b x \right )}}{b} + x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24342, size = 46, normalized size = 1.48 \begin{align*} x + \frac{4 \, a \log \left ({\left | b x - a \right |}\right )}{b} - \frac{4 \, a^{2}}{{\left (b x - a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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