Optimal. Leaf size=17 \[ -\frac{2 a \log (a-b x)}{b}-x \]
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Rubi [A] time = 0.0142805, antiderivative size = 17, 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} \[ -\frac{2 a \log (a-b x)}{b}-x \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{a^2-b^2 x^2} \, dx &=\int \frac{a+b x}{a-b x} \, dx\\ &=\int \left (-1+\frac{2 a}{a-b x}\right ) \, dx\\ &=-x-\frac{2 a \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0029428, size = 17, normalized size = 1. \[ -\frac{2 a \log (a-b x)}{b}-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 19, normalized size = 1.1 \begin{align*} -x-2\,{\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 = 1.03761, size = 24, normalized size = 1.41 \begin{align*} -x - \frac{2 \, 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 = 1.83442, size = 39, normalized size = 2.29 \begin{align*} -\frac{b x + 2 \, a \log \left (b x - a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.269772, size = 14, normalized size = 0.82 \begin{align*} - \frac{2 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 = 1.18335, size = 26, normalized size = 1.53 \begin{align*} -x - \frac{2 \, a \log \left ({\left | b x - a \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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