Optimal. Leaf size=17 \[ -\frac {2 a \log (a-b x)}{b}-x \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 43
Rule 627
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*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {2 a \log (a-b x)}{b}-x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 20, normalized size = 1.18 \[ -\frac {b x + 2 \, a \log \left (b x - a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 19, normalized size = 1.12 \[ -x - \frac {2 \, a \log \left ({\left | b x - a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 19, normalized size = 1.12 \[ -\frac {2 a \ln \left (b x -a \right )}{b}-x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 18, normalized size = 1.06 \[ -x - \frac {2 \, a \log \left (b x - a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 18, normalized size = 1.06 \[ -x-\frac {2\,a\,\ln \left (b\,x-a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 14, normalized size = 0.82 \[ - \frac {2 a \log {\left (- a + b x \right )}}{b} - x \]
Verification of antiderivative is not currently implemented for this CAS.
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