Optimal. Leaf size=28 \[ -\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2} \]
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Rubi [A] time = 0.0146225, antiderivative size = 28, 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} \[ -\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{a^2-b^2 x^2} \, dx &=\int \frac{(a+b x)^2}{a-b x} \, dx\\ &=\int \left (-3 a-b x+\frac{4 a^2}{a-b x}\right ) \, dx\\ &=-3 a x-\frac{b x^2}{2}-\frac{4 a^2 \log (a-b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0048804, size = 28, normalized size = 1. \[ -\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 28, normalized size = 1. \begin{align*} -{\frac{b{x}^{2}}{2}}-3\,ax-4\,{\frac{{a}^{2}\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 = 1.23439, size = 36, normalized size = 1.29 \begin{align*} -\frac{1}{2} \, b x^{2} - 3 \, a x - \frac{4 \, a^{2} \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 = 1.62271, size = 66, normalized size = 2.36 \begin{align*} -\frac{b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.291171, size = 26, normalized size = 0.93 \begin{align*} - \frac{4 a^{2} \log{\left (- a + b x \right )}}{b} - 3 a x - \frac{b x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23165, size = 51, normalized size = 1.82 \begin{align*} -\frac{4 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{b^{3} x^{2} + 6 \, a b^{2} x}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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