Optimal. Leaf size=43 \[ -\frac{4 a^2 \log (a-b x)}{b c}-\frac{(a+b x)^2}{2 b c}-\frac{2 a x}{c} \]
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Rubi [A] time = 0.0137355, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ -\frac{4 a^2 \log (a-b x)}{b c}-\frac{(a+b x)^2}{2 b c}-\frac{2 a x}{c} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{a c-b c x} \, dx &=\int \left (-\frac{2 a}{c}-\frac{a+b x}{c}+\frac{4 a^2}{a c-b c x}\right ) \, dx\\ &=-\frac{2 a x}{c}-\frac{(a+b x)^2}{2 b c}-\frac{4 a^2 \log (a-b x)}{b c}\\ \end{align*}
Mathematica [A] time = 0.0069963, size = 37, normalized size = 0.86 \[ -\frac{4 a^2 \log (a-b x)}{b c}-\frac{3 a x}{c}-\frac{b x^2}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 37, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{2}}{2\,c}}-3\,{\frac{ax}{c}}-4\,{\frac{{a}^{2}\ln \left ( bx-a \right ) }{bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01021, size = 47, normalized size = 1.09 \begin{align*} -\frac{4 \, a^{2} \log \left (b x - a\right )}{b c} - \frac{b x^{2} + 6 \, a x}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51219, size = 72, normalized size = 1.67 \begin{align*} -\frac{b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.297853, size = 31, normalized size = 0.72 \begin{align*} - \frac{4 a^{2} \log{\left (- a + b x \right )}}{b c} - \frac{3 a x}{c} - \frac{b x^{2}}{2 c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04645, size = 62, normalized size = 1.44 \begin{align*} -\frac{4 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b c} - \frac{b^{3} c x^{2} + 6 \, a b^{2} c x}{2 \, b^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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