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 = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.44, size = 34, normalized size = 0.79 \[ -\frac {b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 46, normalized size = 1.07 \[ -\frac {4 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b c} - \frac {b^{3} c x^{2} + 6 \, a b^{2} c x}{2 \, b^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 0.86 \[ -\frac {b \,x^{2}}{2 c}-\frac {4 a^{2} \ln \left (b x -a \right )}{b c}-\frac {3 a x}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 35, normalized size = 0.81 \[ -\frac {4 \, a^{2} \log \left (b x - a\right )}{b c} - \frac {b x^{2} + 6 \, a x}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 34, normalized size = 0.79 \[ -\frac {8\,a^2\,\ln \left (b\,x-a\right )+b^2\,x^2+6\,a\,b\,x}{2\,b\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 31, normalized size = 0.72 \[ - \frac {4 a^{2} \log {\left (- a + b x \right )}}{b c} - \frac {3 a x}{c} - \frac {b x^{2}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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