Optimal. Leaf size=52 \[ \frac{2 a^2}{b c^3 (a-b x)^2}-\frac{4 a}{b c^3 (a-b x)}-\frac{\log (a-b x)}{b c^3} \]
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Rubi [A] time = 0.0280616, antiderivative size = 52, 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} \[ \frac{2 a^2}{b c^3 (a-b x)^2}-\frac{4 a}{b c^3 (a-b x)}-\frac{\log (a-b x)}{b c^3} \]
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)^3} \, dx &=\int \left (\frac{4 a^2}{c^3 (a-b x)^3}-\frac{4 a}{c^3 (a-b x)^2}+\frac{1}{c^3 (a-b x)}\right ) \, dx\\ &=\frac{2 a^2}{b c^3 (a-b x)^2}-\frac{4 a}{b c^3 (a-b x)}-\frac{\log (a-b x)}{b c^3}\\ \end{align*}
Mathematica [A] time = 0.0244032, size = 33, normalized size = 0.63 \[ -\frac{\frac{2 a (a-2 b x)}{(a-b x)^2}+\log (a-b x)}{b c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 1.1 \begin{align*} 4\,{\frac{a}{{c}^{3}b \left ( bx-a \right ) }}+2\,{\frac{{a}^{2}}{{c}^{3}b \left ( bx-a \right ) ^{2}}}-{\frac{\ln \left ( bx-a \right ) }{{c}^{3}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03558, size = 82, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (2 \, a b x - a^{2}\right )}}{b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{3} x + a^{2} b c^{3}} - \frac{\log \left (b x - a\right )}{b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57548, size = 138, normalized size = 2.65 \begin{align*} \frac{4 \, a b x - 2 \, a^{2} -{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{3} x + a^{2} b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.447033, size = 53, normalized size = 1.02 \begin{align*} \frac{- 2 a^{2} + 4 a b x}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} - \frac{\log{\left (- a + b x \right )}}{b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05647, size = 62, normalized size = 1.19 \begin{align*} -\frac{\log \left ({\left | b x - a \right |}\right )}{b c^{3}} + \frac{2 \,{\left (2 \, a b x - a^{2}\right )}}{{\left (b x - a\right )}^{2} b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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