Optimal. Leaf size=41 \[ \frac{4 a^2}{b c^2 (a-b x)}+\frac{4 a \log (a-b x)}{b c^2}+\frac{x}{c^2} \]
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Rubi [A] time = 0.0221189, antiderivative size = 41, 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} \[ \frac{4 a^2}{b c^2 (a-b x)}+\frac{4 a \log (a-b x)}{b c^2}+\frac{x}{c^2} \]
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)^2} \, dx &=\int \left (\frac{1}{c^2}+\frac{4 a^2}{c^2 (a-b x)^2}-\frac{4 a}{c^2 (a-b x)}\right ) \, dx\\ &=\frac{x}{c^2}+\frac{4 a^2}{b c^2 (a-b x)}+\frac{4 a \log (a-b x)}{b c^2}\\ \end{align*}
Mathematica [A] time = 0.0295893, size = 35, normalized size = 0.85 \[ \frac{\frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x}{c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 44, normalized size = 1.1 \begin{align*}{\frac{x}{{c}^{2}}}-4\,{\frac{{a}^{2}}{{c}^{2}b \left ( bx-a \right ) }}+4\,{\frac{a\ln \left ( bx-a \right ) }{{c}^{2}b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00933, size = 62, normalized size = 1.51 \begin{align*} -\frac{4 \, a^{2}}{b^{2} c^{2} x - a b c^{2}} + \frac{x}{c^{2}} + \frac{4 \, a \log \left (b x - a\right )}{b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51304, size = 108, normalized size = 2.63 \begin{align*} \frac{b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \,{\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} c^{2} x - a b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.358625, size = 39, normalized size = 0.95 \begin{align*} - \frac{4 a^{2}}{- a b c^{2} + b^{2} c^{2} x} + \frac{4 a \log{\left (- a + b x \right )}}{b c^{2}} + \frac{x}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08186, size = 107, normalized size = 2.61 \begin{align*} -\frac{4 \, a^{2}}{{\left (b c x - a c\right )} b c} - \frac{4 \, a \log \left (\frac{{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2}{\left | b \right |}{\left | c \right |}}\right )}{b c^{2}} + \frac{b c x - a c}{b c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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