Optimal. Leaf size=39 \[ -\frac{4 a^2 c^2}{b (a+b x)}-\frac{4 a c^2 \log (a+b x)}{b}+c^2 x \]
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Rubi [A] time = 0.0209701, antiderivative size = 39, 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} \[ -\frac{4 a^2 c^2}{b (a+b x)}-\frac{4 a c^2 \log (a+b x)}{b}+c^2 x \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a c-b c x)^2}{(a+b x)^2} \, dx &=\int \left (c^2+\frac{4 a^2 c^2}{(a+b x)^2}-\frac{4 a c^2}{a+b x}\right ) \, dx\\ &=c^2 x-\frac{4 a^2 c^2}{b (a+b x)}-\frac{4 a c^2 \log (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0179474, size = 33, normalized size = 0.85 \[ c^2 \left (-\frac{4 a^2}{b (a+b x)}-\frac{4 a \log (a+b x)}{b}+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 40, normalized size = 1. \begin{align*}{c}^{2}x-4\,{\frac{{a}^{2}{c}^{2}}{b \left ( bx+a \right ) }}-4\,{\frac{a{c}^{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 = 1.04708, size = 54, normalized size = 1.38 \begin{align*} -\frac{4 \, a^{2} c^{2}}{b^{2} x + a b} + c^{2} x - \frac{4 \, a c^{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 = 1.5316, size = 124, normalized size = 3.18 \begin{align*} \frac{b^{2} c^{2} x^{2} + a b c^{2} x - 4 \, a^{2} c^{2} - 4 \,{\left (a b c^{2} x + a^{2} c^{2}\right )} \log \left (b x + a\right )}{b^{2} x + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.344966, size = 36, normalized size = 0.92 \begin{align*} - \frac{4 a^{2} c^{2}}{a b + b^{2} x} - \frac{4 a c^{2} \log{\left (a + b x \right )}}{b} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07997, size = 80, normalized size = 2.05 \begin{align*} \frac{4 \, a c^{2} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} + \frac{{\left (b x + a\right )} c^{2}}{b} - \frac{4 \, a^{2} c^{2}}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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