Optimal. Leaf size=39 \[ c^2 x-\frac {4 a^2 c^2}{b (a+b x)}-\frac {4 a c^2 \log (a+b x)}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 39, 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 = {45}
\begin {gather*} -\frac {4 a^2 c^2}{b (a+b x)}-\frac {4 a c^2 \log (a+b x)}{b}+c^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 45
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*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.85 \begin {gather*} c^2 \left (x-\frac {4 a^2}{b (a+b x)}-\frac {4 a \log (a+b x)}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 34, normalized size = 0.87
method | result | size |
default | \(c^{2} \left (x -\frac {4 a^{2}}{b \left (b x +a \right )}-\frac {4 a \ln \left (b x +a \right )}{b}\right )\) | \(34\) |
risch | \(c^{2} x -\frac {4 a^{2} c^{2}}{b \left (b x +a \right )}-\frac {4 a \,c^{2} \ln \left (b x +a \right )}{b}\) | \(40\) |
norman | \(\frac {b \,c^{2} x^{2}+5 a \,c^{2} x}{b x +a}-\frac {4 a \,c^{2} \ln \left (b x +a \right )}{b}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 40, normalized size = 1.03 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 61, normalized size = 1.56 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 36, normalized size = 0.92 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.30, size = 59, normalized size = 1.51 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 39, normalized size = 1.00 \begin {gather*} c^2\,x-\frac {4\,a\,c^2\,\ln \left (a+b\,x\right )}{b}-\frac {4\,a^2\,c^2}{b\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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