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 = 0.02, antiderivative size = 41, 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} \[ \frac {4 a^2}{b c^2 (a-b x)}+\frac {4 a \log (a-b x)}{b c^2}+\frac {x}{c^2} \]
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.41, size = 57, normalized size = 1.39 \[ \frac {b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \, {\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} c^{2} x - a b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 79, normalized size = 1.93 \[ -\frac {4 \, a^{2}}{{\left (b c x - a c\right )} b c} - \frac {4 \, a \log \left (\frac {{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2} {\left | b \right |} {\left | c \right |}}\right )}{b c^{2}} + \frac {b c x - a c}{b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 1.07 \[ -\frac {4 a^{2}}{\left (b x -a \right ) b \,c^{2}}+\frac {4 a \ln \left (b x -a \right )}{b \,c^{2}}+\frac {x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 46, normalized size = 1.12 \[ -\frac {4 \, a^{2}}{b^{2} c^{2} x - a b c^{2}} + \frac {x}{c^{2}} + \frac {4 \, a \log \left (b x - a\right )}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 46, normalized size = 1.12 \[ \frac {x}{c^2}+\frac {4\,a^2}{b\,\left (a\,c^2-b\,c^2\,x\right )}+\frac {4\,a\,\ln \left (b\,x-a\right )}{b\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 39, normalized size = 0.95 \[ - \frac {4 a^{2}}{- a b c^{2} + b^{2} c^{2} x} + \frac {4 a \log {\left (- a + b x \right )}}{b c^{2}} + \frac {x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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