Optimal. Leaf size=43 \[ \frac {4 a^2 c^2 \log (a+b x)}{b}+\frac {c^2 (a-b x)^2}{2 b}-2 a c^2 x \]
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Rubi [A] time = 0.01, antiderivative size = 43, 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} \[ \frac {4 a^2 c^2 \log (a+b x)}{b}+\frac {c^2 (a-b x)^2}{2 b}-2 a c^2 x \]
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} \, dx &=\int \left (-2 a c^2+\frac {4 a^2 c^2}{a+b x}-c (a c-b c x)\right ) \, dx\\ &=-2 a c^2 x+\frac {c^2 (a-b x)^2}{2 b}+\frac {4 a^2 c^2 \log (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.72 \[ c^2 \left (\frac {4 a^2 \log (a+b x)}{b}-3 a x+\frac {b x^2}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 38, normalized size = 0.88 \[ \frac {b^{2} c^{2} x^{2} - 6 \, a b c^{2} x + 8 \, a^{2} c^{2} \log \left (b x + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 45, normalized size = 1.05 \[ \frac {4 \, a^{2} c^{2} \log \left ({\left | b x + a \right |}\right )}{b} + \frac {b^{3} c^{2} x^{2} - 6 \, a b^{2} c^{2} x}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.81 \[ \frac {b \,c^{2} x^{2}}{2}+\frac {4 a^{2} c^{2} \ln \left (b x +a \right )}{b}-3 a \,c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 34, normalized size = 0.79 \[ \frac {1}{2} \, b c^{2} x^{2} - 3 \, a c^{2} x + \frac {4 \, a^{2} c^{2} \log \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 32, normalized size = 0.74 \[ \frac {c^2\,\left (8\,a^2\,\ln \left (a+b\,x\right )+b^2\,x^2-6\,a\,b\,x\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 34, normalized size = 0.79 \[ \frac {4 a^{2} c^{2} \log {\left (a + b x \right )}}{b} - 3 a c^{2} x + \frac {b c^{2} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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