Optimal. Leaf size=43 \[ -\frac {2 a x}{c}-\frac {(a+b x)^2}{2 b c}-\frac {4 a^2 \log (a-b x)}{b c} \]
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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 = {45}
\begin {gather*} -\frac {4 a^2 \log (a-b x)}{b c}-\frac {(a+b x)^2}{2 b c}-\frac {2 a x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{a c-b c x} \, dx &=\int \left (-\frac {2 a}{c}-\frac {a+b x}{c}+\frac {4 a^2}{a c-b c x}\right ) \, dx\\ &=-\frac {2 a x}{c}-\frac {(a+b x)^2}{2 b c}-\frac {4 a^2 \log (a-b x)}{b c}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 0.86 \begin {gather*} -\frac {3 a x}{c}-\frac {b x^2}{2 c}-\frac {4 a^2 \log (a-b x)}{b c} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.73, size = 33, normalized size = 0.77 \begin {gather*} \frac {-8 a^2 \text {Log}\left [-a+b x\right ]+b x \left (-6 a-b x\right )}{2 b c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 31, normalized size = 0.72
method | result | size |
default | \(\frac {-\frac {x^{2} b}{2}-3 a x -\frac {4 a^{2} \ln \left (-b x +a \right )}{b}}{c}\) | \(31\) |
norman | \(-\frac {3 a x}{c}-\frac {b \,x^{2}}{2 c}-\frac {4 a^{2} \ln \left (-b x +a \right )}{b c}\) | \(36\) |
risch | \(-\frac {3 a x}{c}-\frac {b \,x^{2}}{2 c}-\frac {4 a^{2} \ln \left (-b x +a \right )}{b c}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 35, normalized size = 0.81 \begin {gather*} -\frac {4 \, a^{2} \log \left (b x - a\right )}{b c} - \frac {b x^{2} + 6 \, a x}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 34, normalized size = 0.79 \begin {gather*} -\frac {b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 31, normalized size = 0.72 \begin {gather*} - \frac {4 a^{2} \log {\left (- a + b x \right )}}{b c} - \frac {3 a x}{c} - \frac {b x^{2}}{2 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 50, normalized size = 1.16 \begin {gather*} \frac {-\frac {1}{2} x^{2} b^{3} c-3 x b^{2} a c}{b^{2} c^{2}}-\frac {4 a^{2} \ln \left |x b-a\right |}{b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 34, normalized size = 0.79 \begin {gather*} -\frac {8\,a^2\,\ln \left (b\,x-a\right )+b^2\,x^2+6\,a\,b\,x}{2\,b\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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