Optimal. Leaf size=28 \[ -3 a x-\frac {b x^2}{2}-\frac {4 a^2 \log (a-b x)}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641, 45}
\begin {gather*} -\frac {4 a^2 \log (a-b x)}{b}-3 a x-\frac {b x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 641
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{a^2-b^2 x^2} \, dx &=\int \frac {(a+b x)^2}{a-b x} \, dx\\ &=\int \left (-3 a-b x+\frac {4 a^2}{a-b x}\right ) \, dx\\ &=-3 a x-\frac {b x^2}{2}-\frac {4 a^2 \log (a-b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} -3 a x-\frac {b x^2}{2}-\frac {4 a^2 \log (a-b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 27, normalized size = 0.96
method | result | size |
default | \(-3 a x -\frac {b \,x^{2}}{2}-\frac {4 a^{2} \ln \left (-b x +a \right )}{b}\) | \(27\) |
norman | \(-3 a x -\frac {b \,x^{2}}{2}-\frac {4 a^{2} \ln \left (-b x +a \right )}{b}\) | \(27\) |
risch | \(-3 a x -\frac {b \,x^{2}}{2}-\frac {4 a^{2} \ln \left (-b x +a \right )}{b}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 27, normalized size = 0.96 \begin {gather*} -\frac {1}{2} \, b x^{2} - 3 \, a x - \frac {4 \, a^{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 = 3.12, size = 31, normalized size = 1.11 \begin {gather*} -\frac {b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 26, normalized size = 0.93 \begin {gather*} - \frac {4 a^{2} \log {\left (- a + b x \right )}}{b} - 3 a x - \frac {b x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 38, normalized size = 1.36 \begin {gather*} -\frac {4 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {b^{3} x^{2} + 6 \, a b^{2} x}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 27, normalized size = 0.96 \begin {gather*} -3\,a\,x-\frac {b\,x^2}{2}-\frac {4\,a^2\,\ln \left (b\,x-a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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