Optimal. Leaf size=49 \[ -\frac {8 a^3 \log (a-b x)}{b}-4 a^2 x-\frac {a (a+b x)^2}{b}-\frac {(a+b x)^3}{3 b} \]
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Rubi [A] time = 0.02, antiderivative size = 49, 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 = {627, 43} \begin {gather*} -\frac {8 a^3 \log (a-b x)}{b}-4 a^2 x-\frac {a (a+b x)^2}{b}-\frac {(a+b x)^3}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(a+b x)^4}{a^2-b^2 x^2} \, dx &=\int \frac {(a+b x)^3}{a-b x} \, dx\\ &=\int \left (-4 a^2+\frac {8 a^3}{a-b x}-2 a (a+b x)-(a+b x)^2\right ) \, dx\\ &=-4 a^2 x-\frac {a (a+b x)^2}{b}-\frac {(a+b x)^3}{3 b}-\frac {8 a^3 \log (a-b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.80 \begin {gather*} -\frac {8 a^3 \log (a-b x)}{b}-7 a^2 x-2 a b x^2-\frac {b^2 x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^4}{a^2-b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 42, normalized size = 0.86 \begin {gather*} -\frac {b^{3} x^{3} + 6 \, a b^{2} x^{2} + 21 \, a^{2} b x + 24 \, a^{3} \log \left (b x - a\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 49, normalized size = 1.00 \begin {gather*} -\frac {8 \, a^{3} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {b^{5} x^{3} + 6 \, a b^{4} x^{2} + 21 \, a^{2} b^{3} x}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 0.80 \begin {gather*} -\frac {b^{2} x^{3}}{3}-2 a b \,x^{2}-\frac {8 a^{3} \ln \left (b x -a \right )}{b}-7 a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 38, normalized size = 0.78 \begin {gather*} -\frac {1}{3} \, b^{2} x^{3} - 2 \, a b x^{2} - 7 \, a^{2} x - \frac {8 \, a^{3} \log \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 = 0.40, size = 38, normalized size = 0.78 \begin {gather*} -7\,a^2\,x-\frac {b^2\,x^3}{3}-\frac {8\,a^3\,\ln \left (b\,x-a\right )}{b}-2\,a\,b\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 37, normalized size = 0.76 \begin {gather*} - \frac {8 a^{3} \log {\left (- a + b x \right )}}{b} - 7 a^{2} x - 2 a b x^{2} - \frac {b^{2} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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