Optimal. Leaf size=44 \[ \frac {8 a^3}{b (a-b x)}+\frac {12 a^2 \log (a-b x)}{b}+5 a x+\frac {b x^2}{2} \]
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Rubi [A] time = 0.03, antiderivative size = 44, 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}{b (a-b x)}+\frac {12 a^2 \log (a-b x)}{b}+5 a x+\frac {b x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {(a+b x)^3}{(a-b x)^2} \, dx\\ &=\int \left (5 a+b x+\frac {8 a^3}{(a-b x)^2}-\frac {12 a^2}{a-b x}\right ) \, dx\\ &=5 a x+\frac {b x^2}{2}+\frac {8 a^3}{b (a-b x)}+\frac {12 a^2 \log (a-b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 1.02 \begin {gather*} -\frac {8 a^3}{b (b x-a)}+\frac {12 a^2 \log (a-b x)}{b}+5 a x+\frac {b x^2}{2} \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)^5}{\left (a^2-b^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 65, normalized size = 1.48 \begin {gather*} \frac {b^{3} x^{3} + 9 \, a b^{2} x^{2} - 10 \, a^{2} b x - 16 \, a^{3} + 24 \, {\left (a^{2} b x - a^{3}\right )} \log \left (b x - a\right )}{2 \, {\left (b^{2} x - a b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 55, normalized size = 1.25 \begin {gather*} \frac {12 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {8 \, a^{3}}{{\left (b x - a\right )} b} + \frac {b^{5} x^{2} + 10 \, a b^{4} x}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 45, normalized size = 1.02 \begin {gather*} \frac {b \,x^{2}}{2}-\frac {8 a^{3}}{\left (b x -a \right ) b}+\frac {12 a^{2} \ln \left (b x -a \right )}{b}+5 a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 44, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, b x^{2} - \frac {8 \, a^{3}}{b^{2} x - a b} + 5 \, a x + \frac {12 \, 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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mupad [B] time = 0.05, size = 42, normalized size = 0.95 \begin {gather*} 5\,a\,x+\frac {b\,x^2}{2}+\frac {8\,a^3}{b\,\left (a-b\,x\right )}+\frac {12\,a^2\,\ln \left (a-b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 37, normalized size = 0.84 \begin {gather*} - \frac {8 a^{3}}{- a b + b^{2} x} + \frac {12 a^{2} \log {\left (- a + b x \right )}}{b} + 5 a x + \frac {b x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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