Optimal. Leaf size=31 \[ \frac {4 a^2}{b (a-b x)}+\frac {4 a \log (a-b x)}{b}+x \]
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Rubi [A] time = 0.02, antiderivative size = 31, 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 {4 a^2}{b (a-b x)}+\frac {4 a \log (a-b x)}{b}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(a+b x)^4}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {(a+b x)^2}{(a-b x)^2} \, dx\\ &=\int \left (1+\frac {4 a^2}{(a-b x)^2}-\frac {4 a}{a-b x}\right ) \, dx\\ &=x+\frac {4 a^2}{b (a-b x)}+\frac {4 a \log (a-b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 1.03 \begin {gather*} -\frac {4 a^2}{b (b x-a)}+\frac {4 a \log (a-b x)}{b}+x \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}{\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 = 0.40, size = 51, normalized size = 1.65 \begin {gather*} \frac {b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \, {\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} x - a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 34, normalized size = 1.10 \begin {gather*} x + \frac {4 \, a \log \left ({\left | b x - a \right |}\right )}{b} - \frac {4 \, a^{2}}{{\left (b x - a\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 34, normalized size = 1.10 \begin {gather*} -\frac {4 a^{2}}{\left (b x -a \right ) b}+\frac {4 a \ln \left (b x -a \right )}{b}+x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 33, normalized size = 1.06 \begin {gather*} -\frac {4 \, a^{2}}{b^{2} x - a b} + x + \frac {4 \, a \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 = 0.04, size = 31, normalized size = 1.00 \begin {gather*} x+\frac {4\,a^2}{b\,\left (a-b\,x\right )}+\frac {4\,a\,\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 = 0.20, size = 26, normalized size = 0.84 \begin {gather*} - \frac {4 a^{2}}{- a b + b^{2} x} + \frac {4 a \log {\left (- a + b x \right )}}{b} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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