Optimal. Leaf size=26 \[ \frac {2 a}{b (a-b x)}+\frac {\log (a-b x)}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 26, 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 {2 a}{b (a-b x)}+\frac {\log (a-b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {a+b x}{(a-b x)^2} \, dx\\ &=\int \left (\frac {2 a}{(a-b x)^2}+\frac {1}{-a+b x}\right ) \, dx\\ &=\frac {2 a}{b (a-b x)}+\frac {\log (a-b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.88 \begin {gather*} \frac {\frac {2 a}{a-b x}+\log (a-b x)}{b} \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)^3}{\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 = 33, normalized size = 1.27 \begin {gather*} \frac {{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{b^{2} x - a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 29, normalized size = 1.12 \begin {gather*} \frac {\log \left ({\left | b x - a \right |}\right )}{b} - \frac {2 \, a}{{\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 = 0.06, size = 29, normalized size = 1.12 \begin {gather*} -\frac {2 a}{\left (b x -a \right ) b}+\frac {\ln \left (b x -a \right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 28, normalized size = 1.08 \begin {gather*} -\frac {2 \, a}{b^{2} x - a b} + \frac {\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 = 26, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a-b\,x\right )}{b}+\frac {2\,a}{b\,\left (a-b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 19, normalized size = 0.73 \begin {gather*} - \frac {2 a}{- a b + b^{2} x} + \frac {\log {\left (- a + b x \right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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