Optimal. Leaf size=29 \[ -\frac {\log (a x+b)}{b^2}+\frac {1}{b (a x+b)}+\frac {\log (x)}{b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 44} \[ -\frac {\log (a x+b)}{b^2}+\frac {1}{b (a x+b)}+\frac {\log (x)}{b^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^3} \, dx &=\int \frac {1}{x (b+a x)^2} \, dx\\ &=\int \left (\frac {1}{b^2 x}-\frac {a}{b (b+a x)^2}-\frac {a}{b^2 (b+a x)}\right ) \, dx\\ &=\frac {1}{b (b+a x)}+\frac {\log (x)}{b^2}-\frac {\log (b+a x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.83 \[ \frac {\frac {b}{a x+b}-\log (a x+b)+\log (x)}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 39, normalized size = 1.34 \[ -\frac {{\left (a x + b\right )} \log \left (a x + b\right ) - {\left (a x + b\right )} \log \relax (x) - b}{a b^{2} x + b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 31, normalized size = 1.07 \[ -\frac {\log \left ({\left | a x + b \right |}\right )}{b^{2}} + \frac {\log \left ({\left | x \right |}\right )}{b^{2}} + \frac {1}{{\left (a x + b\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 1.03 \[ \frac {1}{\left (a x +b \right ) b}+\frac {\ln \relax (x )}{b^{2}}-\frac {\ln \left (a x +b \right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 28, normalized size = 0.97 \[ \frac {1}{a b x + b^{2}} - \frac {\log \left (a x + b\right )}{b^{2}} + \frac {\log \relax (x)}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 26, normalized size = 0.90 \[ \frac {1}{b^2+a\,x\,b}-\frac {2\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 22, normalized size = 0.76 \[ \frac {1}{a b x + b^{2}} + \frac {\log {\relax (x )} - \log {\left (x + \frac {b}{a} \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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