Optimal. Leaf size=58 \[ \frac {3 a^2 \log (x)}{b^4}-\frac {3 a^2 \log (a x+b)}{b^4}+\frac {a^2}{b^3 (a x+b)}+\frac {2 a}{b^3 x}-\frac {1}{2 b^2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 58, 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} \[ \frac {a^2}{b^3 (a x+b)}+\frac {3 a^2 \log (x)}{b^4}-\frac {3 a^2 \log (a x+b)}{b^4}+\frac {2 a}{b^3 x}-\frac {1}{2 b^2 x^2} \]
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^5} \, dx &=\int \frac {1}{x^3 (b+a x)^2} \, dx\\ &=\int \left (\frac {1}{b^2 x^3}-\frac {2 a}{b^3 x^2}+\frac {3 a^2}{b^4 x}-\frac {a^3}{b^3 (b+a x)^2}-\frac {3 a^3}{b^4 (b+a x)}\right ) \, dx\\ &=-\frac {1}{2 b^2 x^2}+\frac {2 a}{b^3 x}+\frac {a^2}{b^3 (b+a x)}+\frac {3 a^2 \log (x)}{b^4}-\frac {3 a^2 \log (b+a x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 53, normalized size = 0.91 \[ \frac {b \left (\frac {2 a^2}{a x+b}+\frac {4 a}{x}-\frac {b}{x^2}\right )-6 a^2 \log (a x+b)+6 a^2 \log (x)}{2 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 86, normalized size = 1.48 \[ \frac {6 \, a^{2} b x^{2} + 3 \, a b^{2} x - b^{3} - 6 \, {\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \log \left (a x + b\right ) + 6 \, {\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \log \relax (x)}{2 \, {\left (a b^{4} x^{3} + b^{5} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 64, normalized size = 1.10 \[ -\frac {3 \, a^{2} \log \left ({\left | a x + b \right |}\right )}{b^{4}} + \frac {3 \, a^{2} \log \left ({\left | x \right |}\right )}{b^{4}} + \frac {6 \, a^{2} b x^{2} + 3 \, a b^{2} x - b^{3}}{2 \, {\left (a x + b\right )} b^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.98 \[ \frac {a^{2}}{\left (a x +b \right ) b^{3}}+\frac {3 a^{2} \ln \relax (x )}{b^{4}}-\frac {3 a^{2} \ln \left (a x +b \right )}{b^{4}}+\frac {2 a}{b^{3} x}-\frac {1}{2 b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 64, normalized size = 1.10 \[ \frac {6 \, a^{2} x^{2} + 3 \, a b x - b^{2}}{2 \, {\left (a b^{3} x^{3} + b^{4} x^{2}\right )}} - \frac {3 \, a^{2} \log \left (a x + b\right )}{b^{4}} + \frac {3 \, a^{2} \log \relax (x)}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 57, normalized size = 0.98 \[ \frac {\frac {3\,a^2\,x^2}{b^3}-\frac {1}{2\,b}+\frac {3\,a\,x}{2\,b^2}}{a\,x^3+b\,x^2}-\frac {6\,a^2\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 54, normalized size = 0.93 \[ \frac {3 a^{2} \left (\log {\relax (x )} - \log {\left (x + \frac {b}{a} \right )}\right )}{b^{4}} + \frac {6 a^{2} x^{2} + 3 a b x - b^{2}}{2 a b^{3} x^{3} + 2 b^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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