Optimal. Leaf size=57 \[ -\frac {3 a \log (x)}{b^4}+\frac {3 a \log (a x+b)}{b^4}-\frac {2 a}{b^3 (a x+b)}-\frac {a}{2 b^2 (a x+b)^2}-\frac {1}{b^3 x} \]
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Rubi [A] time = 0.03, antiderivative size = 57, 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} \[ -\frac {2 a}{b^3 (a x+b)}-\frac {a}{2 b^2 (a x+b)^2}-\frac {3 a \log (x)}{b^4}+\frac {3 a \log (a x+b)}{b^4}-\frac {1}{b^3 x} \]
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 )^3 x^5} \, dx &=\int \frac {1}{x^2 (b+a x)^3} \, dx\\ &=\int \left (\frac {1}{b^3 x^2}-\frac {3 a}{b^4 x}+\frac {a^2}{b^2 (b+a x)^3}+\frac {2 a^2}{b^3 (b+a x)^2}+\frac {3 a^2}{b^4 (b+a x)}\right ) \, dx\\ &=-\frac {1}{b^3 x}-\frac {a}{2 b^2 (b+a x)^2}-\frac {2 a}{b^3 (b+a x)}-\frac {3 a \log (x)}{b^4}+\frac {3 a \log (b+a x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 53, normalized size = 0.93 \[ -\frac {\frac {b \left (6 a^2 x^2+9 a b x+2 b^2\right )}{x (a x+b)^2}-6 a \log (a x+b)+6 a \log (x)}{2 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 109, normalized size = 1.91 \[ -\frac {6 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3} - 6 \, {\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \log \left (a x + b\right ) + 6 \, {\left (a^{3} x^{3} + 2 \, a^{2} b x^{2} + a b^{2} x\right )} \log \relax (x)}{2 \, {\left (a^{2} b^{4} x^{3} + 2 \, a b^{5} x^{2} + b^{6} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 60, normalized size = 1.05 \[ \frac {3 \, a \log \left ({\left | a x + b \right |}\right )}{b^{4}} - \frac {3 \, a \log \left ({\left | x \right |}\right )}{b^{4}} - \frac {6 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3}}{2 \, {\left (a x + b\right )}^{2} b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.98 \[ -\frac {a}{2 \left (a x +b \right )^{2} b^{2}}-\frac {2 a}{\left (a x +b \right ) b^{3}}-\frac {3 a \ln \relax (x )}{b^{4}}+\frac {3 a \ln \left (a x +b \right )}{b^{4}}-\frac {1}{b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 69, normalized size = 1.21 \[ -\frac {6 \, a^{2} x^{2} + 9 \, a b x + 2 \, b^{2}}{2 \, {\left (a^{2} b^{3} x^{3} + 2 \, a b^{4} x^{2} + b^{5} x\right )}} + \frac {3 \, a \log \left (a x + b\right )}{b^{4}} - \frac {3 \, a \log \relax (x)}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 63, normalized size = 1.11 \[ \frac {6\,a\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^4}-\frac {\frac {1}{b}+\frac {3\,a^2\,x^2}{b^3}+\frac {9\,a\,x}{2\,b^2}}{a^2\,x^3+2\,a\,b\,x^2+b^2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 66, normalized size = 1.16 \[ \frac {3 a \left (- \log {\relax (x )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{4}} + \frac {- 6 a^{2} x^{2} - 9 a b x - 2 b^{2}}{2 a^{2} b^{3} x^{3} + 4 a b^{4} x^{2} + 2 b^{5} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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