Optimal. Leaf size=56 \[ -\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (a x+b)}{b^4}-\frac {a^2}{b^3 x}+\frac {a}{2 b^2 x^2}-\frac {1}{3 b x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 56, 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} \[ -\frac {a^2}{b^3 x}-\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (a x+b)}{b^4}+\frac {a}{2 b^2 x^2}-\frac {1}{3 b x^3} \]
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 ) x^5} \, dx &=\int \frac {1}{x^4 (b+a x)} \, dx\\ &=\int \left (\frac {1}{b x^4}-\frac {a}{b^2 x^3}+\frac {a^2}{b^3 x^2}-\frac {a^3}{b^4 x}+\frac {a^4}{b^4 (b+a x)}\right ) \, dx\\ &=-\frac {1}{3 b x^3}+\frac {a}{2 b^2 x^2}-\frac {a^2}{b^3 x}-\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (b+a x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 56, normalized size = 1.00 \[ -\frac {a^3 \log (x)}{b^4}+\frac {a^3 \log (a x+b)}{b^4}-\frac {a^2}{b^3 x}+\frac {a}{2 b^2 x^2}-\frac {1}{3 b x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 54, normalized size = 0.96 \[ \frac {6 \, a^{3} x^{3} \log \left (a x + b\right ) - 6 \, a^{3} x^{3} \log \relax (x) - 6 \, a^{2} b x^{2} + 3 \, a b^{2} x - 2 \, b^{3}}{6 \, b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 56, normalized size = 1.00 \[ \frac {a^{3} \log \left ({\left | a x + b \right |}\right )}{b^{4}} - \frac {a^{3} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac {6 \, a^{2} b x^{2} - 3 \, a b^{2} x + 2 \, b^{3}}{6 \, b^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 0.95 \[ -\frac {a^{3} \ln \relax (x )}{b^{4}}+\frac {a^{3} \ln \left (a x +b \right )}{b^{4}}-\frac {a^{2}}{b^{3} x}+\frac {a}{2 b^{2} x^{2}}-\frac {1}{3 b \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 51, normalized size = 0.91 \[ \frac {a^{3} \log \left (a x + b\right )}{b^{4}} - \frac {a^{3} \log \relax (x)}{b^{4}} - \frac {6 \, a^{2} x^{2} - 3 \, a b x + 2 \, b^{2}}{6 \, b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 48, normalized size = 0.86 \[ \frac {2\,a^3\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^4}-\frac {a^2\,b\,x^2-\frac {a\,b^2\,x}{2}+\frac {b^3}{3}}{b^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 44, normalized size = 0.79 \[ \frac {a^{3} \left (- \log {\relax (x )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{4}} + \frac {- 6 a^{2} x^{2} + 3 a b x - 2 b^{2}}{6 b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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