Optimal. Leaf size=68 \[ \frac {b^4 \log (x)}{a^5}-\frac {b^4 \log (a+b x)}{a^5}+\frac {b^3}{a^4 x}-\frac {b^2}{2 a^3 x^2}+\frac {b}{3 a^2 x^3}-\frac {1}{4 a x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {44} \begin {gather*} -\frac {b^2}{2 a^3 x^2}+\frac {b^3}{a^4 x}+\frac {b^4 \log (x)}{a^5}-\frac {b^4 \log (a+b x)}{a^5}+\frac {b}{3 a^2 x^3}-\frac {1}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^5 (a+b x)} \, dx &=\int \left (\frac {1}{a x^5}-\frac {b}{a^2 x^4}+\frac {b^2}{a^3 x^3}-\frac {b^3}{a^4 x^2}+\frac {b^4}{a^5 x}-\frac {b^5}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac {1}{4 a x^4}+\frac {b}{3 a^2 x^3}-\frac {b^2}{2 a^3 x^2}+\frac {b^3}{a^4 x}+\frac {b^4 \log (x)}{a^5}-\frac {b^4 \log (a+b x)}{a^5}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 68, normalized size = 1.00 \begin {gather*} \frac {b^4 \log (x)}{a^5}-\frac {b^4 \log (a+b x)}{a^5}+\frac {b^3}{a^4 x}-\frac {b^2}{2 a^3 x^2}+\frac {b}{3 a^2 x^3}-\frac {1}{4 a x^4} \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 {1}{x^5 (a+b x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.22, size = 65, normalized size = 0.96 \begin {gather*} -\frac {12 \, b^{4} x^{4} \log \left (b x + a\right ) - 12 \, b^{4} x^{4} \log \relax (x) - 12 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 3 \, a^{4}}{12 \, a^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 67, normalized size = 0.99 \begin {gather*} -\frac {b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{5}} + \frac {b^{4} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {12 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x - 3 \, a^{4}}{12 \, a^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.93 \begin {gather*} \frac {b^{4} \ln \relax (x )}{a^{5}}-\frac {b^{4} \ln \left (b x +a \right )}{a^{5}}+\frac {b^{3}}{a^{4} x}-\frac {b^{2}}{2 a^{3} x^{2}}+\frac {b}{3 a^{2} x^{3}}-\frac {1}{4 a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 62, normalized size = 0.91 \begin {gather*} -\frac {b^{4} \log \left (b x + a\right )}{a^{5}} + \frac {b^{4} \log \relax (x)}{a^{5}} + \frac {12 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 4 \, a^{2} b x - 3 \, a^{3}}{12 \, a^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 60, normalized size = 0.88 \begin {gather*} -\frac {\frac {a^4}{4}-\frac {a^3\,b\,x}{3}+\frac {a^2\,b^2\,x^2}{2}-a\,b^3\,x^3}{a^5\,x^4}-\frac {2\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 56, normalized size = 0.82 \begin {gather*} \frac {- 3 a^{3} + 4 a^{2} b x - 6 a b^{2} x^{2} + 12 b^{3} x^{3}}{12 a^{4} x^{4}} + \frac {b^{4} \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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