Optimal. Leaf size=97 \[ \frac {15 b^4 \log (x)}{a^7}-\frac {15 b^4 \log (a+b x)}{a^7}+\frac {5 b^4}{a^6 (a+b x)}+\frac {10 b^3}{a^6 x}+\frac {b^4}{2 a^5 (a+b x)^2}-\frac {3 b^2}{a^5 x^2}+\frac {b}{a^4 x^3}-\frac {1}{4 a^3 x^4} \]
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Rubi [A] time = 0.05, antiderivative size = 97, 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 {3 b^2}{a^5 x^2}+\frac {5 b^4}{a^6 (a+b x)}+\frac {b^4}{2 a^5 (a+b x)^2}+\frac {10 b^3}{a^6 x}+\frac {15 b^4 \log (x)}{a^7}-\frac {15 b^4 \log (a+b x)}{a^7}+\frac {b}{a^4 x^3}-\frac {1}{4 a^3 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)^3} \, dx &=\int \left (\frac {1}{a^3 x^5}-\frac {3 b}{a^4 x^4}+\frac {6 b^2}{a^5 x^3}-\frac {10 b^3}{a^6 x^2}+\frac {15 b^4}{a^7 x}-\frac {b^5}{a^5 (a+b x)^3}-\frac {5 b^5}{a^6 (a+b x)^2}-\frac {15 b^5}{a^7 (a+b x)}\right ) \, dx\\ &=-\frac {1}{4 a^3 x^4}+\frac {b}{a^4 x^3}-\frac {3 b^2}{a^5 x^2}+\frac {10 b^3}{a^6 x}+\frac {b^4}{2 a^5 (a+b x)^2}+\frac {5 b^4}{a^6 (a+b x)}+\frac {15 b^4 \log (x)}{a^7}-\frac {15 b^4 \log (a+b x)}{a^7}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 90, normalized size = 0.93 \begin {gather*} \frac {\frac {a \left (-a^5+2 a^4 b x-5 a^3 b^2 x^2+20 a^2 b^3 x^3+90 a b^4 x^4+60 b^5 x^5\right )}{x^4 (a+b x)^2}-60 b^4 \log (a+b x)+60 b^4 \log (x)}{4 a^7} \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)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.95, size = 152, normalized size = 1.57 \begin {gather*} \frac {60 \, a b^{5} x^{5} + 90 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} + 2 \, a^{5} b x - a^{6} - 60 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{5} + a^{2} b^{4} x^{4}\right )} \log \relax (x)}{4 \, {\left (a^{7} b^{2} x^{6} + 2 \, a^{8} b x^{5} + a^{9} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 97, normalized size = 1.00 \begin {gather*} -\frac {15 \, b^{4} \log \left ({\left | b x + a \right |}\right )}{a^{7}} + \frac {15 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{5} + 90 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{4} b^{2} x^{2} + 2 \, a^{5} b x - a^{6}}{4 \, {\left (b x + a\right )}^{2} a^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 0.97 \begin {gather*} \frac {b^{4}}{2 \left (b x +a \right )^{2} a^{5}}+\frac {5 b^{4}}{\left (b x +a \right ) a^{6}}+\frac {15 b^{4} \ln \relax (x )}{a^{7}}-\frac {15 b^{4} \ln \left (b x +a \right )}{a^{7}}+\frac {10 b^{3}}{a^{6} x}-\frac {3 b^{2}}{a^{5} x^{2}}+\frac {b}{a^{4} x^{3}}-\frac {1}{4 a^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 108, normalized size = 1.11 \begin {gather*} \frac {60 \, b^{5} x^{5} + 90 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} - 5 \, a^{3} b^{2} x^{2} + 2 \, a^{4} b x - a^{5}}{4 \, {\left (a^{6} b^{2} x^{6} + 2 \, a^{7} b x^{5} + a^{8} x^{4}\right )}} - \frac {15 \, b^{4} \log \left (b x + a\right )}{a^{7}} + \frac {15 \, b^{4} \log \relax (x)}{a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 101, normalized size = 1.04 \begin {gather*} \frac {\frac {5\,b^3\,x^3}{a^4}-\frac {5\,b^2\,x^2}{4\,a^3}-\frac {1}{4\,a}+\frac {45\,b^4\,x^4}{2\,a^5}+\frac {15\,b^5\,x^5}{a^6}+\frac {b\,x}{2\,a^2}}{a^2\,x^4+2\,a\,b\,x^5+b^2\,x^6}-\frac {30\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 102, normalized size = 1.05 \begin {gather*} \frac {- a^{5} + 2 a^{4} b x - 5 a^{3} b^{2} x^{2} + 20 a^{2} b^{3} x^{3} + 90 a b^{4} x^{4} + 60 b^{5} x^{5}}{4 a^{8} x^{4} + 8 a^{7} b x^{5} + 4 a^{6} b^{2} x^{6}} + \frac {15 b^{4} \left (\log {\relax (x )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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