Optimal. Leaf size=97 \[ \frac {15 a^4 \log (x)}{b^7}-\frac {15 a^4 \log (a x+b)}{b^7}+\frac {5 a^4}{b^6 (a x+b)}+\frac {a^4}{2 b^5 (a x+b)^2}+\frac {10 a^3}{b^6 x}-\frac {3 a^2}{b^5 x^2}+\frac {a}{b^4 x^3}-\frac {1}{4 b^3 x^4} \]
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Rubi [A] time = 0.06, antiderivative size = 97, 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} \[ -\frac {3 a^2}{b^5 x^2}+\frac {5 a^4}{b^6 (a x+b)}+\frac {a^4}{2 b^5 (a x+b)^2}+\frac {10 a^3}{b^6 x}+\frac {15 a^4 \log (x)}{b^7}-\frac {15 a^4 \log (a x+b)}{b^7}+\frac {a}{b^4 x^3}-\frac {1}{4 b^3 x^4} \]
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^8} \, dx &=\int \frac {1}{x^5 (b+a x)^3} \, dx\\ &=\int \left (\frac {1}{b^3 x^5}-\frac {3 a}{b^4 x^4}+\frac {6 a^2}{b^5 x^3}-\frac {10 a^3}{b^6 x^2}+\frac {15 a^4}{b^7 x}-\frac {a^5}{b^5 (b+a x)^3}-\frac {5 a^5}{b^6 (b+a x)^2}-\frac {15 a^5}{b^7 (b+a x)}\right ) \, dx\\ &=-\frac {1}{4 b^3 x^4}+\frac {a}{b^4 x^3}-\frac {3 a^2}{b^5 x^2}+\frac {10 a^3}{b^6 x}+\frac {a^4}{2 b^5 (b+a x)^2}+\frac {5 a^4}{b^6 (b+a x)}+\frac {15 a^4 \log (x)}{b^7}-\frac {15 a^4 \log (b+a x)}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 90, normalized size = 0.93 \[ \frac {-60 a^4 \log (a x+b)+60 a^4 \log (x)+\frac {b \left (60 a^5 x^5+90 a^4 b x^4+20 a^3 b^2 x^3-5 a^2 b^3 x^2+2 a b^4 x-b^5\right )}{x^4 (a x+b)^2}}{4 b^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 152, normalized size = 1.57 \[ \frac {60 \, a^{5} b x^{5} + 90 \, a^{4} b^{2} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{2} b^{4} x^{2} + 2 \, a b^{5} x - b^{6} - 60 \, {\left (a^{6} x^{6} + 2 \, a^{5} b x^{5} + a^{4} b^{2} x^{4}\right )} \log \left (a x + b\right ) + 60 \, {\left (a^{6} x^{6} + 2 \, a^{5} b x^{5} + a^{4} b^{2} x^{4}\right )} \log \relax (x)}{4 \, {\left (a^{2} b^{7} x^{6} + 2 \, a b^{8} x^{5} + b^{9} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 97, normalized size = 1.00 \[ -\frac {15 \, a^{4} \log \left ({\left | a x + b \right |}\right )}{b^{7}} + \frac {15 \, a^{4} \log \left ({\left | x \right |}\right )}{b^{7}} + \frac {60 \, a^{5} b x^{5} + 90 \, a^{4} b^{2} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{2} b^{4} x^{2} + 2 \, a b^{5} x - b^{6}}{4 \, {\left (a x + b\right )}^{2} b^{7} x^{4}} \]
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 \[ \frac {a^{4}}{2 \left (a x +b \right )^{2} b^{5}}+\frac {5 a^{4}}{\left (a x +b \right ) b^{6}}+\frac {15 a^{4} \ln \relax (x )}{b^{7}}-\frac {15 a^{4} \ln \left (a x +b \right )}{b^{7}}+\frac {10 a^{3}}{b^{6} x}-\frac {3 a^{2}}{b^{5} x^{2}}+\frac {a}{b^{4} x^{3}}-\frac {1}{4 b^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 108, normalized size = 1.11 \[ \frac {60 \, a^{5} x^{5} + 90 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} - 5 \, a^{2} b^{3} x^{2} + 2 \, a b^{4} x - b^{5}}{4 \, {\left (a^{2} b^{6} x^{6} + 2 \, a b^{7} x^{5} + b^{8} x^{4}\right )}} - \frac {15 \, a^{4} \log \left (a x + b\right )}{b^{7}} + \frac {15 \, a^{4} \log \relax (x)}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 101, normalized size = 1.04 \[ \frac {\frac {5\,a^3\,x^3}{b^4}-\frac {5\,a^2\,x^2}{4\,b^3}-\frac {1}{4\,b}+\frac {45\,a^4\,x^4}{2\,b^5}+\frac {15\,a^5\,x^5}{b^6}+\frac {a\,x}{2\,b^2}}{a^2\,x^6+2\,a\,b\,x^5+b^2\,x^4}-\frac {30\,a^4\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 102, normalized size = 1.05 \[ \frac {15 a^{4} \left (\log {\relax (x )} - \log {\left (x + \frac {b}{a} \right )}\right )}{b^{7}} + \frac {60 a^{5} x^{5} + 90 a^{4} b x^{4} + 20 a^{3} b^{2} x^{3} - 5 a^{2} b^{3} x^{2} + 2 a b^{4} x - b^{5}}{4 a^{2} b^{6} x^{6} + 8 a b^{7} x^{5} + 4 b^{8} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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