Integrand size = 13, antiderivative size = 111 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^9} \, dx=-\frac {1}{5 b^3 x^5}+\frac {3 a}{4 b^4 x^4}-\frac {2 a^2}{b^5 x^3}+\frac {5 a^3}{b^6 x^2}-\frac {15 a^4}{b^7 x}-\frac {a^5}{2 b^6 (b+a x)^2}-\frac {6 a^5}{b^7 (b+a x)}-\frac {21 a^5 \log (x)}{b^8}+\frac {21 a^5 \log (b+a x)}{b^8} \]
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Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 46} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^9} \, dx=-\frac {21 a^5 \log (x)}{b^8}+\frac {21 a^5 \log (a x+b)}{b^8}-\frac {6 a^5}{b^7 (a x+b)}-\frac {a^5}{2 b^6 (a x+b)^2}-\frac {15 a^4}{b^7 x}+\frac {5 a^3}{b^6 x^2}-\frac {2 a^2}{b^5 x^3}+\frac {3 a}{4 b^4 x^4}-\frac {1}{5 b^3 x^5} \]
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Rule 46
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^6 (b+a x)^3} \, dx \\ & = \int \left (\frac {1}{b^3 x^6}-\frac {3 a}{b^4 x^5}+\frac {6 a^2}{b^5 x^4}-\frac {10 a^3}{b^6 x^3}+\frac {15 a^4}{b^7 x^2}-\frac {21 a^5}{b^8 x}+\frac {a^6}{b^6 (b+a x)^3}+\frac {6 a^6}{b^7 (b+a x)^2}+\frac {21 a^6}{b^8 (b+a x)}\right ) \, dx \\ & = -\frac {1}{5 b^3 x^5}+\frac {3 a}{4 b^4 x^4}-\frac {2 a^2}{b^5 x^3}+\frac {5 a^3}{b^6 x^2}-\frac {15 a^4}{b^7 x}-\frac {a^5}{2 b^6 (b+a x)^2}-\frac {6 a^5}{b^7 (b+a x)}-\frac {21 a^5 \log (x)}{b^8}+\frac {21 a^5 \log (b+a x)}{b^8} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^9} \, dx=-\frac {\frac {b \left (4 b^6-7 a b^5 x+14 a^2 b^4 x^2-35 a^3 b^3 x^3+140 a^4 b^2 x^4+630 a^5 b x^5+420 a^6 x^6\right )}{x^5 (b+a x)^2}+420 a^5 \log (x)-420 a^5 \log (b+a x)}{20 b^8} \]
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Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {1}{5 b^{3} x^{5}}+\frac {3 a}{4 b^{4} x^{4}}-\frac {2 a^{2}}{b^{5} x^{3}}+\frac {5 a^{3}}{b^{6} x^{2}}-\frac {15 a^{4}}{b^{7} x}-\frac {a^{5}}{2 b^{6} \left (a x +b \right )^{2}}-\frac {6 a^{5}}{b^{7} \left (a x +b \right )}-\frac {21 a^{5} \ln \left (x \right )}{b^{8}}+\frac {21 a^{5} \ln \left (a x +b \right )}{b^{8}}\) | \(106\) |
risch | \(\frac {-\frac {21 a^{6} x^{6}}{b^{7}}-\frac {63 a^{5} x^{5}}{2 b^{6}}-\frac {7 a^{4} x^{4}}{b^{5}}+\frac {7 a^{3} x^{3}}{4 b^{4}}-\frac {7 a^{2} x^{2}}{10 b^{3}}+\frac {7 a x}{20 b^{2}}-\frac {1}{5 b}}{x^{5} \left (a x +b \right )^{2}}+\frac {21 a^{5} \ln \left (-a x -b \right )}{b^{8}}-\frac {21 a^{5} \ln \left (x \right )}{b^{8}}\) | \(108\) |
norman | \(\frac {-\frac {x^{3}}{5 b}+\frac {7 a \,x^{4}}{20 b^{2}}-\frac {7 a^{2} x^{5}}{10 b^{3}}+\frac {7 a^{3} x^{6}}{4 b^{4}}-\frac {7 a^{4} x^{7}}{b^{5}}+\frac {42 a^{6} x^{9}}{b^{7}}+\frac {63 a^{7} x^{10}}{2 b^{8}}}{\left (a x +b \right )^{2} x^{8}}-\frac {21 a^{5} \ln \left (x \right )}{b^{8}}+\frac {21 a^{5} \ln \left (a x +b \right )}{b^{8}}\) | \(110\) |
parallelrisch | \(-\frac {420 \ln \left (x \right ) x^{7} a^{7}-420 \ln \left (a x +b \right ) x^{7} a^{7}+840 \ln \left (x \right ) x^{6} a^{6} b -840 \ln \left (a x +b \right ) x^{6} a^{6} b -630 a^{7} x^{7}+420 \ln \left (x \right ) x^{5} a^{5} b^{2}-420 \ln \left (a x +b \right ) x^{5} a^{5} b^{2}-840 a^{6} b \,x^{6}+140 a^{4} b^{3} x^{4}-35 a^{3} b^{4} x^{3}+14 b^{5} x^{2} a^{2}-7 b^{6} x a +4 b^{7}}{20 b^{8} x^{5} \left (a x +b \right )^{2}}\) | \(159\) |
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Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^9} \, dx=-\frac {420 \, a^{6} b x^{6} + 630 \, a^{5} b^{2} x^{5} + 140 \, a^{4} b^{3} x^{4} - 35 \, a^{3} b^{4} x^{3} + 14 \, a^{2} b^{5} x^{2} - 7 \, a b^{6} x + 4 \, b^{7} - 420 \, {\left (a^{7} x^{7} + 2 \, a^{6} b x^{6} + a^{5} b^{2} x^{5}\right )} \log \left (a x + b\right ) + 420 \, {\left (a^{7} x^{7} + 2 \, a^{6} b x^{6} + a^{5} b^{2} x^{5}\right )} \log \left (x\right )}{20 \, {\left (a^{2} b^{8} x^{7} + 2 \, a b^{9} x^{6} + b^{10} x^{5}\right )}} \]
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Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^9} \, dx=\frac {21 a^{5} \left (- \log {\left (x \right )} + \log {\left (x + \frac {b}{a} \right )}\right )}{b^{8}} + \frac {- 420 a^{6} x^{6} - 630 a^{5} b x^{5} - 140 a^{4} b^{2} x^{4} + 35 a^{3} b^{3} x^{3} - 14 a^{2} b^{4} x^{2} + 7 a b^{5} x - 4 b^{6}}{20 a^{2} b^{7} x^{7} + 40 a b^{8} x^{6} + 20 b^{9} x^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^9} \, dx=-\frac {420 \, a^{6} x^{6} + 630 \, a^{5} b x^{5} + 140 \, a^{4} b^{2} x^{4} - 35 \, a^{3} b^{3} x^{3} + 14 \, a^{2} b^{4} x^{2} - 7 \, a b^{5} x + 4 \, b^{6}}{20 \, {\left (a^{2} b^{7} x^{7} + 2 \, a b^{8} x^{6} + b^{9} x^{5}\right )}} + \frac {21 \, a^{5} \log \left (a x + b\right )}{b^{8}} - \frac {21 \, a^{5} \log \left (x\right )}{b^{8}} \]
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Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^9} \, dx=\frac {21 \, a^{5} \log \left ({\left | a x + b \right |}\right )}{b^{8}} - \frac {21 \, a^{5} \log \left ({\left | x \right |}\right )}{b^{8}} - \frac {420 \, a^{6} b x^{6} + 630 \, a^{5} b^{2} x^{5} + 140 \, a^{4} b^{3} x^{4} - 35 \, a^{3} b^{4} x^{3} + 14 \, a^{2} b^{5} x^{2} - 7 \, a b^{6} x + 4 \, b^{7}}{20 \, {\left (a x + b\right )}^{2} b^{8} x^{5}} \]
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Time = 6.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^9} \, dx=\frac {42\,a^5\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^8}-\frac {\frac {1}{5\,b}+\frac {7\,a^2\,x^2}{10\,b^3}-\frac {7\,a^3\,x^3}{4\,b^4}+\frac {7\,a^4\,x^4}{b^5}+\frac {63\,a^5\,x^5}{2\,b^6}+\frac {21\,a^6\,x^6}{b^7}-\frac {7\,a\,x}{20\,b^2}}{a^2\,x^7+2\,a\,b\,x^6+b^2\,x^5} \]
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