Integrand size = 13, antiderivative size = 97 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=-\frac {1}{4 b^3 x^4}+\frac {a}{b^4 x^3}-\frac {3 a^2}{b^5 x^2}+\frac {10 a^3}{b^6 x}+\frac {a^4}{2 b^5 (b+a x)^2}+\frac {5 a^4}{b^6 (b+a x)}+\frac {15 a^4 \log (x)}{b^7}-\frac {15 a^4 \log (b+a x)}{b^7} \]
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Time = 0.04 (sec) , antiderivative size = 97, 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} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {15 a^4 \log (x)}{b^7}-\frac {15 a^4 \log (a x+b)}{b^7}+\frac {5 a^4}{b^6 (a x+b)}+\frac {a^4}{2 b^5 (a x+b)^2}+\frac {10 a^3}{b^6 x}-\frac {3 a^2}{b^5 x^2}+\frac {a}{b^4 x^3}-\frac {1}{4 b^3 x^4} \]
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Rule 46
Rule 269
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {\frac {b \left (-b^5+2 a b^4 x-5 a^2 b^3 x^2+20 a^3 b^2 x^3+90 a^4 b x^4+60 a^5 x^5\right )}{x^4 (b+a x)^2}+60 a^4 \log (x)-60 a^4 \log (b+a x)}{4 b^7} \]
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Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {1}{4 b^{3} x^{4}}+\frac {a}{b^{4} x^{3}}-\frac {3 a^{2}}{b^{5} x^{2}}+\frac {10 a^{3}}{b^{6} x}+\frac {a^{4}}{2 b^{5} \left (a x +b \right )^{2}}+\frac {5 a^{4}}{b^{6} \left (a x +b \right )}+\frac {15 a^{4} \ln \left (x \right )}{b^{7}}-\frac {15 a^{4} \ln \left (a x +b \right )}{b^{7}}\) | \(94\) |
risch | \(\frac {\frac {15 a^{5} x^{5}}{b^{6}}+\frac {45 a^{4} x^{4}}{2 b^{5}}+\frac {5 a^{3} x^{3}}{b^{4}}-\frac {5 a^{2} x^{2}}{4 b^{3}}+\frac {a x}{2 b^{2}}-\frac {1}{4 b}}{x^{4} \left (a x +b \right )^{2}}-\frac {15 a^{4} \ln \left (a x +b \right )}{b^{7}}+\frac {15 a^{4} \ln \left (-x \right )}{b^{7}}\) | \(96\) |
norman | \(\frac {-\frac {x^{3}}{4 b}+\frac {a \,x^{4}}{2 b^{2}}-\frac {5 a^{2} x^{5}}{4 b^{3}}+\frac {5 a^{3} x^{6}}{b^{4}}-\frac {30 a^{5} x^{8}}{b^{6}}-\frac {45 a^{6} x^{9}}{2 b^{7}}}{\left (a x +b \right )^{2} x^{7}}+\frac {15 a^{4} \ln \left (x \right )}{b^{7}}-\frac {15 a^{4} \ln \left (a x +b \right )}{b^{7}}\) | \(99\) |
parallelrisch | \(\frac {60 \ln \left (x \right ) x^{6} a^{6}-60 \ln \left (a x +b \right ) x^{6} a^{6}+120 \ln \left (x \right ) x^{5} a^{5} b -120 \ln \left (a x +b \right ) x^{5} a^{5} b -90 a^{6} x^{6}+60 \ln \left (x \right ) x^{4} a^{4} b^{2}-60 \ln \left (a x +b \right ) x^{4} a^{4} b^{2}-120 a^{5} b \,x^{5}+20 a^{3} x^{3} b^{3}-5 b^{4} x^{2} a^{2}+2 a x \,b^{5}-b^{6}}{4 b^{7} x^{4} \left (a x +b \right )^{2}}\) | \(148\) |
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Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {60 \, a^{5} b x^{5} + 90 \, a^{4} b^{2} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{2} b^{4} x^{2} + 2 \, a b^{5} x - b^{6} - 60 \, {\left (a^{6} x^{6} + 2 \, a^{5} b x^{5} + a^{4} b^{2} x^{4}\right )} \log \left (a x + b\right ) + 60 \, {\left (a^{6} x^{6} + 2 \, a^{5} b x^{5} + a^{4} b^{2} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{2} b^{7} x^{6} + 2 \, a b^{8} x^{5} + b^{9} x^{4}\right )}} \]
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Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {15 a^{4} \left (\log {\left (x \right )} - \log {\left (x + \frac {b}{a} \right )}\right )}{b^{7}} + \frac {60 a^{5} x^{5} + 90 a^{4} b x^{4} + 20 a^{3} b^{2} x^{3} - 5 a^{2} b^{3} x^{2} + 2 a b^{4} x - b^{5}}{4 a^{2} b^{6} x^{6} + 8 a b^{7} x^{5} + 4 b^{8} x^{4}} \]
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Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {60 \, a^{5} x^{5} + 90 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} - 5 \, a^{2} b^{3} x^{2} + 2 \, a b^{4} x - b^{5}}{4 \, {\left (a^{2} b^{6} x^{6} + 2 \, a b^{7} x^{5} + b^{8} x^{4}\right )}} - \frac {15 \, a^{4} \log \left (a x + b\right )}{b^{7}} + \frac {15 \, a^{4} \log \left (x\right )}{b^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=-\frac {15 \, a^{4} \log \left ({\left | a x + b \right |}\right )}{b^{7}} + \frac {15 \, a^{4} \log \left ({\left | x \right |}\right )}{b^{7}} + \frac {60 \, a^{5} b x^{5} + 90 \, a^{4} b^{2} x^{4} + 20 \, a^{3} b^{3} x^{3} - 5 \, a^{2} b^{4} x^{2} + 2 \, a b^{5} x - b^{6}}{4 \, {\left (a x + b\right )}^{2} b^{7} x^{4}} \]
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Time = 6.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^8} \, dx=\frac {\frac {5\,a^3\,x^3}{b^4}-\frac {5\,a^2\,x^2}{4\,b^3}-\frac {1}{4\,b}+\frac {45\,a^4\,x^4}{2\,b^5}+\frac {15\,a^5\,x^5}{b^6}+\frac {a\,x}{2\,b^2}}{a^2\,x^6+2\,a\,b\,x^5+b^2\,x^4}-\frac {30\,a^4\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{b^7} \]
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