Integrand size = 24, antiderivative size = 77 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx=-\frac {a^6}{10 x^{10}}-\frac {3 a^5 b}{4 x^8}-\frac {5 a^4 b^2}{2 x^6}-\frac {5 a^3 b^3}{x^4}-\frac {15 a^2 b^4}{2 x^2}+\frac {b^6 x^2}{2}+6 a b^5 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 45} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx=-\frac {a^6}{10 x^{10}}-\frac {3 a^5 b}{4 x^8}-\frac {5 a^4 b^2}{2 x^6}-\frac {5 a^3 b^3}{x^4}-\frac {15 a^2 b^4}{2 x^2}+6 a b^5 \log (x)+\frac {b^6 x^2}{2} \]
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Rule 28
Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{11}} \, dx}{b^6} \\ & = \frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^6} \, dx,x,x^2\right )}{2 b^6} \\ & = \frac {\text {Subst}\left (\int \left (b^{12}+\frac {a^6 b^6}{x^6}+\frac {6 a^5 b^7}{x^5}+\frac {15 a^4 b^8}{x^4}+\frac {20 a^3 b^9}{x^3}+\frac {15 a^2 b^{10}}{x^2}+\frac {6 a b^{11}}{x}\right ) \, dx,x,x^2\right )}{2 b^6} \\ & = -\frac {a^6}{10 x^{10}}-\frac {3 a^5 b}{4 x^8}-\frac {5 a^4 b^2}{2 x^6}-\frac {5 a^3 b^3}{x^4}-\frac {15 a^2 b^4}{2 x^2}+\frac {b^6 x^2}{2}+6 a b^5 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx=-\frac {a^6}{10 x^{10}}-\frac {3 a^5 b}{4 x^8}-\frac {5 a^4 b^2}{2 x^6}-\frac {5 a^3 b^3}{x^4}-\frac {15 a^2 b^4}{2 x^2}+\frac {b^6 x^2}{2}+6 a b^5 \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {a^{6}}{10 x^{10}}-\frac {3 a^{5} b}{4 x^{8}}-\frac {5 a^{4} b^{2}}{2 x^{6}}-\frac {5 a^{3} b^{3}}{x^{4}}-\frac {15 a^{2} b^{4}}{2 x^{2}}+\frac {b^{6} x^{2}}{2}+6 a \,b^{5} \ln \left (x \right )\) | \(68\) |
norman | \(\frac {-\frac {1}{10} a^{6}+\frac {1}{2} b^{6} x^{12}-\frac {15}{2} a^{2} b^{4} x^{8}-5 a^{3} b^{3} x^{6}-\frac {5}{2} a^{4} b^{2} x^{4}-\frac {3}{4} a^{5} b \,x^{2}}{x^{10}}+6 a \,b^{5} \ln \left (x \right )\) | \(70\) |
risch | \(\frac {b^{6} x^{2}}{2}+\frac {-\frac {15}{2} a^{2} b^{4} x^{8}-5 a^{3} b^{3} x^{6}-\frac {5}{2} a^{4} b^{2} x^{4}-\frac {3}{4} a^{5} b \,x^{2}-\frac {1}{10} a^{6}}{x^{10}}+6 a \,b^{5} \ln \left (x \right )\) | \(70\) |
parallelrisch | \(\frac {10 b^{6} x^{12}+120 b^{5} a \ln \left (x \right ) x^{10}-150 a^{2} b^{4} x^{8}-100 a^{3} b^{3} x^{6}-50 a^{4} b^{2} x^{4}-15 a^{5} b \,x^{2}-2 a^{6}}{20 x^{10}}\) | \(73\) |
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Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx=\frac {10 \, b^{6} x^{12} + 120 \, a b^{5} x^{10} \log \left (x\right ) - 150 \, a^{2} b^{4} x^{8} - 100 \, a^{3} b^{3} x^{6} - 50 \, a^{4} b^{2} x^{4} - 15 \, a^{5} b x^{2} - 2 \, a^{6}}{20 \, x^{10}} \]
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Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx=6 a b^{5} \log {\left (x \right )} + \frac {b^{6} x^{2}}{2} + \frac {- 2 a^{6} - 15 a^{5} b x^{2} - 50 a^{4} b^{2} x^{4} - 100 a^{3} b^{3} x^{6} - 150 a^{2} b^{4} x^{8}}{20 x^{10}} \]
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Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx=\frac {1}{2} \, b^{6} x^{2} + 3 \, a b^{5} \log \left (x^{2}\right ) - \frac {150 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 50 \, a^{4} b^{2} x^{4} + 15 \, a^{5} b x^{2} + 2 \, a^{6}}{20 \, x^{10}} \]
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Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx=\frac {1}{2} \, b^{6} x^{2} + 3 \, a b^{5} \log \left (x^{2}\right ) - \frac {137 \, a b^{5} x^{10} + 150 \, a^{2} b^{4} x^{8} + 100 \, a^{3} b^{3} x^{6} + 50 \, a^{4} b^{2} x^{4} + 15 \, a^{5} b x^{2} + 2 \, a^{6}}{20 \, x^{10}} \]
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Time = 13.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{11}} \, dx=\frac {b^6\,x^2}{2}-\frac {\frac {a^6}{10}+\frac {3\,a^5\,b\,x^2}{4}+\frac {5\,a^4\,b^2\,x^4}{2}+5\,a^3\,b^3\,x^6+\frac {15\,a^2\,b^4\,x^8}{2}}{x^{10}}+6\,a\,b^5\,\ln \left (x\right ) \]
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