Integrand size = 24, antiderivative size = 79 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^7} \, dx=-\frac {a^6}{6 x^6}-\frac {3 a^5 b}{2 x^4}-\frac {15 a^4 b^2}{2 x^2}+\frac {15}{2} a^2 b^4 x^2+\frac {3}{2} a b^5 x^4+\frac {b^6 x^6}{6}+20 a^3 b^3 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 79, 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} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^7} \, dx=-\frac {a^6}{6 x^6}-\frac {3 a^5 b}{2 x^4}-\frac {15 a^4 b^2}{2 x^2}+20 a^3 b^3 \log (x)+\frac {15}{2} a^2 b^4 x^2+\frac {3}{2} a b^5 x^4+\frac {b^6 x^6}{6} \]
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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^7} \, dx}{b^6} \\ & = \frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^4} \, dx,x,x^2\right )}{2 b^6} \\ & = \frac {\text {Subst}\left (\int \left (15 a^2 b^{10}+\frac {a^6 b^6}{x^4}+\frac {6 a^5 b^7}{x^3}+\frac {15 a^4 b^8}{x^2}+\frac {20 a^3 b^9}{x}+6 a b^{11} x+b^{12} x^2\right ) \, dx,x,x^2\right )}{2 b^6} \\ & = -\frac {a^6}{6 x^6}-\frac {3 a^5 b}{2 x^4}-\frac {15 a^4 b^2}{2 x^2}+\frac {15}{2} a^2 b^4 x^2+\frac {3}{2} a b^5 x^4+\frac {b^6 x^6}{6}+20 a^3 b^3 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^7} \, dx=-\frac {a^6}{6 x^6}-\frac {3 a^5 b}{2 x^4}-\frac {15 a^4 b^2}{2 x^2}+\frac {15}{2} a^2 b^4 x^2+\frac {3}{2} a b^5 x^4+\frac {b^6 x^6}{6}+20 a^3 b^3 \log (x) \]
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Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {a^{6}}{6 x^{6}}-\frac {3 a^{5} b}{2 x^{4}}-\frac {15 a^{4} b^{2}}{2 x^{2}}+\frac {15 a^{2} b^{4} x^{2}}{2}+\frac {3 a \,b^{5} x^{4}}{2}+\frac {b^{6} x^{6}}{6}+20 a^{3} b^{3} \ln \left (x \right )\) | \(68\) |
norman | \(\frac {-\frac {1}{6} a^{6}+\frac {1}{6} b^{6} x^{12}+\frac {3}{2} a \,b^{5} x^{10}+\frac {15}{2} a^{2} b^{4} x^{8}-\frac {15}{2} a^{4} b^{2} x^{4}-\frac {3}{2} a^{5} b \,x^{2}}{x^{6}}+20 a^{3} b^{3} \ln \left (x \right )\) | \(70\) |
risch | \(\frac {b^{6} x^{6}}{6}+\frac {3 a \,b^{5} x^{4}}{2}+\frac {15 a^{2} b^{4} x^{2}}{2}+\frac {-\frac {15}{2} a^{4} b^{2} x^{4}-\frac {3}{2} a^{5} b \,x^{2}-\frac {1}{6} a^{6}}{x^{6}}+20 a^{3} b^{3} \ln \left (x \right )\) | \(70\) |
parallelrisch | \(\frac {b^{6} x^{12}+9 a \,b^{5} x^{10}+45 a^{2} b^{4} x^{8}+120 a^{3} b^{3} \ln \left (x \right ) x^{6}-45 a^{4} b^{2} x^{4}-9 a^{5} b \,x^{2}-a^{6}}{6 x^{6}}\) | \(72\) |
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Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^7} \, dx=\frac {b^{6} x^{12} + 9 \, a b^{5} x^{10} + 45 \, a^{2} b^{4} x^{8} + 120 \, a^{3} b^{3} x^{6} \log \left (x\right ) - 45 \, a^{4} b^{2} x^{4} - 9 \, a^{5} b x^{2} - a^{6}}{6 \, x^{6}} \]
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Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^7} \, dx=20 a^{3} b^{3} \log {\left (x \right )} + \frac {15 a^{2} b^{4} x^{2}}{2} + \frac {3 a b^{5} x^{4}}{2} + \frac {b^{6} x^{6}}{6} + \frac {- a^{6} - 9 a^{5} b x^{2} - 45 a^{4} b^{2} x^{4}}{6 x^{6}} \]
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Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^7} \, dx=\frac {1}{6} \, b^{6} x^{6} + \frac {3}{2} \, a b^{5} x^{4} + \frac {15}{2} \, a^{2} b^{4} x^{2} + 10 \, a^{3} b^{3} \log \left (x^{2}\right ) - \frac {45 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{6 \, x^{6}} \]
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Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^7} \, dx=\frac {1}{6} \, b^{6} x^{6} + \frac {3}{2} \, a b^{5} x^{4} + \frac {15}{2} \, a^{2} b^{4} x^{2} + 10 \, a^{3} b^{3} \log \left (x^{2}\right ) - \frac {110 \, a^{3} b^{3} x^{6} + 45 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{6 \, x^{6}} \]
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Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^7} \, dx=\frac {b^6\,x^6}{6}-\frac {\frac {a^6}{6}+\frac {3\,a^5\,b\,x^2}{2}+\frac {15\,a^4\,b^2\,x^4}{2}}{x^6}+\frac {3\,a\,b^5\,x^4}{2}+\frac {15\,a^2\,b^4\,x^2}{2}+20\,a^3\,b^3\,\ln \left (x\right ) \]
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