Integrand size = 24, antiderivative size = 76 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{13}} \, dx=-\frac {a^6}{12 x^{12}}-\frac {3 a^5 b}{5 x^{10}}-\frac {15 a^4 b^2}{8 x^8}-\frac {10 a^3 b^3}{3 x^6}-\frac {15 a^2 b^4}{4 x^4}-\frac {3 a b^5}{x^2}+b^6 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 76, 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} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{13}} \, dx=-\frac {a^6}{12 x^{12}}-\frac {3 a^5 b}{5 x^{10}}-\frac {15 a^4 b^2}{8 x^8}-\frac {10 a^3 b^3}{3 x^6}-\frac {15 a^2 b^4}{4 x^4}-\frac {3 a b^5}{x^2}+b^6 \log (x) \]
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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^{13}} \, dx}{b^6} \\ & = \frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^7} \, dx,x,x^2\right )}{2 b^6} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^6 b^6}{x^7}+\frac {6 a^5 b^7}{x^6}+\frac {15 a^4 b^8}{x^5}+\frac {20 a^3 b^9}{x^4}+\frac {15 a^2 b^{10}}{x^3}+\frac {6 a b^{11}}{x^2}+\frac {b^{12}}{x}\right ) \, dx,x,x^2\right )}{2 b^6} \\ & = -\frac {a^6}{12 x^{12}}-\frac {3 a^5 b}{5 x^{10}}-\frac {15 a^4 b^2}{8 x^8}-\frac {10 a^3 b^3}{3 x^6}-\frac {15 a^2 b^4}{4 x^4}-\frac {3 a b^5}{x^2}+b^6 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{13}} \, dx=-\frac {a^6}{12 x^{12}}-\frac {3 a^5 b}{5 x^{10}}-\frac {15 a^4 b^2}{8 x^8}-\frac {10 a^3 b^3}{3 x^6}-\frac {15 a^2 b^4}{4 x^4}-\frac {3 a b^5}{x^2}+b^6 \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {a^{6}}{12 x^{12}}-\frac {3 a^{5} b}{5 x^{10}}-\frac {15 a^{4} b^{2}}{8 x^{8}}-\frac {10 a^{3} b^{3}}{3 x^{6}}-\frac {15 a^{2} b^{4}}{4 x^{4}}-\frac {3 a \,b^{5}}{x^{2}}+b^{6} \ln \left (x \right )\) | \(67\) |
norman | \(\frac {-\frac {1}{12} a^{6}-3 a \,b^{5} x^{10}-\frac {15}{4} a^{2} b^{4} x^{8}-\frac {10}{3} a^{3} b^{3} x^{6}-\frac {15}{8} a^{4} b^{2} x^{4}-\frac {3}{5} a^{5} b \,x^{2}}{x^{12}}+b^{6} \ln \left (x \right )\) | \(69\) |
risch | \(\frac {-\frac {1}{12} a^{6}-3 a \,b^{5} x^{10}-\frac {15}{4} a^{2} b^{4} x^{8}-\frac {10}{3} a^{3} b^{3} x^{6}-\frac {15}{8} a^{4} b^{2} x^{4}-\frac {3}{5} a^{5} b \,x^{2}}{x^{12}}+b^{6} \ln \left (x \right )\) | \(69\) |
parallelrisch | \(\frac {120 b^{6} \ln \left (x \right ) x^{12}-360 a \,b^{5} x^{10}-450 a^{2} b^{4} x^{8}-400 a^{3} b^{3} x^{6}-225 a^{4} b^{2} x^{4}-72 a^{5} b \,x^{2}-10 a^{6}}{120 x^{12}}\) | \(73\) |
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Time = 0.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{13}} \, dx=\frac {120 \, b^{6} x^{12} \log \left (x\right ) - 360 \, a b^{5} x^{10} - 450 \, a^{2} b^{4} x^{8} - 400 \, a^{3} b^{3} x^{6} - 225 \, a^{4} b^{2} x^{4} - 72 \, a^{5} b x^{2} - 10 \, a^{6}}{120 \, x^{12}} \]
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Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{13}} \, dx=b^{6} \log {\left (x \right )} + \frac {- 10 a^{6} - 72 a^{5} b x^{2} - 225 a^{4} b^{2} x^{4} - 400 a^{3} b^{3} x^{6} - 450 a^{2} b^{4} x^{8} - 360 a b^{5} x^{10}}{120 x^{12}} \]
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Time = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{13}} \, dx=\frac {1}{2} \, b^{6} \log \left (x^{2}\right ) - \frac {360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \]
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Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{13}} \, dx=\frac {1}{2} \, b^{6} \log \left (x^{2}\right ) - \frac {147 \, b^{6} x^{12} + 360 \, a b^{5} x^{10} + 450 \, a^{2} b^{4} x^{8} + 400 \, a^{3} b^{3} x^{6} + 225 \, a^{4} b^{2} x^{4} + 72 \, a^{5} b x^{2} + 10 \, a^{6}}{120 \, x^{12}} \]
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Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{13}} \, dx=b^6\,\ln \left (x\right )-\frac {\frac {a^6}{12}+\frac {3\,a^5\,b\,x^2}{5}+\frac {15\,a^4\,b^2\,x^4}{8}+\frac {10\,a^3\,b^3\,x^6}{3}+\frac {15\,a^2\,b^4\,x^8}{4}+3\,a\,b^5\,x^{10}}{x^{12}} \]
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