Integrand size = 24, antiderivative size = 83 \[ \int \frac {x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {2 a^3 x^2}{b^5}+\frac {3 a^2 x^4}{4 b^4}-\frac {a x^6}{3 b^3}+\frac {x^8}{8 b^2}+\frac {a^5}{2 b^6 \left (a+b x^2\right )}+\frac {5 a^4 \log \left (a+b x^2\right )}{2 b^6} \]
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Time = 0.06 (sec) , antiderivative size = 83, 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} \[ \int \frac {x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^5}{2 b^6 \left (a+b x^2\right )}+\frac {5 a^4 \log \left (a+b x^2\right )}{2 b^6}-\frac {2 a^3 x^2}{b^5}+\frac {3 a^2 x^4}{4 b^4}-\frac {a x^6}{3 b^3}+\frac {x^8}{8 b^2} \]
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Rule 28
Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {x^{11}}{\left (a b+b^2 x^2\right )^2} \, dx \\ & = \frac {1}{2} b^2 \text {Subst}\left (\int \frac {x^5}{\left (a b+b^2 x\right )^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} b^2 \text {Subst}\left (\int \left (-\frac {4 a^3}{b^7}+\frac {3 a^2 x}{b^6}-\frac {2 a x^2}{b^5}+\frac {x^3}{b^4}-\frac {a^5}{b^7 (a+b x)^2}+\frac {5 a^4}{b^7 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {2 a^3 x^2}{b^5}+\frac {3 a^2 x^4}{4 b^4}-\frac {a x^6}{3 b^3}+\frac {x^8}{8 b^2}+\frac {a^5}{2 b^6 \left (a+b x^2\right )}+\frac {5 a^4 \log \left (a+b x^2\right )}{2 b^6} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87 \[ \int \frac {x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {-48 a^3 b x^2+18 a^2 b^2 x^4-8 a b^3 x^6+3 b^4 x^8+\frac {12 a^5}{a+b x^2}+60 a^4 \log \left (a+b x^2\right )}{24 b^6} \]
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Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {2 a^{3} x^{2}}{b^{5}}+\frac {3 a^{2} x^{4}}{4 b^{4}}-\frac {a \,x^{6}}{3 b^{3}}+\frac {x^{8}}{8 b^{2}}+\frac {a^{5}}{2 b^{6} \left (b \,x^{2}+a \right )}+\frac {5 a^{4} \ln \left (b \,x^{2}+a \right )}{2 b^{6}}\) | \(74\) |
default | \(-\frac {-\frac {1}{4} b^{3} x^{8}+\frac {2}{3} b^{2} x^{6} a -\frac {3}{2} a^{2} b \,x^{4}+4 a^{3} x^{2}}{2 b^{5}}+\frac {5 a^{4} \ln \left (b \,x^{2}+a \right )}{2 b^{6}}+\frac {a^{5}}{2 b^{6} \left (b \,x^{2}+a \right )}\) | \(75\) |
norman | \(\frac {\frac {x^{10}}{8 b}-\frac {5 a \,x^{8}}{24 b^{2}}+\frac {5 a^{5}}{2 b^{6}}+\frac {5 a^{2} x^{6}}{12 b^{3}}-\frac {5 a^{3} x^{4}}{4 b^{4}}}{b \,x^{2}+a}+\frac {5 a^{4} \ln \left (b \,x^{2}+a \right )}{2 b^{6}}\) | \(76\) |
parallelrisch | \(\frac {3 x^{10} b^{5}-5 a \,x^{8} b^{4}+10 a^{2} x^{6} b^{3}-30 a^{3} x^{4} b^{2}+60 \ln \left (b \,x^{2}+a \right ) x^{2} a^{4} b +60 \ln \left (b \,x^{2}+a \right ) a^{5}+60 a^{5}}{24 b^{6} \left (b \,x^{2}+a \right )}\) | \(90\) |
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Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.12 \[ \int \frac {x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {3 \, b^{5} x^{10} - 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} - 30 \, a^{3} b^{2} x^{4} - 48 \, a^{4} b x^{2} + 12 \, a^{5} + 60 \, {\left (a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{24 \, {\left (b^{7} x^{2} + a b^{6}\right )}} \]
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Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \frac {x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^{5}}{2 a b^{6} + 2 b^{7} x^{2}} + \frac {5 a^{4} \log {\left (a + b x^{2} \right )}}{2 b^{6}} - \frac {2 a^{3} x^{2}}{b^{5}} + \frac {3 a^{2} x^{4}}{4 b^{4}} - \frac {a x^{6}}{3 b^{3}} + \frac {x^{8}}{8 b^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93 \[ \int \frac {x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^{5}}{2 \, {\left (b^{7} x^{2} + a b^{6}\right )}} + \frac {5 \, a^{4} \log \left (b x^{2} + a\right )}{2 \, b^{6}} + \frac {3 \, b^{3} x^{8} - 8 \, a b^{2} x^{6} + 18 \, a^{2} b x^{4} - 48 \, a^{3} x^{2}}{24 \, b^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.11 \[ \int \frac {x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {5 \, a^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac {5 \, a^{4} b x^{2} + 4 \, a^{5}}{2 \, {\left (b x^{2} + a\right )} b^{6}} + \frac {3 \, b^{6} x^{8} - 8 \, a b^{5} x^{6} + 18 \, a^{2} b^{4} x^{4} - 48 \, a^{3} b^{3} x^{2}}{24 \, b^{8}} \]
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Time = 13.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int \frac {x^{11}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {x^8}{8\,b^2}+\frac {a^5}{2\,b\,\left (b^6\,x^2+a\,b^5\right )}-\frac {a\,x^6}{3\,b^3}+\frac {5\,a^4\,\ln \left (b\,x^2+a\right )}{2\,b^6}+\frac {3\,a^2\,x^4}{4\,b^4}-\frac {2\,a^3\,x^2}{b^5} \]
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