Integrand size = 24, antiderivative size = 57 \[ \int \frac {x^7}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {a x^2}{b^3}+\frac {x^4}{4 b^2}+\frac {a^3}{2 b^4 \left (a+b x^2\right )}+\frac {3 a^2 \log \left (a+b x^2\right )}{2 b^4} \]
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Time = 0.04 (sec) , antiderivative size = 57, 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} \[ \int \frac {x^7}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^3}{2 b^4 \left (a+b x^2\right )}+\frac {3 a^2 \log \left (a+b x^2\right )}{2 b^4}-\frac {a x^2}{b^3}+\frac {x^4}{4 b^2} \]
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Rule 28
Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {x^7}{\left (a b+b^2 x^2\right )^2} \, dx \\ & = \frac {1}{2} b^2 \text {Subst}\left (\int \frac {x^3}{\left (a b+b^2 x\right )^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} b^2 \text {Subst}\left (\int \left (-\frac {2 a}{b^5}+\frac {x}{b^4}-\frac {a^3}{b^5 (a+b x)^2}+\frac {3 a^2}{b^5 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a x^2}{b^3}+\frac {x^4}{4 b^2}+\frac {a^3}{2 b^4 \left (a+b x^2\right )}+\frac {3 a^2 \log \left (a+b x^2\right )}{2 b^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {x^7}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {-4 a b x^2+b^2 x^4+\frac {2 a^3}{a+b x^2}+6 a^2 \log \left (a+b x^2\right )}{4 b^4} \]
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Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {-\frac {1}{2} b \,x^{4}+2 a \,x^{2}}{2 b^{3}}+\frac {3 a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{4}}+\frac {a^{3}}{2 b^{4} \left (b \,x^{2}+a \right )}\) | \(53\) |
norman | \(\frac {\frac {x^{6}}{4 b}-\frac {3 a \,x^{4}}{4 b^{2}}+\frac {3 a^{3}}{2 b^{4}}}{b \,x^{2}+a}+\frac {3 a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(54\) |
risch | \(\frac {x^{4}}{4 b^{2}}-\frac {a \,x^{2}}{b^{3}}+\frac {a^{2}}{b^{4}}+\frac {a^{3}}{2 b^{4} \left (b \,x^{2}+a \right )}+\frac {3 a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(59\) |
parallelrisch | \(\frac {b^{3} x^{6}-3 b^{2} x^{4} a +6 \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b +6 \ln \left (b \,x^{2}+a \right ) a^{3}+6 a^{3}}{4 b^{4} \left (b \,x^{2}+a \right )}\) | \(67\) |
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Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.23 \[ \int \frac {x^7}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {b^{3} x^{6} - 3 \, a b^{2} x^{4} - 4 \, a^{2} b x^{2} + 2 \, a^{3} + 6 \, {\left (a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {x^7}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^{3}}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {3 a^{2} \log {\left (a + b x^{2} \right )}}{2 b^{4}} - \frac {a x^{2}}{b^{3}} + \frac {x^{4}}{4 b^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {x^7}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^{3}}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {3 \, a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac {b x^{4} - 4 \, a x^{2}}{4 \, b^{3}} \]
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Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.18 \[ \int \frac {x^7}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {3 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac {b^{2} x^{4} - 4 \, a b x^{2}}{4 \, b^{4}} - \frac {3 \, a^{2} b x^{2} + 2 \, a^{3}}{2 \, {\left (b x^{2} + a\right )} b^{4}} \]
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Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {x^7}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {x^4}{4\,b^2}+\frac {a^3}{2\,b\,\left (b^4\,x^2+a\,b^3\right )}-\frac {a\,x^2}{b^3}+\frac {3\,a^2\,\ln \left (b\,x^2+a\right )}{2\,b^4} \]
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