Integrand size = 24, antiderivative size = 79 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}+\frac {7 x^5}{10 b^2}-\frac {x^7}{2 b \left (a+b x^2\right )}-\frac {7 a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}} \]
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Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 294, 308, 211} \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {7 a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}+\frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}-\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7 x^5}{10 b^2} \]
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Rule 28
Rule 211
Rule 294
Rule 308
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {x^8}{\left (a b+b^2 x^2\right )^2} \, dx \\ & = -\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7}{2} \int \frac {x^6}{a b+b^2 x^2} \, dx \\ & = -\frac {x^7}{2 b \left (a+b x^2\right )}+\frac {7}{2} \int \left (\frac {a^2}{b^4}-\frac {a x^2}{b^3}+\frac {x^4}{b^2}-\frac {a^3}{b^3 \left (a b+b^2 x^2\right )}\right ) \, dx \\ & = \frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}+\frac {7 x^5}{10 b^2}-\frac {x^7}{2 b \left (a+b x^2\right )}-\frac {\left (7 a^3\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{2 b^3} \\ & = \frac {7 a^2 x}{2 b^4}-\frac {7 a x^3}{6 b^3}+\frac {7 x^5}{10 b^2}-\frac {x^7}{2 b \left (a+b x^2\right )}-\frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {x \left (90 a^2-20 a b x^2+6 b^2 x^4+\frac {15 a^3}{a+b x^2}\right )}{30 b^4}-\frac {7 a^{5/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}} \]
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Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {\frac {1}{5} b^{2} x^{5}-\frac {2}{3} a b \,x^{3}+3 a^{2} x}{b^{4}}-\frac {a^{3} \left (-\frac {x}{2 \left (b \,x^{2}+a \right )}+\frac {7 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{4}}\) | \(65\) |
risch | \(\frac {x^{5}}{5 b^{2}}-\frac {2 a \,x^{3}}{3 b^{3}}+\frac {3 a^{2} x}{b^{4}}+\frac {a^{3} x}{2 b^{4} \left (b \,x^{2}+a \right )}+\frac {7 \sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right )}{4 b^{5}}-\frac {7 \sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right )}{4 b^{5}}\) | \(101\) |
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Time = 0.26 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.41 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\left [\frac {12 \, b^{3} x^{7} - 28 \, a b^{2} x^{5} + 140 \, a^{2} b x^{3} + 210 \, a^{3} x + 105 \, {\left (a^{2} b x^{2} + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{60 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, \frac {6 \, b^{3} x^{7} - 14 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 105 \, a^{3} x - 105 \, {\left (a^{2} b x^{2} + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{30 \, {\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \]
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Time = 0.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.57 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^{3} x}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {3 a^{2} x}{b^{4}} - \frac {2 a x^{3}}{3 b^{3}} + \frac {7 \sqrt {- \frac {a^{5}}{b^{9}}} \log {\left (x - \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} - \frac {7 \sqrt {- \frac {a^{5}}{b^{9}}} \log {\left (x + \frac {b^{4} \sqrt {- \frac {a^{5}}{b^{9}}}}{a^{2}} \right )}}{4} + \frac {x^{5}}{5 b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {a^{3} x}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} - \frac {7 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {3 \, b^{2} x^{5} - 10 \, a b x^{3} + 45 \, a^{2} x}{15 \, b^{4}} \]
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.92 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {7 \, a^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {a^{3} x}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {3 \, b^{8} x^{5} - 10 \, a b^{7} x^{3} + 45 \, a^{2} b^{6} x}{15 \, b^{10}} \]
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Time = 13.51 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int \frac {x^8}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {x^5}{5\,b^2}-\frac {2\,a\,x^3}{3\,b^3}+\frac {3\,a^2\,x}{b^4}-\frac {7\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{9/2}}+\frac {a^3\,x}{2\,\left (b^5\,x^2+a\,b^4\right )} \]
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