Integrand size = 24, antiderivative size = 92 \[ \int \frac {x^{10}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {9 a^3 x}{2 b^5}+\frac {3 a^2 x^3}{2 b^4}-\frac {9 a x^5}{10 b^3}+\frac {9 x^7}{14 b^2}-\frac {x^9}{2 b \left (a+b x^2\right )}+\frac {9 a^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \]
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Time = 0.04 (sec) , antiderivative size = 92, 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} \[ \int \frac {x^{10}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {9 a^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}}-\frac {9 a^3 x}{2 b^5}+\frac {3 a^2 x^3}{2 b^4}-\frac {9 a x^5}{10 b^3}-\frac {x^9}{2 b \left (a+b x^2\right )}+\frac {9 x^7}{14 b^2} \]
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Rule 28
Rule 211
Rule 294
Rule 308
Rubi steps \begin{align*} \text {integral}& = b^2 \int \frac {x^{10}}{\left (a b+b^2 x^2\right )^2} \, dx \\ & = -\frac {x^9}{2 b \left (a+b x^2\right )}+\frac {9}{2} \int \frac {x^8}{a b+b^2 x^2} \, dx \\ & = -\frac {x^9}{2 b \left (a+b x^2\right )}+\frac {9}{2} \int \left (-\frac {a^3}{b^5}+\frac {a^2 x^2}{b^4}-\frac {a x^4}{b^3}+\frac {x^6}{b^2}+\frac {a^4}{b^4 \left (a b+b^2 x^2\right )}\right ) \, dx \\ & = -\frac {9 a^3 x}{2 b^5}+\frac {3 a^2 x^3}{2 b^4}-\frac {9 a x^5}{10 b^3}+\frac {9 x^7}{14 b^2}-\frac {x^9}{2 b \left (a+b x^2\right )}+\frac {\left (9 a^4\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{2 b^4} \\ & = -\frac {9 a^3 x}{2 b^5}+\frac {3 a^2 x^3}{2 b^4}-\frac {9 a x^5}{10 b^3}+\frac {9 x^7}{14 b^2}-\frac {x^9}{2 b \left (a+b x^2\right )}+\frac {9 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {x^{10}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {x \left (-280 a^3+70 a^2 b x^2-28 a b^2 x^4+10 b^3 x^6-\frac {35 a^4}{a+b x^2}\right )}{70 b^5}+\frac {9 a^{7/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}} \]
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Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {-\frac {1}{7} b^{3} x^{7}+\frac {2}{5} b^{2} x^{5} a -a^{2} b \,x^{3}+4 a^{3} x}{b^{5}}+\frac {a^{4} \left (-\frac {x}{2 \left (b \,x^{2}+a \right )}+\frac {9 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{5}}\) | \(76\) |
risch | \(\frac {x^{7}}{7 b^{2}}-\frac {2 a \,x^{5}}{5 b^{3}}+\frac {a^{2} x^{3}}{b^{4}}-\frac {4 a^{3} x}{b^{5}}-\frac {a^{4} x}{2 b^{5} \left (b \,x^{2}+a \right )}+\frac {9 \sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x +a \right )}{4 b^{6}}-\frac {9 \sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x +a \right )}{4 b^{6}}\) | \(107\) |
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Time = 0.26 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.30 \[ \int \frac {x^{10}}{a^2+2 a b x^2+b^2 x^4} \, dx=\left [\frac {20 \, b^{4} x^{9} - 36 \, a b^{3} x^{7} + 84 \, a^{2} b^{2} x^{5} - 420 \, a^{3} b x^{3} - 630 \, a^{4} x + 315 \, {\left (a^{3} b x^{2} + a^{4}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{140 \, {\left (b^{6} x^{2} + a b^{5}\right )}}, \frac {10 \, b^{4} x^{9} - 18 \, a b^{3} x^{7} + 42 \, a^{2} b^{2} x^{5} - 210 \, a^{3} b x^{3} - 315 \, a^{4} x + 315 \, {\left (a^{3} b x^{2} + a^{4}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{70 \, {\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \]
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Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.46 \[ \int \frac {x^{10}}{a^2+2 a b x^2+b^2 x^4} \, dx=- \frac {a^{4} x}{2 a b^{5} + 2 b^{6} x^{2}} - \frac {4 a^{3} x}{b^{5}} + \frac {a^{2} x^{3}}{b^{4}} - \frac {2 a x^{5}}{5 b^{3}} - \frac {9 \sqrt {- \frac {a^{7}}{b^{11}}} \log {\left (x - \frac {b^{5} \sqrt {- \frac {a^{7}}{b^{11}}}}{a^{3}} \right )}}{4} + \frac {9 \sqrt {- \frac {a^{7}}{b^{11}}} \log {\left (x + \frac {b^{5} \sqrt {- \frac {a^{7}}{b^{11}}}}{a^{3}} \right )}}{4} + \frac {x^{7}}{7 b^{2}} \]
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Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {x^{10}}{a^2+2 a b x^2+b^2 x^4} \, dx=-\frac {a^{4} x}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} + \frac {9 \, a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {5 \, b^{3} x^{7} - 14 \, a b^{2} x^{5} + 35 \, a^{2} b x^{3} - 140 \, a^{3} x}{35 \, b^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {x^{10}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {9 \, a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} - \frac {a^{4} x}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {5 \, b^{12} x^{7} - 14 \, a b^{11} x^{5} + 35 \, a^{2} b^{10} x^{3} - 140 \, a^{3} b^{9} x}{35 \, b^{14}} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.84 \[ \int \frac {x^{10}}{a^2+2 a b x^2+b^2 x^4} \, dx=\frac {x^7}{7\,b^2}-\frac {2\,a\,x^5}{5\,b^3}-\frac {4\,a^3\,x}{b^5}+\frac {9\,a^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{11/2}}+\frac {a^2\,x^3}{b^4}-\frac {a^4\,x}{2\,\left (b^6\,x^2+a\,b^5\right )} \]
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