Integrand size = 24, antiderivative size = 121 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {x^9}{10 b \left (a+b x^2\right )^5}-\frac {9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac {21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac {21 x^3}{128 b^4 \left (a+b x^2\right )^2}-\frac {63 x}{256 b^5 \left (a+b x^2\right )}+\frac {63 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 \sqrt {a} b^{11/2}} \]
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Time = 0.05 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 294, 211} \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {63 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 \sqrt {a} b^{11/2}}-\frac {63 x}{256 b^5 \left (a+b x^2\right )}-\frac {21 x^3}{128 b^4 \left (a+b x^2\right )^2}-\frac {21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac {9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac {x^9}{10 b \left (a+b x^2\right )^5} \]
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Rule 28
Rule 211
Rule 294
Rubi steps \begin{align*} \text {integral}& = b^6 \int \frac {x^{10}}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = -\frac {x^9}{10 b \left (a+b x^2\right )^5}+\frac {1}{10} \left (9 b^4\right ) \int \frac {x^8}{\left (a b+b^2 x^2\right )^5} \, dx \\ & = -\frac {x^9}{10 b \left (a+b x^2\right )^5}-\frac {9 x^7}{80 b^2 \left (a+b x^2\right )^4}+\frac {1}{80} \left (63 b^2\right ) \int \frac {x^6}{\left (a b+b^2 x^2\right )^4} \, dx \\ & = -\frac {x^9}{10 b \left (a+b x^2\right )^5}-\frac {9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac {21 x^5}{160 b^3 \left (a+b x^2\right )^3}+\frac {21}{32} \int \frac {x^4}{\left (a b+b^2 x^2\right )^3} \, dx \\ & = -\frac {x^9}{10 b \left (a+b x^2\right )^5}-\frac {9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac {21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac {21 x^3}{128 b^4 \left (a+b x^2\right )^2}+\frac {63 \int \frac {x^2}{\left (a b+b^2 x^2\right )^2} \, dx}{128 b^2} \\ & = -\frac {x^9}{10 b \left (a+b x^2\right )^5}-\frac {9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac {21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac {21 x^3}{128 b^4 \left (a+b x^2\right )^2}-\frac {63 x}{256 b^5 \left (a+b x^2\right )}+\frac {63 \int \frac {1}{a b+b^2 x^2} \, dx}{256 b^4} \\ & = -\frac {x^9}{10 b \left (a+b x^2\right )^5}-\frac {9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac {21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac {21 x^3}{128 b^4 \left (a+b x^2\right )^2}-\frac {63 x}{256 b^5 \left (a+b x^2\right )}+\frac {63 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 \sqrt {a} b^{11/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.73 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {x \left (315 a^4+1470 a^3 b x^2+2688 a^2 b^2 x^4+2370 a b^3 x^6+965 b^4 x^8\right )}{1280 b^5 \left (a+b x^2\right )^5}+\frac {63 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 \sqrt {a} b^{11/2}} \]
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Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {-\frac {193 x^{9}}{256 b}-\frac {237 a \,x^{7}}{128 b^{2}}-\frac {21 a^{2} x^{5}}{10 b^{3}}-\frac {147 a^{3} x^{3}}{128 b^{4}}-\frac {63 a^{4} x}{256 b^{5}}}{\left (b \,x^{2}+a \right )^{5}}+\frac {63 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 b^{5} \sqrt {a b}}\) | \(80\) |
risch | \(\frac {-\frac {193 x^{9}}{256 b}-\frac {237 a \,x^{7}}{128 b^{2}}-\frac {21 a^{2} x^{5}}{10 b^{3}}-\frac {147 a^{3} x^{3}}{128 b^{4}}-\frac {63 a^{4} x}{256 b^{5}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}-\frac {63 \ln \left (b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b^{5}}+\frac {63 \ln \left (-b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b^{5}}\) | \(126\) |
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Time = 0.27 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.19 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [-\frac {1930 \, a b^{5} x^{9} + 4740 \, a^{2} b^{4} x^{7} + 5376 \, a^{3} b^{3} x^{5} + 2940 \, a^{4} b^{2} x^{3} + 630 \, a^{5} b x + 315 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2560 \, {\left (a b^{11} x^{10} + 5 \, a^{2} b^{10} x^{8} + 10 \, a^{3} b^{9} x^{6} + 10 \, a^{4} b^{8} x^{4} + 5 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\right )}}, -\frac {965 \, a b^{5} x^{9} + 2370 \, a^{2} b^{4} x^{7} + 2688 \, a^{3} b^{3} x^{5} + 1470 \, a^{4} b^{2} x^{3} + 315 \, a^{5} b x - 315 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{1280 \, {\left (a b^{11} x^{10} + 5 \, a^{2} b^{10} x^{8} + 10 \, a^{3} b^{9} x^{6} + 10 \, a^{4} b^{8} x^{4} + 5 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\right )}}\right ] \]
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Time = 0.35 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.50 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=- \frac {63 \sqrt {- \frac {1}{a b^{11}}} \log {\left (- a b^{5} \sqrt {- \frac {1}{a b^{11}}} + x \right )}}{512} + \frac {63 \sqrt {- \frac {1}{a b^{11}}} \log {\left (a b^{5} \sqrt {- \frac {1}{a b^{11}}} + x \right )}}{512} + \frac {- 315 a^{4} x - 1470 a^{3} b x^{3} - 2688 a^{2} b^{2} x^{5} - 2370 a b^{3} x^{7} - 965 b^{4} x^{9}}{1280 a^{5} b^{5} + 6400 a^{4} b^{6} x^{2} + 12800 a^{3} b^{7} x^{4} + 12800 a^{2} b^{8} x^{6} + 6400 a b^{9} x^{8} + 1280 b^{10} x^{10}} \]
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Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.03 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {965 \, b^{4} x^{9} + 2370 \, a b^{3} x^{7} + 2688 \, a^{2} b^{2} x^{5} + 1470 \, a^{3} b x^{3} + 315 \, a^{4} x}{1280 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} + \frac {63 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {63 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{5}} - \frac {965 \, b^{4} x^{9} + 2370 \, a b^{3} x^{7} + 2688 \, a^{2} b^{2} x^{5} + 1470 \, a^{3} b x^{3} + 315 \, a^{4} x}{1280 \, {\left (b x^{2} + a\right )}^{5} b^{5}} \]
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Time = 14.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01 \[ \int \frac {x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {63\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,\sqrt {a}\,b^{11/2}}-\frac {\frac {193\,x^9}{256\,b}+\frac {237\,a\,x^7}{128\,b^2}+\frac {63\,a^4\,x}{256\,b^5}+\frac {21\,a^2\,x^5}{10\,b^3}+\frac {147\,a^3\,x^3}{128\,b^4}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}} \]
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