Integrand size = 24, antiderivative size = 125 \[ \int \frac {x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {x}{10 b \left (a+b x^2\right )^5}+\frac {x}{80 a b \left (a+b x^2\right )^4}+\frac {7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac {7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac {7 x}{256 a^4 b \left (a+b x^2\right )}+\frac {7 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{9/2} b^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 294, 205, 211} \[ \int \frac {x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {7 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{9/2} b^{3/2}}+\frac {7 x}{256 a^4 b \left (a+b x^2\right )}+\frac {7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac {7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac {x}{80 a b \left (a+b x^2\right )^4}-\frac {x}{10 b \left (a+b x^2\right )^5} \]
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Rule 28
Rule 205
Rule 211
Rule 294
Rubi steps \begin{align*} \text {integral}& = b^6 \int \frac {x^2}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = -\frac {x}{10 b \left (a+b x^2\right )^5}+\frac {1}{10} b^4 \int \frac {1}{\left (a b+b^2 x^2\right )^5} \, dx \\ & = -\frac {x}{10 b \left (a+b x^2\right )^5}+\frac {x}{80 a b \left (a+b x^2\right )^4}+\frac {\left (7 b^3\right ) \int \frac {1}{\left (a b+b^2 x^2\right )^4} \, dx}{80 a} \\ & = -\frac {x}{10 b \left (a+b x^2\right )^5}+\frac {x}{80 a b \left (a+b x^2\right )^4}+\frac {7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac {\left (7 b^2\right ) \int \frac {1}{\left (a b+b^2 x^2\right )^3} \, dx}{96 a^2} \\ & = -\frac {x}{10 b \left (a+b x^2\right )^5}+\frac {x}{80 a b \left (a+b x^2\right )^4}+\frac {7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac {7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac {(7 b) \int \frac {1}{\left (a b+b^2 x^2\right )^2} \, dx}{128 a^3} \\ & = -\frac {x}{10 b \left (a+b x^2\right )^5}+\frac {x}{80 a b \left (a+b x^2\right )^4}+\frac {7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac {7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac {7 x}{256 a^4 b \left (a+b x^2\right )}+\frac {7 \int \frac {1}{a b+b^2 x^2} \, dx}{256 a^4} \\ & = -\frac {x}{10 b \left (a+b x^2\right )^5}+\frac {x}{80 a b \left (a+b x^2\right )^4}+\frac {7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac {7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac {7 x}{256 a^4 b \left (a+b x^2\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{9/2} b^{3/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \frac {x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {-105 a^4 x+790 a^3 b x^3+896 a^2 b^2 x^5+490 a b^3 x^7+105 b^4 x^9}{3840 a^4 b \left (a+b x^2\right )^5}+\frac {7 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 a^{9/2} b^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {\frac {7 b^{3} x^{9}}{256 a^{4}}+\frac {49 b^{2} x^{7}}{384 a^{3}}+\frac {7 b \,x^{5}}{30 a^{2}}+\frac {79 x^{3}}{384 a}-\frac {7 x}{256 b}}{\left (b \,x^{2}+a \right )^{5}}+\frac {7 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 b \,a^{4} \sqrt {a b}}\) | \(80\) |
risch | \(\frac {\frac {7 b^{3} x^{9}}{256 a^{4}}+\frac {49 b^{2} x^{7}}{384 a^{3}}+\frac {7 b \,x^{5}}{30 a^{2}}+\frac {79 x^{3}}{384 a}-\frac {7 x}{256 b}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}-\frac {7 \ln \left (b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b \,a^{4}}+\frac {7 \ln \left (-b x +\sqrt {-a b}\right )}{512 \sqrt {-a b}\, b \,a^{4}}\) | \(129\) |
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Time = 0.27 (sec) , antiderivative size = 390, normalized size of antiderivative = 3.12 \[ \int \frac {x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [\frac {210 \, a b^{5} x^{9} + 980 \, a^{2} b^{4} x^{7} + 1792 \, a^{3} b^{3} x^{5} + 1580 \, a^{4} b^{2} x^{3} - 210 \, a^{5} b x - 105 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{7680 \, {\left (a^{5} b^{7} x^{10} + 5 \, a^{6} b^{6} x^{8} + 10 \, a^{7} b^{5} x^{6} + 10 \, a^{8} b^{4} x^{4} + 5 \, a^{9} b^{3} x^{2} + a^{10} b^{2}\right )}}, \frac {105 \, a b^{5} x^{9} + 490 \, a^{2} b^{4} x^{7} + 896 \, a^{3} b^{3} x^{5} + 790 \, a^{4} b^{2} x^{3} - 105 \, a^{5} b x + 105 \, {\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{3840 \, {\left (a^{5} b^{7} x^{10} + 5 \, a^{6} b^{6} x^{8} + 10 \, a^{7} b^{5} x^{6} + 10 \, a^{8} b^{4} x^{4} + 5 \, a^{9} b^{3} x^{2} + a^{10} b^{2}\right )}}\right ] \]
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Time = 0.28 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.52 \[ \int \frac {x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=- \frac {7 \sqrt {- \frac {1}{a^{9} b^{3}}} \log {\left (- a^{5} b \sqrt {- \frac {1}{a^{9} b^{3}}} + x \right )}}{512} + \frac {7 \sqrt {- \frac {1}{a^{9} b^{3}}} \log {\left (a^{5} b \sqrt {- \frac {1}{a^{9} b^{3}}} + x \right )}}{512} + \frac {- 105 a^{4} x + 790 a^{3} b x^{3} + 896 a^{2} b^{2} x^{5} + 490 a b^{3} x^{7} + 105 b^{4} x^{9}}{3840 a^{9} b + 19200 a^{8} b^{2} x^{2} + 38400 a^{7} b^{3} x^{4} + 38400 a^{6} b^{4} x^{6} + 19200 a^{5} b^{5} x^{8} + 3840 a^{4} b^{6} x^{10}} \]
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Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {105 \, b^{4} x^{9} + 490 \, a b^{3} x^{7} + 896 \, a^{2} b^{2} x^{5} + 790 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )}} + \frac {7 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{4} b} \]
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Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.67 \[ \int \frac {x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {7 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{4} b} + \frac {105 \, b^{4} x^{9} + 490 \, a b^{3} x^{7} + 896 \, a^{2} b^{2} x^{5} + 790 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \, {\left (b x^{2} + a\right )}^{5} a^{4} b} \]
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Time = 14.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {79\,x^3}{384\,a}-\frac {7\,x}{256\,b}+\frac {7\,b\,x^5}{30\,a^2}+\frac {49\,b^2\,x^7}{384\,a^3}+\frac {7\,b^3\,x^9}{256\,a^4}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}}+\frac {7\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{9/2}\,b^{3/2}} \]
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