Integrand size = 13, antiderivative size = 65 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^4} \, dx=-\frac {x}{4 a \left (b+a x^2\right )^2}+\frac {x}{8 a b \left (b+a x^2\right )}+\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {269, 294, 205, 211} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2}}+\frac {x}{8 a b \left (a x^2+b\right )}-\frac {x}{4 a \left (a x^2+b\right )^2} \]
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Rule 205
Rule 211
Rule 269
Rule 294
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{\left (b+a x^2\right )^3} \, dx \\ & = -\frac {x}{4 a \left (b+a x^2\right )^2}+\frac {\int \frac {1}{\left (b+a x^2\right )^2} \, dx}{4 a} \\ & = -\frac {x}{4 a \left (b+a x^2\right )^2}+\frac {x}{8 a b \left (b+a x^2\right )}+\frac {\int \frac {1}{b+a x^2} \, dx}{8 a b} \\ & = -\frac {x}{4 a \left (b+a x^2\right )^2}+\frac {x}{8 a b \left (b+a x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^4} \, dx=\frac {\frac {\sqrt {a} \sqrt {b} x \left (-b+a x^2\right )}{\left (b+a x^2\right )^2}+\arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{3/2} b^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\frac {x^{3}}{8 b}-\frac {x}{8 a}}{\left (a \,x^{2}+b \right )^{2}}+\frac {\arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 b a \sqrt {a b}}\) | \(49\) |
risch | \(\frac {\frac {x^{3}}{8 b}-\frac {x}{8 a}}{\left (a \,x^{2}+b \right )^{2}}-\frac {\ln \left (a x +\sqrt {-a b}\right )}{16 \sqrt {-a b}\, b a}+\frac {\ln \left (-a x +\sqrt {-a b}\right )}{16 \sqrt {-a b}\, b a}\) | \(78\) |
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none
Time = 0.28 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.92 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^4} \, dx=\left [\frac {2 \, a^{2} b x^{3} - 2 \, a b^{2} x - {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {-a b} \log \left (\frac {a x^{2} - 2 \, \sqrt {-a b} x - b}{a x^{2} + b}\right )}{16 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{2} b^{4}\right )}}, \frac {a^{2} b x^{3} - a b^{2} x + {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{b}\right )}{8 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{2} b^{4}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (51) = 102\).
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^4} \, dx=- \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \log {\left (- a b^{2} \sqrt {- \frac {1}{a^{3} b^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \log {\left (a b^{2} \sqrt {- \frac {1}{a^{3} b^{3}}} + x \right )}}{16} + \frac {a x^{3} - b x}{8 a^{3} b x^{4} + 16 a^{2} b^{2} x^{2} + 8 a b^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^4} \, dx=\frac {a x^{3} - b x}{8 \, {\left (a^{3} b x^{4} + 2 \, a^{2} b^{2} x^{2} + a b^{3}\right )}} + \frac {\arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b} \]
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Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^4} \, dx=\frac {\arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b} + \frac {a x^{3} - b x}{8 \, {\left (a x^{2} + b\right )}^{2} a b} \]
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Time = 5.47 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x^4} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{8\,a^{3/2}\,b^{3/2}}-\frac {\frac {x}{8\,a}-\frac {x^3}{8\,b}}{a^2\,x^4+2\,a\,b\,x^2+b^2} \]
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