Integrand size = 22, antiderivative size = 83 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx=\frac {c x}{e^2}+\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{2 d \left (d+e x^2\right )}-\frac {\left (3 c d^2-e (b d+a e)\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1171, 396, 211} \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (3 c d^2-e (a e+b d)\right )}{2 d^{3/2} e^{5/2}}+\frac {x \left (a e^2-b d e+c d^2\right )}{2 d e^2 \left (d+e x^2\right )}+\frac {c x}{e^2} \]
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Rule 211
Rule 396
Rule 1171
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c d^2-b d e+a e^2\right ) x}{2 d e^2 \left (d+e x^2\right )}-\frac {\int \frac {\frac {c d^2-e (b d+a e)}{e^2}-\frac {2 c d x^2}{e}}{d+e x^2} \, dx}{2 d} \\ & = \frac {c x}{e^2}+\frac {\left (c d^2-b d e+a e^2\right ) x}{2 d e^2 \left (d+e x^2\right )}-\frac {\left (3 c d^2-e (b d+a e)\right ) \int \frac {1}{d+e x^2} \, dx}{2 d e^2} \\ & = \frac {c x}{e^2}+\frac {\left (c d^2-b d e+a e^2\right ) x}{2 d e^2 \left (d+e x^2\right )}-\frac {\left (3 c d^2-e (b d+a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{5/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.06 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx=\frac {c x}{e^2}+\frac {\left (c d^2-b d e+a e^2\right ) x}{2 d e^2 \left (d+e x^2\right )}-\frac {\left (3 c d^2-b d e-a e^2\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{5/2}} \]
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Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {c x}{e^{2}}+\frac {\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) x}{2 d \left (e \,x^{2}+d \right )}+\frac {\left (a \,e^{2}+b d e -3 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \sqrt {e d}}}{e^{2}}\) | \(79\) |
risch | \(\frac {c x}{e^{2}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) x}{2 d \,e^{2} \left (e \,x^{2}+d \right )}-\frac {\ln \left (e x +\sqrt {-e d}\right ) a}{4 \sqrt {-e d}\, d}-\frac {\ln \left (e x +\sqrt {-e d}\right ) b}{4 e \sqrt {-e d}}+\frac {3 d \ln \left (e x +\sqrt {-e d}\right ) c}{4 e^{2} \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) a}{4 \sqrt {-e d}\, d}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) b}{4 e \sqrt {-e d}}-\frac {3 d \ln \left (-e x +\sqrt {-e d}\right ) c}{4 e^{2} \sqrt {-e d}}\) | \(185\) |
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Time = 0.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 3.23 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx=\left [\frac {4 \, c d^{2} e^{2} x^{3} + {\left (3 \, c d^{3} - b d^{2} e - a d e^{2} + {\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 2 \, {\left (3 \, c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} x}{4 \, {\left (d^{2} e^{4} x^{2} + d^{3} e^{3}\right )}}, \frac {2 \, c d^{2} e^{2} x^{3} - {\left (3 \, c d^{3} - b d^{2} e - a d e^{2} + {\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + {\left (3 \, c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} x}{2 \, {\left (d^{2} e^{4} x^{2} + d^{3} e^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (75) = 150\).
Time = 0.39 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.84 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx=\frac {c x}{e^{2}} + \frac {x \left (a e^{2} - b d e + c d^{2}\right )}{2 d^{2} e^{2} + 2 d e^{3} x^{2}} - \frac {\sqrt {- \frac {1}{d^{3} e^{5}}} \left (a e^{2} + b d e - 3 c d^{2}\right ) \log {\left (- d^{2} e^{2} \sqrt {- \frac {1}{d^{3} e^{5}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{d^{3} e^{5}}} \left (a e^{2} + b d e - 3 c d^{2}\right ) \log {\left (d^{2} e^{2} \sqrt {- \frac {1}{d^{3} e^{5}}} + x \right )}}{4} \]
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Exception generated. \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx=\frac {c x}{e^{2}} - \frac {{\left (3 \, c d^{2} - b d e - a e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d e^{2}} + \frac {c d^{2} x - b d e x + a e^{2} x}{2 \, {\left (e x^{2} + d\right )} d e^{2}} \]
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Time = 7.56 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^2} \, dx=\frac {c\,x}{e^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (-3\,c\,d^2+b\,d\,e+a\,e^2\right )}{2\,d^{3/2}\,e^{5/2}}+\frac {x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{2\,d\,\left (e^3\,x^2+d\,e^2\right )} \]
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