Integrand size = 24, antiderivative size = 81 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{e+f x^2} \, dx=-\frac {(3 b d e-3 b c f-2 a d f) x}{3 f^2}+\frac {d x \left (a+b x^2\right )}{3 f}+\frac {(b e-a f) (d e-c f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {542, 396, 211} \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{e+f x^2} \, dx=\frac {(b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f)}{\sqrt {e} f^{5/2}}-\frac {x (-2 a d f-3 b c f+3 b d e)}{3 f^2}+\frac {d x \left (a+b x^2\right )}{3 f} \]
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Rule 211
Rule 396
Rule 542
Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (a+b x^2\right )}{3 f}+\frac {\int \frac {-a (d e-3 c f)-(3 b d e-3 b c f-2 a d f) x^2}{e+f x^2} \, dx}{3 f} \\ & = -\frac {(3 b d e-3 b c f-2 a d f) x}{3 f^2}+\frac {d x \left (a+b x^2\right )}{3 f}+\frac {((b e-a f) (d e-c f)) \int \frac {1}{e+f x^2} \, dx}{f^2} \\ & = -\frac {(3 b d e-3 b c f-2 a d f) x}{3 f^2}+\frac {d x \left (a+b x^2\right )}{3 f}+\frac {(b e-a f) (d e-c f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{5/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{e+f x^2} \, dx=\frac {(-b d e+b c f+a d f) x}{f^2}+\frac {b d x^3}{3 f}+\frac {(b e-a f) (d e-c f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} f^{5/2}} \]
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Time = 3.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\frac {1}{3} b d f \,x^{3}+a d f x +b c f x -b d e x}{f^{2}}+\frac {\left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{f^{2} \sqrt {e f}}\) | \(74\) |
risch | \(\frac {b d \,x^{3}}{3 f}+\frac {a d x}{f}+\frac {b c x}{f}-\frac {b d e x}{f^{2}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) a c}{2 \sqrt {-e f}}+\frac {\ln \left (f x +\sqrt {-e f}\right ) a d e}{2 f \sqrt {-e f}}+\frac {\ln \left (f x +\sqrt {-e f}\right ) b c e}{2 f \sqrt {-e f}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) b d \,e^{2}}{2 f^{2} \sqrt {-e f}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) a c}{2 \sqrt {-e f}}-\frac {\ln \left (-f x +\sqrt {-e f}\right ) a d e}{2 f \sqrt {-e f}}-\frac {\ln \left (-f x +\sqrt {-e f}\right ) b c e}{2 f \sqrt {-e f}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) b d \,e^{2}}{2 f^{2} \sqrt {-e f}}\) | \(235\) |
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Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.36 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{e+f x^2} \, dx=\left [\frac {2 \, b d e f^{2} x^{3} - 3 \, {\left (b d e^{2} + a c f^{2} - {\left (b c + a d\right )} e f\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right ) - 6 \, {\left (b d e^{2} f - {\left (b c + a d\right )} e f^{2}\right )} x}{6 \, e f^{3}}, \frac {b d e f^{2} x^{3} + 3 \, {\left (b d e^{2} + a c f^{2} - {\left (b c + a d\right )} e f\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) - 3 \, {\left (b d e^{2} f - {\left (b c + a d\right )} e f^{2}\right )} x}{3 \, e f^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (73) = 146\).
Time = 0.31 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.54 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{e+f x^2} \, dx=\frac {b d x^{3}}{3 f} + x \left (\frac {a d}{f} + \frac {b c}{f} - \frac {b d e}{f^{2}}\right ) - \frac {\sqrt {- \frac {1}{e f^{5}}} \left (a f - b e\right ) \left (c f - d e\right ) \log {\left (- \frac {e f^{2} \sqrt {- \frac {1}{e f^{5}}} \left (a f - b e\right ) \left (c f - d e\right )}{a c f^{2} - a d e f - b c e f + b d e^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{e f^{5}}} \left (a f - b e\right ) \left (c f - d e\right ) \log {\left (\frac {e f^{2} \sqrt {- \frac {1}{e f^{5}}} \left (a f - b e\right ) \left (c f - d e\right )}{a c f^{2} - a d e f - b c e f + b d e^{2}} + x \right )}}{2} \]
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Exception generated. \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{e+f x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{e+f x^2} \, dx=\frac {{\left (b d e^{2} - b c e f - a d e f + a c f^{2}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{\sqrt {e f} f^{2}} + \frac {b d f^{2} x^{3} - 3 \, b d e f x + 3 \, b c f^{2} x + 3 \, a d f^{2} x}{3 \, f^{3}} \]
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Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{e+f x^2} \, dx=x\,\left (\frac {a\,d+b\,c}{f}-\frac {b\,d\,e}{f^2}\right )+\frac {b\,d\,x^3}{3\,f}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,\left (a\,f-b\,e\right )\,\left (c\,f-d\,e\right )}{\sqrt {e}\,\left (a\,c\,f^2+b\,d\,e^2-a\,d\,e\,f-b\,c\,e\,f\right )}\right )\,\left (a\,f-b\,e\right )\,\left (c\,f-d\,e\right )}{\sqrt {e}\,f^{5/2}} \]
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