Integrand size = 24, antiderivative size = 130 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=-\frac {(d e-c f) x \left (a+b x^2\right )}{4 e f \left (e+f x^2\right )^2}-\frac {(b e (3 d e+c f)-a f (d e+3 c f)) x}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {(b e (3 d e+c f)+a f (d e+3 c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 130, 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 = {540, 393, 211} \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (a f (3 c f+d e)+b e (c f+3 d e))}{8 e^{5/2} f^{5/2}}-\frac {x (b e (c f+3 d e)-a f (3 c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}-\frac {x \left (a+b x^2\right ) (d e-c f)}{4 e f \left (e+f x^2\right )^2} \]
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Rule 211
Rule 393
Rule 540
Rubi steps \begin{align*} \text {integral}& = -\frac {(d e-c f) x \left (a+b x^2\right )}{4 e f \left (e+f x^2\right )^2}-\frac {\int \frac {-a (d e+3 c f)-b (3 d e+c f) x^2}{\left (e+f x^2\right )^2} \, dx}{4 e f} \\ & = -\frac {(d e-c f) x \left (a+b x^2\right )}{4 e f \left (e+f x^2\right )^2}-\frac {(b e (3 d e+c f)-a f (d e+3 c f)) x}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {(b e (3 d e+c f)+a f (d e+3 c f)) \int \frac {1}{e+f x^2} \, dx}{8 e^2 f^2} \\ & = -\frac {(d e-c f) x \left (a+b x^2\right )}{4 e f \left (e+f x^2\right )^2}-\frac {(b e (3 d e+c f)-a f (d e+3 c f)) x}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {(b e (3 d e+c f)+a f (d e+3 c f)) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{5/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=\frac {(b e-a f) (d e-c f) x}{4 e f^2 \left (e+f x^2\right )^2}+\frac {(b e (-5 d e+c f)+a f (d e+3 c f)) x}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {(b e (3 d e+c f)+a f (d e+3 c f)) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{5/2}} \]
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Time = 3.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\frac {\left (3 a c \,f^{2}+a d e f +b c e f -5 b d \,e^{2}\right ) x^{3}}{8 e^{2} f}+\frac {\left (5 a c \,f^{2}-a d e f -b c e f -3 b d \,e^{2}\right ) x}{8 e \,f^{2}}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a c \,f^{2}+a d e f +b c e f +3 b d \,e^{2}\right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 e^{2} f^{2} \sqrt {e f}}\) | \(132\) |
risch | \(\frac {\frac {\left (3 a c \,f^{2}+a d e f +b c e f -5 b d \,e^{2}\right ) x^{3}}{8 e^{2} f}+\frac {\left (5 a c \,f^{2}-a d e f -b c e f -3 b d \,e^{2}\right ) x}{8 e \,f^{2}}}{\left (f \,x^{2}+e \right )^{2}}-\frac {3 \ln \left (f x +\sqrt {-e f}\right ) a c}{16 \sqrt {-e f}\, e^{2}}-\frac {\ln \left (f x +\sqrt {-e f}\right ) a d}{16 \sqrt {-e f}\, f e}-\frac {\ln \left (f x +\sqrt {-e f}\right ) b c}{16 \sqrt {-e f}\, f e}-\frac {3 \ln \left (f x +\sqrt {-e f}\right ) b d}{16 \sqrt {-e f}\, f^{2}}+\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) a c}{16 \sqrt {-e f}\, e^{2}}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) a d}{16 \sqrt {-e f}\, f e}+\frac {\ln \left (-f x +\sqrt {-e f}\right ) b c}{16 \sqrt {-e f}\, f e}+\frac {3 \ln \left (-f x +\sqrt {-e f}\right ) b d}{16 \sqrt {-e f}\, f^{2}}\) | \(293\) |
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Time = 0.31 (sec) , antiderivative size = 471, normalized size of antiderivative = 3.62 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=\left [-\frac {2 \, {\left (5 \, b d e^{3} f^{2} - 3 \, a c e f^{4} - {\left (b c + a d\right )} e^{2} f^{3}\right )} x^{3} + {\left (3 \, b d e^{4} + 3 \, a c e^{2} f^{2} + {\left (b c + a d\right )} e^{3} f + {\left (3 \, b d e^{2} f^{2} + 3 \, a c f^{4} + {\left (b c + a d\right )} e f^{3}\right )} x^{4} + 2 \, {\left (3 \, b d e^{3} f + 3 \, a c e f^{3} + {\left (b c + a d\right )} e^{2} f^{2}\right )} x^{2}\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right ) + 2 \, {\left (3 \, b d e^{4} f - 5 \, a c e^{2} f^{3} + {\left (b c + a d\right )} e^{3} f^{2}\right )} x}{16 \, {\left (e^{3} f^{5} x^{4} + 2 \, e^{4} f^{4} x^{2} + e^{5} f^{3}\right )}}, -\frac {{\left (5 \, b d e^{3} f^{2} - 3 \, a c e f^{4} - {\left (b c + a d\right )} e^{2} f^{3}\right )} x^{3} - {\left (3 \, b d e^{4} + 3 \, a c e^{2} f^{2} + {\left (b c + a d\right )} e^{3} f + {\left (3 \, b d e^{2} f^{2} + 3 \, a c f^{4} + {\left (b c + a d\right )} e f^{3}\right )} x^{4} + 2 \, {\left (3 \, b d e^{3} f + 3 \, a c e f^{3} + {\left (b c + a d\right )} e^{2} f^{2}\right )} x^{2}\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right ) + {\left (3 \, b d e^{4} f - 5 \, a c e^{2} f^{3} + {\left (b c + a d\right )} e^{3} f^{2}\right )} x}{8 \, {\left (e^{3} f^{5} x^{4} + 2 \, e^{4} f^{4} x^{2} + e^{5} f^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (122) = 244\).
Time = 1.39 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=- \frac {\sqrt {- \frac {1}{e^{5} f^{5}}} \cdot \left (3 a c f^{2} + a d e f + b c e f + 3 b d e^{2}\right ) \log {\left (- e^{3} f^{2} \sqrt {- \frac {1}{e^{5} f^{5}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{e^{5} f^{5}}} \cdot \left (3 a c f^{2} + a d e f + b c e f + 3 b d e^{2}\right ) \log {\left (e^{3} f^{2} \sqrt {- \frac {1}{e^{5} f^{5}}} + x \right )}}{16} + \frac {x^{3} \cdot \left (3 a c f^{3} + a d e f^{2} + b c e f^{2} - 5 b d e^{2} f\right ) + x \left (5 a c e f^{2} - a d e^{2} f - b c e^{2} f - 3 b d e^{3}\right )}{8 e^{4} f^{2} + 16 e^{3} f^{3} x^{2} + 8 e^{2} f^{4} x^{4}} \]
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Exception generated. \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=\frac {{\left (3 \, b d e^{2} + b c e f + a d e f + 3 \, a c f^{2}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, \sqrt {e f} e^{2} f^{2}} - \frac {5 \, b d e^{2} f x^{3} - b c e f^{2} x^{3} - a d e f^{2} x^{3} - 3 \, a c f^{3} x^{3} + 3 \, b d e^{3} x + b c e^{2} f x + a d e^{2} f x - 5 \, a c e f^{2} x}{8 \, {\left (f x^{2} + e\right )}^{2} e^{2} f^{2}} \]
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Time = 5.46 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (3\,a\,c\,f^2+3\,b\,d\,e^2+a\,d\,e\,f+b\,c\,e\,f\right )}{8\,e^{5/2}\,f^{5/2}}-\frac {\frac {x\,\left (3\,b\,d\,e^2-5\,a\,c\,f^2+a\,d\,e\,f+b\,c\,e\,f\right )}{8\,e\,f^2}-\frac {x^3\,\left (3\,a\,c\,f^2-5\,b\,d\,e^2+a\,d\,e\,f+b\,c\,e\,f\right )}{8\,e^2\,f}}{e^2+2\,e\,f\,x^2+f^2\,x^4} \]
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