Optimal. Leaf size=108 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) (b e (3 d e-c f)-a f (c f+d e))}{2 e^{3/2} f^{5/2}}-\frac{x \left (a+b x^2\right ) (d e-c f)}{2 e f \left (e+f x^2\right )}+\frac{b x (3 d e-c f)}{2 e f^2} \]
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Rubi [A] time = 0.0849137, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {526, 388, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) (b e (3 d e-c f)-a f (c f+d e))}{2 e^{3/2} f^{5/2}}-\frac{x \left (a+b x^2\right ) (d e-c f)}{2 e f \left (e+f x^2\right )}+\frac{b x (3 d e-c f)}{2 e f^2} \]
Antiderivative was successfully verified.
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Rule 526
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (c+d x^2\right )}{\left (e+f x^2\right )^2} \, dx &=-\frac{(d e-c f) x \left (a+b x^2\right )}{2 e f \left (e+f x^2\right )}-\frac{\int \frac{-a (d e+c f)-b (3 d e-c f) x^2}{e+f x^2} \, dx}{2 e f}\\ &=\frac{b (3 d e-c f) x}{2 e f^2}-\frac{(d e-c f) x \left (a+b x^2\right )}{2 e f \left (e+f x^2\right )}-\frac{(b e (3 d e-c f)-a f (d e+c f)) \int \frac{1}{e+f x^2} \, dx}{2 e f^2}\\ &=\frac{b (3 d e-c f) x}{2 e f^2}-\frac{(d e-c f) x \left (a+b x^2\right )}{2 e f \left (e+f x^2\right )}-\frac{(b e (3 d e-c f)-a f (d e+c f)) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{2 e^{3/2} f^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0666011, size = 95, normalized size = 0.88 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) (b e (3 d e-c f)-a f (c f+d e))}{2 e^{3/2} f^{5/2}}+\frac{x (b e-a f) (d e-c f)}{2 e f^2 \left (e+f x^2\right )}+\frac{b d x}{f^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 163, normalized size = 1.5 \begin{align*}{\frac{bdx}{{f}^{2}}}+{\frac{axc}{2\,e \left ( f{x}^{2}+e \right ) }}-{\frac{axd}{2\,f \left ( f{x}^{2}+e \right ) }}-{\frac{bcx}{2\,f \left ( f{x}^{2}+e \right ) }}+{\frac{bxed}{2\,{f}^{2} \left ( f{x}^{2}+e \right ) }}+{\frac{ac}{2\,e}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{ad}{2\,f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{bc}{2\,f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{3\,bde}{2\,{f}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46482, size = 660, normalized size = 6.11 \begin{align*} \left [\frac{4 \, b d e^{2} f^{2} x^{3} +{\left (3 \, b d e^{3} - a c e f^{2} -{\left (b c + a d\right )} e^{2} f +{\left (3 \, b d e^{2} f - a c f^{3} -{\left (b c + a d\right )} e 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^{3} f + a c e f^{3} -{\left (b c + a d\right )} e^{2} f^{2}\right )} x}{4 \,{\left (e^{2} f^{4} x^{2} + e^{3} f^{3}\right )}}, \frac{2 \, b d e^{2} f^{2} x^{3} -{\left (3 \, b d e^{3} - a c e f^{2} -{\left (b c + a d\right )} e^{2} f +{\left (3 \, b d e^{2} f - a c f^{3} -{\left (b c + a d\right )} e f^{2}\right )} x^{2}\right )} \sqrt{e f} \arctan \left (\frac{\sqrt{e f} x}{e}\right ) +{\left (3 \, b d e^{3} f + a c e f^{3} -{\left (b c + a d\right )} e^{2} f^{2}\right )} x}{2 \,{\left (e^{2} f^{4} x^{2} + e^{3} f^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.58296, size = 190, normalized size = 1.76 \begin{align*} \frac{b d x}{f^{2}} + \frac{x \left (a c f^{2} - a d e f - b c e f + b d e^{2}\right )}{2 e^{2} f^{2} + 2 e f^{3} x^{2}} - \frac{\sqrt{- \frac{1}{e^{3} f^{5}}} \left (a c f^{2} + a d e f + b c e f - 3 b d e^{2}\right ) \log{\left (- e^{2} f^{2} \sqrt{- \frac{1}{e^{3} f^{5}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{e^{3} f^{5}}} \left (a c f^{2} + a d e f + b c e f - 3 b d e^{2}\right ) \log{\left (e^{2} f^{2} \sqrt{- \frac{1}{e^{3} f^{5}}} + x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14734, size = 128, normalized size = 1.19 \begin{align*} \frac{b d x}{f^{2}} + \frac{{\left (a c f^{2} + b c f e + a d f e - 3 \, b d e^{2}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{3}{2}\right )}}{2 \, f^{\frac{5}{2}}} + \frac{{\left (a c f^{2} x - b c f x e - a d f x e + b d x e^{2}\right )} e^{\left (-1\right )}}{2 \,{\left (f x^{2} + e\right )} f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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