Optimal. Leaf size=142 \[ -\frac{x \left (5 a d f (3 d e-5 c f)-b \left (8 c^2 f^2-25 c d e f+15 d^2 e^2\right )\right )}{15 f^3}-\frac{x \left (c+d x^2\right ) (-5 a d f-4 b c f+5 b d e)}{15 f^2}-\frac{(b e-a f) (d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{7/2}}+\frac{b x \left (c+d x^2\right )^2}{5 f} \]
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Rubi [A] time = 0.208257, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {528, 388, 205} \[ -\frac{x \left (5 a d f (3 d e-5 c f)-b \left (8 c^2 f^2-25 c d e f+15 d^2 e^2\right )\right )}{15 f^3}-\frac{x \left (c+d x^2\right ) (-5 a d f-4 b c f+5 b d e)}{15 f^2}-\frac{(b e-a f) (d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{7/2}}+\frac{b x \left (c+d x^2\right )^2}{5 f} \]
Antiderivative was successfully verified.
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Rule 528
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (c+d x^2\right )^2}{e+f x^2} \, dx &=\frac{b x \left (c+d x^2\right )^2}{5 f}+\frac{\int \frac{\left (c+d x^2\right ) \left (-c (b e-5 a f)+(-5 b d e+4 b c f+5 a d f) x^2\right )}{e+f x^2} \, dx}{5 f}\\ &=-\frac{(5 b d e-4 b c f-5 a d f) x \left (c+d x^2\right )}{15 f^2}+\frac{b x \left (c+d x^2\right )^2}{5 f}+\frac{\int \frac{c (b e (5 d e-7 c f)-5 a f (d e-3 c f))-\left (5 a d f (3 d e-5 c f)-b \left (15 d^2 e^2-25 c d e f+8 c^2 f^2\right )\right ) x^2}{e+f x^2} \, dx}{15 f^2}\\ &=-\frac{\left (5 a d f (3 d e-5 c f)-b \left (15 d^2 e^2-25 c d e f+8 c^2 f^2\right )\right ) x}{15 f^3}-\frac{(5 b d e-4 b c f-5 a d f) x \left (c+d x^2\right )}{15 f^2}+\frac{b x \left (c+d x^2\right )^2}{5 f}-\frac{\left ((b e-a f) (d e-c f)^2\right ) \int \frac{1}{e+f x^2} \, dx}{f^3}\\ &=-\frac{\left (5 a d f (3 d e-5 c f)-b \left (15 d^2 e^2-25 c d e f+8 c^2 f^2\right )\right ) x}{15 f^3}-\frac{(5 b d e-4 b c f-5 a d f) x \left (c+d x^2\right )}{15 f^2}+\frac{b x \left (c+d x^2\right )^2}{5 f}-\frac{(b e-a f) (d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0636626, size = 115, normalized size = 0.81 \[ \frac{d x^3 (a d f+2 b c f-b d e)}{3 f^2}+\frac{x \left (a d f (2 c f-d e)+b (d e-c f)^2\right )}{f^3}-\frac{(b e-a f) (d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{\sqrt{e} f^{7/2}}+\frac{b d^2 x^5}{5 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 243, normalized size = 1.7 \begin{align*}{\frac{b{d}^{2}{x}^{5}}{5\,f}}+{\frac{{x}^{3}a{d}^{2}}{3\,f}}+{\frac{2\,{x}^{3}bcd}{3\,f}}-{\frac{{x}^{3}b{d}^{2}e}{3\,{f}^{2}}}+2\,{\frac{acdx}{f}}-{\frac{a{d}^{2}ex}{{f}^{2}}}+{\frac{b{c}^{2}x}{f}}-2\,{\frac{bcdex}{{f}^{2}}}+{\frac{b{d}^{2}{e}^{2}x}{{f}^{3}}}+{a{c}^{2}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-2\,{\frac{acde}{f\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }+{\frac{a{d}^{2}{e}^{2}}{{f}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{b{c}^{2}e}{f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+2\,{\frac{bcd{e}^{2}}{{f}^{2}\sqrt{ef}}\arctan \left ({\frac{fx}{\sqrt{ef}}} \right ) }-{\frac{b{d}^{2}{e}^{3}}{{f}^{3}}\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 = 1.48085, size = 775, normalized size = 5.46 \begin{align*} \left [\frac{6 \, b d^{2} e f^{3} x^{5} - 10 \,{\left (b d^{2} e^{2} f^{2} -{\left (2 \, b c d + a d^{2}\right )} e f^{3}\right )} x^{3} + 15 \,{\left (b d^{2} e^{3} - a c^{2} f^{3} -{\left (2 \, b c d + a d^{2}\right )} e^{2} f +{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} \sqrt{-e f} \log \left (\frac{f x^{2} - 2 \, \sqrt{-e f} x - e}{f x^{2} + e}\right ) + 30 \,{\left (b d^{2} e^{3} f -{\left (2 \, b c d + a d^{2}\right )} e^{2} f^{2} +{\left (b c^{2} + 2 \, a c d\right )} e f^{3}\right )} x}{30 \, e f^{4}}, \frac{3 \, b d^{2} e f^{3} x^{5} - 5 \,{\left (b d^{2} e^{2} f^{2} -{\left (2 \, b c d + a d^{2}\right )} e f^{3}\right )} x^{3} - 15 \,{\left (b d^{2} e^{3} - a c^{2} f^{3} -{\left (2 \, b c d + a d^{2}\right )} e^{2} f +{\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} \sqrt{e f} \arctan \left (\frac{\sqrt{e f} x}{e}\right ) + 15 \,{\left (b d^{2} e^{3} f -{\left (2 \, b c d + a d^{2}\right )} e^{2} f^{2} +{\left (b c^{2} + 2 \, a c d\right )} e f^{3}\right )} x}{15 \, e f^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.33411, size = 343, normalized size = 2.42 \begin{align*} \frac{b d^{2} x^{5}}{5 f} - \frac{\sqrt{- \frac{1}{e f^{7}}} \left (a f - b e\right ) \left (c f - d e\right )^{2} \log{\left (- \frac{e f^{3} \sqrt{- \frac{1}{e f^{7}}} \left (a f - b e\right ) \left (c f - d e\right )^{2}}{a c^{2} f^{3} - 2 a c d e f^{2} + a d^{2} e^{2} f - b c^{2} e f^{2} + 2 b c d e^{2} f - b d^{2} e^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{e f^{7}}} \left (a f - b e\right ) \left (c f - d e\right )^{2} \log{\left (\frac{e f^{3} \sqrt{- \frac{1}{e f^{7}}} \left (a f - b e\right ) \left (c f - d e\right )^{2}}{a c^{2} f^{3} - 2 a c d e f^{2} + a d^{2} e^{2} f - b c^{2} e f^{2} + 2 b c d e^{2} f - b d^{2} e^{3}} + x \right )}}{2} + \frac{x^{3} \left (a d^{2} f + 2 b c d f - b d^{2} e\right )}{3 f^{2}} + \frac{x \left (2 a c d f^{2} - a d^{2} e f + b c^{2} f^{2} - 2 b c d e f + b d^{2} e^{2}\right )}{f^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17317, size = 240, normalized size = 1.69 \begin{align*} \frac{{\left (a c^{2} f^{3} - b c^{2} f^{2} e - 2 \, a c d f^{2} e + 2 \, b c d f e^{2} + a d^{2} f e^{2} - b d^{2} e^{3}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{1}{2}\right )}}{f^{\frac{7}{2}}} + \frac{3 \, b d^{2} f^{4} x^{5} + 10 \, b c d f^{4} x^{3} + 5 \, a d^{2} f^{4} x^{3} - 5 \, b d^{2} f^{3} x^{3} e + 15 \, b c^{2} f^{4} x + 30 \, a c d f^{4} x - 30 \, b c d f^{3} x e - 15 \, a d^{2} f^{3} x e + 15 \, b d^{2} f^{2} x e^{2}}{15 \, f^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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