Optimal. Leaf size=317 \[ \frac{(2 A c-b C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a C+x^2 (2 A c-b C)+A b}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B x \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \sqrt{c} \left (2 b-\sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{B \sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]
[Out]
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Rubi [A] time = 0.99192, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ \frac{(2 A c-b C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a C+x^2 (2 A c-b C)+A b}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B x \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \sqrt{c} \left (2 b-\sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{B \sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 98.3809, size = 291, normalized size = 0.92 \[ - \frac{\sqrt{2} B \sqrt{c} \left (2 b + \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{2} B \sqrt{c} \left (2 b - \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\left (2 A c - C b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{x \left (- B a b - 2 B a c x^{2} + c x^{3} \left (A b - 2 C a\right ) + x \left (- 2 A a c + A b^{2} - C a b\right )\right )}{2 a \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(C*x**2+B*x+A)/(c*x**4+b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 3.315, size = 335, normalized size = 1.06 \[ \frac{1}{2} \left (\frac{2 a C-A \left (b+2 c x^2\right )+x \left (-b B+b C x-2 B c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{(b C-2 A c) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{(2 A c-b C) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{\sqrt{2} B \sqrt{c} \left (\sqrt{b^2-4 a c}-2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} B \sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4)^2,x]
[Out]
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Maple [B] time = 0.162, size = 1344, normalized size = 4.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(C*x^2+B*x+A)/(c*x^4+b*x^2+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{2 \, B c x^{3} + B b x -{\left (C b - 2 \, A c\right )} x^{2} - 2 \, C a + A b}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )}} - \frac{\int \frac{2 \, B c x^{2} - B b - 2 \,{\left (C b - 2 \, A c\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (b^{2} - 4 \, a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*x/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*x/(c*x^4 + b*x^2 + a)^2,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(C*x**2+B*x+A)/(c*x**4+b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)*x/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")
[Out]