Optimal. Leaf size=223 \[ -\frac{(2 A c-b C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}-\frac{B \sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{B \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{C \log \left (a+b x^2+c x^4\right )}{4 c} \]
[Out]
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Rubi [A] time = 0.520649, antiderivative size = 223, 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 \[ -\frac{(2 A c-b C) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c \sqrt{b^2-4 a c}}-\frac{B \sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{B \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}+\frac{C \log \left (a+b x^2+c x^4\right )}{4 c} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 64.4513, size = 209, normalized size = 0.94 \[ - \frac{\sqrt{2} B \sqrt{b - \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} B \sqrt{b + \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{- 4 a c + b^{2}}} + \frac{C \log{\left (a + b x^{2} + c x^{4} \right )}}{4 c} - \frac{\left (2 A c - C b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c \sqrt{- 4 a c + b^{2}}} \]
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),x)
[Out]
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Mathematica [A] time = 0.790396, size = 240, normalized size = 1.08 \[ \frac{\left (C \left (\sqrt{b^2-4 a c}-b\right )+2 A c\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )-\left (2 A c-C \left (\sqrt{b^2-4 a c}+b\right )\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )-2 \sqrt{2} B \sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )+2 \sqrt{2} B \sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{4 c \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x + C*x^2))/(a + b*x^2 + c*x^4),x]
[Out]
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Maple [B] time = 0.035, size = 728, normalized size = 3.3 \[ \text{result too large to display} \]
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),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + B x + A\right )} x}{c x^{4} + b x^{2} + a}\,{d x} \]
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),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),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),x)
[Out]
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GIAC/XCAS [A] time = 1.2154, size = 1, normalized size = 0. \[ \mathit{Done} \]
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),x, algorithm="giac")
[Out]