Optimal. Leaf size=141 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{4\ 6^{3/4}}-\frac{c \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]
[Out]
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Rubi [A] time = 0.240347, antiderivative size = 123, normalized size of antiderivative = 0.87, number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)/(2 + 3*x^4),x]
[Out]
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Rubi in Sympy [A] time = 27.0423, size = 110, normalized size = 0.78 \[ \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} + \frac{\sqrt [4]{6} c \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} - \frac{\sqrt [4]{6} c \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)/(3*x**4+2),x)
[Out]
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Mathematica [A] time = 0.092438, size = 99, normalized size = 0.7 \[ \frac{-2 \left (\sqrt [4]{6} b+c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \left (c-\sqrt [4]{6} b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+c \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-c \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )}{4\ 6^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)/(2 + 3*x^4),x]
[Out]
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Maple [A] time = 0.004, size = 129, normalized size = 0.9 \[{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)/(3*x^4+2),x)
[Out]
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Maxima [A] time = 1.53638, size = 198, normalized size = 1.4 \[ \frac{1}{24} \, \sqrt{2}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} c - 2 \, \sqrt{3} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \, \sqrt{2}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} c + 2 \, \sqrt{3} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) - \frac{1}{24} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{24} \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} c \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(3*x^4 + 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)/(3*x^4 + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.80539, size = 85, normalized size = 0.6 \[ \operatorname{RootSum}{\left (27648 t^{4} + 576 t^{2} b^{2} + 96 t b c^{2} + 3 b^{4} + 2 c^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 1152 t^{3} c^{2} + 288 t^{2} b^{3} - 36 t b^{2} c^{2} + 3 b^{5} - 3 b c^{4}}{6 b^{4} c - c^{5}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)/(3*x**4+2),x)
[Out]
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GIAC/XCAS [A] time = 0.223479, size = 154, normalized size = 1.09 \[ -\frac{1}{24} \cdot 6^{\frac{1}{4}} c{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \cdot 6^{\frac{1}{4}} c{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{12} \,{\left (\sqrt{6} b - 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{12} \,{\left (\sqrt{6} b + 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)/(3*x^4 + 2),x, algorithm="giac")
[Out]