Optimal. Leaf size=154 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{4\ 6^{3/4}}-\frac{c \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
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Rubi [A] time = 0.323716, antiderivative size = 136, normalized size of antiderivative = 0.88, number of steps used = 16, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ \frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{c \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{4\ 6^{3/4}}-\frac{c \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{2\ 6^{3/4}}+\frac{c \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{2\ 6^{3/4}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2 + d*x^3)/(2 + 3*x^4),x]
[Out]
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Rubi in Sympy [A] time = 36.8169, size = 122, normalized size = 0.79 \[ \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} + \frac{\sqrt [4]{6} c \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} - \frac{\sqrt [4]{6} c \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{24} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{12} + \frac{\sqrt [4]{6} c \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{12} + \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c*x**2+b*x)/(3*x**4+2),x)
[Out]
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Mathematica [A] time = 0.126703, size = 125, normalized size = 0.81 \[ \frac{1}{24} \left (-2 \sqrt [4]{6} \left (\sqrt [4]{6} b+c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \left (c-\sqrt [4]{6} b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+\sqrt [4]{6} c \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )-\sqrt [4]{6} c \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+2 d \log \left (3 x^4+2\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2 + d*x^3)/(2 + 3*x^4),x]
[Out]
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Maple [A] time = 0.005, size = 140, normalized size = 0.9 \[{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{72}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{c\sqrt{3}{6}^{{\frac{3}{4}}}\sqrt{2}}{144}\ln \left ({1 \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c*x^2+b*x)/(3*x^4+2),x)
[Out]
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Maxima [A] time = 1.61724, size = 235, normalized size = 1.53 \[ \frac{1}{72} \, \sqrt{3} \sqrt{2}{\left (3^{\frac{3}{4}} 2^{\frac{3}{4}} c - 6 \, b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{72} \, \sqrt{3} \sqrt{2}{\left (3^{\frac{3}{4}} 2^{\frac{3}{4}} c + 6 \, b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d - \sqrt{3} c\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{72} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} d + \sqrt{3} c\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + 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((d*x^3 + c*x^2 + b*x)/(3*x^4 + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.24829, size = 189, normalized size = 1.23 \[ \operatorname{RootSum}{\left (82944 t^{4} - 27648 t^{3} d + t^{2} \left (1728 b^{2} + 3456 d^{2}\right ) + t \left (- 288 b^{2} d + 288 b c^{2} - 192 d^{3}\right ) + 9 b^{4} + 12 b^{2} d^{2} - 24 b c^{2} d + 6 c^{4} + 4 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 3456 t^{3} c^{2} + 864 t^{2} b^{3} + 864 t^{2} c^{2} d - 144 t b^{3} d - 108 t b^{2} c^{2} - 72 t c^{2} d^{2} + 9 b^{5} + 6 b^{3} d^{2} + 9 b^{2} c^{2} d - 9 b c^{4} + 2 c^{2} d^{3}}{18 b^{4} c - 3 c^{5}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+c*x**2+b*x)/(3*x**4+2),x)
[Out]
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GIAC/XCAS [A] time = 0.225389, size = 167, normalized size = 1.08 \[ -\frac{1}{12} \,{\left (\sqrt{6} b - 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{12} \,{\left (\sqrt{6} b + 6^{\frac{1}{4}} c\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) - \frac{1}{24} \,{\left (6^{\frac{1}{4}} c - 2 \, d\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \,{\left (6^{\frac{1}{4}} c + 2 \, d\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c*x^2 + b*x)/(3*x^4 + 2),x, algorithm="giac")
[Out]