Optimal. Leaf size=154 \[ -\frac{a \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
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Rubi [A] time = 0.262121, antiderivative size = 136, normalized size of antiderivative = 0.88, number of steps used = 15, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + d*x^3)/(2 + 3*x^4),x]
[Out]
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Rubi in Sympy [A] time = 31.1556, size = 122, normalized size = 0.79 \[ - \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} + \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} + \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+b*x+a)/(3*x**4+2),x)
[Out]
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Mathematica [A] time = 0.130159, size = 128, normalized size = 0.83 \[ \frac{1}{48} \left (-2 \sqrt{6} \left (\sqrt [4]{6} a+2 b\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt{6} \left (\sqrt [4]{6} a-2 b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )-6^{3/4} a \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} a \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+4 d \log \left (3 x^4+2\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + d*x^3)/(2 + 3*x^4),x]
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Maple [A] time = 0.003, size = 140, normalized size = 0.9 \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+b*x+a)/(3*x^4+2),x)
[Out]
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Maxima [A] time = 1.55219, size = 231, normalized size = 1.5 \[ \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d + 3 \, a\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{24} \, \sqrt{3}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} a - 2 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \, \sqrt{3}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + b*x + a)/(3*x^4 + 2),x, algorithm="maxima")
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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 + b*x + a)/(3*x^4 + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.2686, size = 199, normalized size = 1.29 \[ \operatorname{RootSum}{\left (165888 t^{4} - 55296 t^{3} d + t^{2} \left (3456 b^{2} + 6912 d^{2}\right ) + t \left (- 864 a^{2} b - 576 b^{2} d - 384 d^{3}\right ) + 27 a^{4} + 72 a^{2} b d + 18 b^{4} + 24 b^{2} d^{2} + 8 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{27648 t^{3} b^{2} + 1728 t^{2} a^{2} b - 6912 t^{2} b^{2} d + 216 t a^{4} - 288 t a^{2} b d + 288 t b^{4} + 576 t b^{2} d^{2} - 18 a^{4} d - 90 a^{2} b^{3} + 12 a^{2} b d^{2} - 24 b^{4} d - 16 b^{2} d^{3}}{27 a^{5} - 72 a b^{4}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+b*x+a)/(3*x**4+2),x)
[Out]
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GIAC/XCAS [A] time = 0.225803, size = 169, normalized size = 1.1 \[ \frac{1}{24} \,{\left (6^{\frac{3}{4}} a - 2 \, \sqrt{6} b\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \, \sqrt{6} b\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a + 4 \, d\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 4 \, d\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + b*x + a)/(3*x^4 + 2),x, algorithm="giac")
[Out]