Optimal. Leaf size=141 \[ -\frac{a \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]
[Out]
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Rubi [A] time = 0.227079, antiderivative size = 123, normalized size of antiderivative = 0.87, number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/(2 + 3*x^4),x]
[Out]
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Rubi in Sympy [A] time = 23.5077, size = 110, normalized size = 0.78 \[ - \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} + \frac{\sqrt{6} b \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(3*x**4+2),x)
[Out]
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Mathematica [A] time = 0.109898, size = 107, normalized size = 0.76 \[ \frac{-2 \left (\sqrt [4]{6} a+2 b\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \left (\sqrt [4]{6} a-2 b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+\sqrt [4]{6} a \left (\log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )\right )}{8 \sqrt{6}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/(2 + 3*x^4),x]
[Out]
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Maple [A] time = 0.004, size = 129, normalized size = 0.9 \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(3*x^4+2),x)
[Out]
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Maxima [A] time = 1.52984, size = 198, normalized size = 1.4 \[ \frac{1}{48} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) - \frac{1}{48} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{24} \, \sqrt{3}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} a - 2 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \, \sqrt{3}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(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((b*x + a)/(3*x^4 + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.796032, size = 88, normalized size = 0.62 \[ \operatorname{RootSum}{\left (18432 t^{4} + 384 t^{2} b^{2} - 96 t a^{2} b + 3 a^{4} + 2 b^{4}, \left ( t \mapsto t \log{\left (x + \frac{3072 t^{3} b^{2} + 192 t^{2} a^{2} b + 24 t a^{4} + 32 t b^{4} - 10 a^{2} b^{3}}{3 a^{5} - 8 a b^{4}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(3*x**4+2),x)
[Out]
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GIAC/XCAS [A] time = 0.222151, size = 155, normalized size = 1.1 \[ \frac{1}{48} \cdot 6^{\frac{3}{4}} a{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \cdot 6^{\frac{3}{4}} a{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a - 2 \, \sqrt{6} b\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \, \sqrt{6} b\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(3*x^4 + 2),x, algorithm="giac")
[Out]