Optimal. Leaf size=132 \[ -\frac{a \log \left (\sqrt{3} x^2-2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{3} x^2+2^{3/4} \sqrt [4]{3} x+\sqrt{2}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
[Out]
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Rubi [A] time = 0.208991, antiderivative size = 114, normalized size of antiderivative = 0.86, number of steps used = 12, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + d*x^3)/(2 + 3*x^4),x]
[Out]
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Rubi in Sympy [A] time = 23.7964, size = 102, normalized size = 0.77 \[ - \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} - 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \log{\left (3 x^{2} + 6^{\frac{3}{4}} x + \sqrt{6} \right )}}{48} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac{6^{\frac{3}{4}} a \operatorname{atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24} + \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+a)/(3*x**4+2),x)
[Out]
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Mathematica [A] time = 0.0610137, size = 108, normalized size = 0.82 \[ \frac{1}{48} \left (-6^{3/4} a \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} a \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )-2\ 6^{3/4} a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )+4 d \log \left (3 x^4+2\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + d*x^3)/(2 + 3*x^4),x]
[Out]
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Maple [A] time = 0.004, size = 125, normalized size = 1. \[{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({1 \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+a)/(3*x^4+2),x)
[Out]
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Maxima [A] time = 1.54513, size = 201, normalized size = 1.52 \[ \frac{1}{24} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}} a \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d + 3 \, a\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + a)/(3*x^4 + 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246206, size = 389, normalized size = 2.95 \[ \frac{1}{576} \cdot 24^{\frac{3}{4}}{\left ({\left (2 \cdot 24^{\frac{1}{4}} d + 3 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}\right )} \log \left (2 \, \sqrt{6} a^{2} x^{2} + 2 \cdot 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{a^{4}}\right ) +{\left (2 \cdot 24^{\frac{1}{4}} d - 3 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}\right )} \log \left (2 \, \sqrt{6} a^{2} x^{2} - 2 \cdot 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 4 \, \sqrt{a^{4}}\right ) - 12 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}{24^{\frac{1}{4}} a x + 24^{\frac{1}{4}} \sqrt{\frac{1}{6}} a \sqrt{\frac{\sqrt{6}{\left (\sqrt{6} a^{2} x^{2} + 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 2 \, \sqrt{a^{4}}\right )}}{a^{2}}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}\right ) - 12 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}{24^{\frac{1}{4}} a x + 24^{\frac{1}{4}} \sqrt{\frac{1}{6}} a \sqrt{\frac{\sqrt{6}{\left (\sqrt{6} a^{2} x^{2} - 24^{\frac{1}{4}} \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a x + 2 \, \sqrt{a^{4}}\right )}}{a^{2}}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + a)/(3*x^4 + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.503587, size = 51, normalized size = 0.39 \[ \operatorname{RootSum}{\left (165888 t^{4} - 55296 t^{3} d + 6912 t^{2} d^{2} - 384 t d^{3} + 27 a^{4} + 8 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{24 t - 2 d}{3 a} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+a)/(3*x**4+2),x)
[Out]
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GIAC/XCAS [A] time = 0.223524, size = 147, normalized size = 1.11 \[ \frac{1}{24} \cdot 6^{\frac{3}{4}} a \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \cdot 6^{\frac{3}{4}} a \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a + 4 \, d\right )}{\rm ln}\left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 4 \, d\right )}{\rm ln}\left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + a)/(3*x^4 + 2),x, algorithm="giac")
[Out]