Optimal. Leaf size=93 \[ \frac{27 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}}-\frac{27 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}}+\frac{1}{8} \sqrt [4]{4 x^4+3} x^7+\frac{3}{128} \sqrt [4]{4 x^4+3} x^3 \]
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Rubi [A] time = 0.0819841, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{27 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}}-\frac{27 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}}+\frac{1}{8} \sqrt [4]{4 x^4+3} x^7+\frac{3}{128} \sqrt [4]{4 x^4+3} x^3 \]
Antiderivative was successfully verified.
[In] Int[x^6*(3 + 4*x^4)^(1/4),x]
[Out]
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Rubi in Sympy [A] time = 3.99397, size = 85, normalized size = 0.91 \[ \frac{x^{7} \sqrt [4]{4 x^{4} + 3}}{8} + \frac{3 x^{3} \sqrt [4]{4 x^{4} + 3}}{128} + \frac{27 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^{4} + 3}} \right )}}{1024} - \frac{27 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^{4} + 3}} \right )}}{1024} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6*(4*x**4+3)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0275489, size = 51, normalized size = 0.55 \[ \frac{1}{128} x^3 \left (\sqrt [4]{4 x^4+3} \left (16 x^4+3\right )-3 \sqrt [4]{3} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{4 x^4}{3}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^6*(3 + 4*x^4)^(1/4),x]
[Out]
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Maple [C] time = 0.035, size = 42, normalized size = 0.5 \[{\frac{{x}^{3} \left ( 16\,{x}^{4}+3 \right ) }{128}\sqrt [4]{4\,{x}^{4}+3}}-{\frac{3\,\sqrt [4]{3}{x}^{3}}{128}{\mbox{$_2$F$_1$}({\frac{3}{4}},{\frac{3}{4}};\,{\frac{7}{4}};\,-{\frac{4\,{x}^{4}}{3}})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6*(4*x^4+3)^(1/4),x)
[Out]
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Maxima [A] time = 1.69122, size = 176, normalized size = 1.89 \[ -\frac{27}{1024} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{27}{2048} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \sqrt{2} + \frac{2 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) - \frac{9 \,{\left (\frac{12 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{5}{4}}}{x^{5}}\right )}}{128 \,{\left (\frac{8 \,{\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac{{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^4 + 3)^(1/4)*x^6,x, algorithm="maxima")
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Fricas [A] time = 0.215886, size = 147, normalized size = 1.58 \[ \frac{1}{2048} \, \sqrt{2}{\left (8 \, \sqrt{2}{\left (16 \, x^{7} + 3 \, x^{3}\right )}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} - 54 \, \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) + 27 \, \log \left (-\frac{2 \, \sqrt{2} x^{2} - 4 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} x + \sqrt{2} \sqrt{4 \, x^{4} + 3}}{2 \, x^{2} - \sqrt{4 \, x^{4} + 3}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^4 + 3)^(1/4)*x^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.68289, size = 39, normalized size = 0.42 \[ \frac{\sqrt [4]{3} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{4 x^{4} e^{i \pi }}{3}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6*(4*x**4+3)**(1/4),x)
[Out]
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GIAC/XCAS [A] time = 0.210533, size = 147, normalized size = 1.58 \[ \frac{1}{128} \, x^{8}{\left (\frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}{\left (\frac{3}{x^{4}} + 4\right )}}{x} + \frac{12 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}\right )} - \frac{27}{1024} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{27}{2048} \, \sqrt{2}{\rm ln}\left (-\frac{\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}{\sqrt{2} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^4 + 3)^(1/4)*x^6,x, algorithm="giac")
[Out]