3.305 \(\int x^6 \sqrt [4]{3+4 x^4} \, dx\)

Optimal. Leaf size=93 \[ \frac{27 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}}-\frac{27 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}}+\frac{1}{8} \sqrt [4]{4 x^4+3} x^7+\frac{3}{128} \sqrt [4]{4 x^4+3} x^3 \]

[Out]

(3*x^3*(3 + 4*x^4)^(1/4))/128 + (x^7*(3 + 4*x^4)^(1/4))/8 + (27*ArcTan[(Sqrt[2]*
x)/(3 + 4*x^4)^(1/4)])/(512*Sqrt[2]) - (27*ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)
])/(512*Sqrt[2])

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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]

(3*x^3*(3 + 4*x^4)^(1/4))/128 + (x^7*(3 + 4*x^4)^(1/4))/8 + (27*ArcTan[(Sqrt[2]*
x)/(3 + 4*x^4)^(1/4)])/(512*Sqrt[2]) - (27*ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)
])/(512*Sqrt[2])

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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]

x**7*(4*x**4 + 3)**(1/4)/8 + 3*x**3*(4*x**4 + 3)**(1/4)/128 + 27*sqrt(2)*atan(sq
rt(2)*x/(4*x**4 + 3)**(1/4))/1024 - 27*sqrt(2)*atanh(sqrt(2)*x/(4*x**4 + 3)**(1/
4))/1024

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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]

(x^3*((3 + 4*x^4)^(1/4)*(3 + 16*x^4) - 3*3^(1/4)*Hypergeometric2F1[3/4, 3/4, 7/4
, (-4*x^4)/3]))/128

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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]

1/128*x^3*(16*x^4+3)*(4*x^4+3)^(1/4)-3/128*3^(1/4)*x^3*hypergeom([3/4,3/4],[7/4]
,-4/3*x^4)

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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")

[Out]

-27/1024*sqrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) + 27/2048*sqrt(2)*log(-
2*(sqrt(2) - (4*x^4 + 3)^(1/4)/x)/((2*sqrt(2)) + 2*(4*x^4 + 3)^(1/4)/x)) - 9/128
*(12*(4*x^4 + 3)^(1/4)/x + (4*x^4 + 3)^(5/4)/x^5)/(8*(4*x^4 + 3)/x^4 - (4*x^4 +
3)^2/x^8 - 16)

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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]

1/2048*sqrt(2)*(8*sqrt(2)*(16*x^7 + 3*x^3)*(4*x^4 + 3)^(1/4) - 54*arctan(1/2*sqr
t(2)*(4*x^4 + 3)^(1/4)/x) + 27*log(-(2*sqrt(2)*x^2 - 4*(4*x^4 + 3)^(1/4)*x + sqr
t(2)*sqrt(4*x^4 + 3))/(2*x^2 - sqrt(4*x^4 + 3))))

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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]

3**(1/4)*x**7*gamma(7/4)*hyper((-1/4, 7/4), (11/4,), 4*x**4*exp_polar(I*pi)/3)/(
4*gamma(11/4))

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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]

1/128*x^8*((4*x^4 + 3)^(1/4)*(3/x^4 + 4)/x + 12*(4*x^4 + 3)^(1/4)/x) - 27/1024*s
qrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) + 27/2048*sqrt(2)*ln(-(sqrt(2) -
(4*x^4 + 3)^(1/4)/x)/(sqrt(2) + (4*x^4 + 3)^(1/4)/x))