3.925 \(\int \frac{x^8}{\sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{5}{21} \sqrt{x^4+1} x+\frac{1}{7} \sqrt{x^4+1} x^5+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{42 \sqrt{x^4+1}} \]

[Out]

(-5*x*Sqrt[1 + x^4])/21 + (x^5*Sqrt[1 + x^4])/7 + (5*(1 + x^2)*Sqrt[(1 + x^4)/(1
+ x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(42*Sqrt[1 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.0456225, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{5}{21} \sqrt{x^4+1} x+\frac{1}{7} \sqrt{x^4+1} x^5+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{42 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^8/Sqrt[1 + x^4],x]

[Out]

(-5*x*Sqrt[1 + x^4])/21 + (x^5*Sqrt[1 + x^4])/7 + (5*(1 + x^2)*Sqrt[(1 + x^4)/(1
+ x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(42*Sqrt[1 + x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.08809, size = 66, normalized size = 0.89 \[ \frac{x^{5} \sqrt{x^{4} + 1}}{7} - \frac{5 x \sqrt{x^{4} + 1}}{21} + \frac{5 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{42 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(x**4+1)**(1/2),x)

[Out]

x**5*sqrt(x**4 + 1)/7 - 5*x*sqrt(x**4 + 1)/21 + 5*sqrt((x**4 + 1)/(x**2 + 1)**2)
*(x**2 + 1)*elliptic_f(2*atan(x), 1/2)/(42*sqrt(x**4 + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0501285, size = 57, normalized size = 0.77 \[ -\frac{-3 x^9+2 x^5+5 \sqrt [4]{-1} \sqrt{x^4+1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+5 x}{21 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/Sqrt[1 + x^4],x]

[Out]

-(5*x + 2*x^5 - 3*x^9 + 5*(-1)^(1/4)*Sqrt[1 + x^4]*EllipticF[I*ArcSinh[(-1)^(1/4
)*x], -1])/(21*Sqrt[1 + x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.008, size = 84, normalized size = 1.1 \[{\frac{{x}^{5}}{7}\sqrt{{x}^{4}+1}}-{\frac{5\,x}{21}\sqrt{{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{{\frac{21\,\sqrt{2}}{2}}+{\frac{21\,i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(x^4+1)^(1/2),x)

[Out]

1/7*x^5*(x^4+1)^(1/2)-5/21*x*(x^4+1)^(1/2)+5/21/(1/2*2^(1/2)+1/2*I*2^(1/2))*(1-I
*x^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticF(x*(1/2*2^(1/2)+1/2*I*2^(1/2)
),I)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^8/sqrt(x^4 + 1), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{8}}{\sqrt{x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(x^4 + 1),x, algorithm="fricas")

[Out]

integral(x^8/sqrt(x^4 + 1), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.5589, size = 29, normalized size = 0.39 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(x**4+1)**(1/2),x)

[Out]

x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), x**4*exp_polar(I*pi))/(4*gamma(13/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^8/sqrt(x^4 + 1), x)