3.1508 \(\int x \sqrt{1+x^8} \, dx\)

Optimal. Leaf size=62 \[ \frac{1}{6} \sqrt{x^8+1} x^2+\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{6 \sqrt{x^8+1}} \]

[Out]

(x^2*Sqrt[1 + x^8])/6 + ((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*EllipticF[2*ArcTa
n[x^2], 1/2])/(6*Sqrt[1 + x^8])

_______________________________________________________________________________________

Rubi [A]  time = 0.0579022, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{1}{6} \sqrt{x^8+1} x^2+\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{6 \sqrt{x^8+1}} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[1 + x^8],x]

[Out]

(x^2*Sqrt[1 + x^8])/6 + ((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*EllipticF[2*ArcTa
n[x^2], 1/2])/(6*Sqrt[1 + x^8])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.81271, size = 53, normalized size = 0.85 \[ \frac{x^{2} \sqrt{x^{8} + 1}}{6} + \frac{\sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) F\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{6 \sqrt{x^{8} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(x**8 + 1)/6 + sqrt((x**8 + 1)/(x**4 + 1)**2)*(x**4 + 1)*elliptic_f(2*a
tan(x**2), 1/2)/(6*sqrt(x**8 + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0253225, size = 34, normalized size = 0.55 \[ \frac{1}{6} x^2 \left (2 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^8\right )+\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[1 + x^8],x]

[Out]

(x^2*(Sqrt[1 + x^8] + 2*Hypergeometric2F1[1/4, 1/2, 5/4, -x^8]))/6

_______________________________________________________________________________________

Maple [C]  time = 0.041, size = 30, normalized size = 0.5 \[{\frac{{x}^{2}}{6}\sqrt{{x}^{8}+1}}+{\frac{{x}^{2}}{3}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{5}{4}};\,-{x}^{8})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(x^8+1)^(1/2),x)

[Out]

1/6*x^2*(x^8+1)^(1/2)+1/3*x^2*hypergeom([1/4,1/2],[5/4],-x^8)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x^8 + 1)*x, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(x^8 + 1)*x, x)

_______________________________________________________________________________________

Sympy [A]  time = 1.74249, size = 31, normalized size = 0.5 \[ \frac{x^{2} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(x**8+1)**(1/2),x)

[Out]

x**2*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), x**8*exp_polar(I*pi))/(8*gamma(5/4))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x^8 + 1)*x, x)