3.1535 \(\int \frac{1}{x^4 \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{15} x^5 \, _2F_1\left (\frac{1}{2},\frac{5}{8};\frac{13}{8};-x^8\right )-\frac{\sqrt{x^8+1}}{3 x^3} \]

[Out]

-Sqrt[1 + x^8]/(3*x^3) + (x^5*Hypergeometric2F1[1/2, 5/8, 13/8, -x^8])/15

_______________________________________________________________________________________

Rubi [A]  time = 0.0366944, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{15} x^5 \, _2F_1\left (\frac{1}{2},\frac{5}{8};\frac{13}{8};-x^8\right )-\frac{\sqrt{x^8+1}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(3*x^3) + (x^5*Hypergeometric2F1[1/2, 5/8, 13/8, -x^8])/15

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.92955, size = 29, normalized size = 0.74 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{8} \\ \frac{13}{8} \end{matrix}\middle |{- x^{8}} \right )}}{15} - \frac{\sqrt{x^{8} + 1}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*hyper((1/2, 5/8), (13/8,), -x**8)/15 - sqrt(x**8 + 1)/(3*x**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0317673, size = 39, normalized size = 1. \[ \frac{1}{15} x^5 \, _2F_1\left (\frac{1}{2},\frac{5}{8};\frac{13}{8};-x^8\right )-\frac{\sqrt{x^8+1}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(3*x^3) + (x^5*Hypergeometric2F1[1/2, 5/8, 13/8, -x^8])/15

_______________________________________________________________________________________

Maple [A]  time = 0.039, size = 30, normalized size = 0.8 \[{\frac{{x}^{5}}{15}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{5}{8}};\,{\frac{13}{8}};\,-{x}^{8})}}-{\frac{1}{3\,{x}^{3}}\sqrt{{x}^{8}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^8+1)^(1/2),x)

[Out]

1/15*x^5*hypergeom([1/2,5/8],[13/8],-x^8)-1/3*(x^8+1)^(1/2)/x^3

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^8 + 1)*x^4), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x^8 + 1)*x^4), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.28194, size = 32, normalized size = 0.82 \[ \frac{\Gamma \left (- \frac{3}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{8}, \frac{1}{2} \\ \frac{5}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 x^{3} \Gamma \left (\frac{5}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-3/8)*hyper((-3/8, 1/2), (5/8,), x**8*exp_polar(I*pi))/(8*x**3*gamma(5/8))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^8 + 1)*x^4), x)