3.1533 \(\int \frac {x^4}{\sqrt {1+x^8}} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{5} x^5 \, _2F_1\left (\frac {1}{2},\frac {5}{8};\frac {13}{8};-x^8\right ) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {364} \[ \frac {1}{5} x^5 \, _2F_1\left (\frac {1}{2},\frac {5}{8};\frac {13}{8};-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[1 + x^8],x]

[Out]

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt {1+x^8}} \, dx &=\frac {1}{5} x^5 \, _2F_1\left (\frac {1}{2},\frac {5}{8};\frac {13}{8};-x^8\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \[ \frac {1}{5} x^5 \, _2F_1\left (\frac {1}{2},\frac {5}{8};\frac {13}{8};-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[1 + x^8],x]

[Out]

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

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fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{\sqrt {x^{8} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

integral(x^4/sqrt(x^8 + 1), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(x^8 + 1), x)

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maple [A]  time = 0.15, size = 17, normalized size = 0.77 \[ \frac {x^{5} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{8}\right ], \left [\frac {13}{8}\right ], -x^{8}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^8+1)^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4/sqrt(x^8 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {x^4}{\sqrt {x^8+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(x^8 + 1)^(1/2),x)

[Out]

int(x^4/(x^8 + 1)^(1/2), x)

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sympy [C]  time = 0.74, size = 29, normalized size = 1.32 \[ \frac {x^{5} \Gamma \left (\frac {5}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{8} \\ \frac {13}{8} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {13}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5*gamma(5/8)*hyper((1/2, 5/8), (13/8,), x**8*exp_polar(I*pi))/(8*gamma(13/8))

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