3.1532 \(\int \frac{x^6}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=22 \[ \frac{1}{7} x^7 \, _2F_1\left (\frac{1}{2},\frac{7}{8};\frac{15}{8};-x^8\right ) \]

[Out]

(x^7*Hypergeometric2F1[1/2, 7/8, 15/8, -x^8])/7

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Rubi [A]  time = 0.0043918, antiderivative size = 22, normalized size of antiderivative = 1., 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} \[ \frac{1}{7} x^7 \, _2F_1\left (\frac{1}{2},\frac{7}{8};\frac{15}{8};-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^6/Sqrt[1 + x^8],x]

[Out]

(x^7*Hypergeometric2F1[1/2, 7/8, 15/8, -x^8])/7

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^6}{\sqrt{1+x^8}} \, dx &=\frac{1}{7} x^7 \, _2F_1\left (\frac{1}{2},\frac{7}{8};\frac{15}{8};-x^8\right )\\ \end{align*}

Mathematica [A]  time = 0.0022879, size = 22, normalized size = 1. \[ \frac{1}{7} x^7 \, _2F_1\left (\frac{1}{2},\frac{7}{8};\frac{15}{8};-x^8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/Sqrt[1 + x^8],x]

[Out]

(x^7*Hypergeometric2F1[1/2, 7/8, 15/8, -x^8])/7

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Maple [A]  time = 0.025, size = 17, normalized size = 0.8 \begin{align*}{\frac{{x}^{7}}{7}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{7}{8}};\,{\frac{15}{8}};\,-{x}^{8})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(x^8+1)^(1/2),x)

[Out]

1/7*x^7*hypergeom([1/2,7/8],[15/8],-x^8)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\sqrt{x^{8} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^6/sqrt(x^8 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{6}}{\sqrt{x^{8} + 1}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

integral(x^6/sqrt(x^8 + 1), x)

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Sympy [C]  time = 0.655305, size = 29, normalized size = 1.32 \begin{align*} \frac{x^{7} \Gamma \left (\frac{7}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{8} \\ \frac{15}{8} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{15}{8}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(x**8+1)**(1/2),x)

[Out]

x**7*gamma(7/8)*hyper((1/2, 7/8), (15/8,), x**8*exp_polar(I*pi))/(8*gamma(15/8))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\sqrt{x^{8} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^6/sqrt(x^8 + 1), x)