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

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

[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])

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Rubi [A]  time = 0.0143709, 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, Rules used = {321, 220} \[ \frac{1}{7} \sqrt{x^4+1} x^5-\frac{5}{21} \sqrt{x^4+1} x+\frac{5 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{42 \sqrt{x^4+1}} \]

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])

Rule 321

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

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{x^8}{\sqrt{1+x^4}} \, dx &=\frac{1}{7} x^5 \sqrt{1+x^4}-\frac{5}{7} \int \frac{x^4}{\sqrt{1+x^4}} \, dx\\ &=-\frac{5}{21} x \sqrt{1+x^4}+\frac{1}{7} x^5 \sqrt{1+x^4}+\frac{5}{21} \int \frac{1}{\sqrt{1+x^4}} \, dx\\ &=-\frac{5}{21} x \sqrt{1+x^4}+\frac{1}{7} x^5 \sqrt{1+x^4}+\frac{5 \left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{42 \sqrt{1+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0109852, size = 40, normalized size = 0.54 \[ \frac{1}{21} x \left (5 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^4\right )+\sqrt{x^4+1} \left (3 x^4-5\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/Sqrt[1 + x^4],x]

[Out]

(x*(Sqrt[1 + x^4]*(-5 + 3*x^4) + 5*Hypergeometric2F1[1/4, 1/2, 5/4, -x^4]))/21

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Maple [C]  time = 0.043, size = 84, normalized size = 1.1 \begin{align*}{\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}}}} \end{align*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^8/sqrt(x^4 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^8/sqrt(x^4 + 1), x)

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Sympy [C]  time = 1.04815, size = 29, normalized size = 0.39 \begin{align*} \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 )} \end{align*}

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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{\sqrt{x^{4} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^8/sqrt(x^4 + 1), x)