3.930 \(\int \frac{x^{10}}{\sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=140 \[ \frac{1}{9} \sqrt{x^4+1} x^7-\frac{7}{45} \sqrt{x^4+1} x^3+\frac{7 \sqrt{x^4+1} x}{15 \left (x^2+1\right )}+\frac{7 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{30 \sqrt{x^4+1}}-\frac{7 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+1}} \]

[Out]

(-7*x^3*Sqrt[1 + x^4])/45 + (x^7*Sqrt[1 + x^4])/9 + (7*x*Sqrt[1 + x^4])/(15*(1 +
x^2)) - (7*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(
15*Sqrt[1 + x^4]) + (7*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[
x], 1/2])/(30*Sqrt[1 + x^4])

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Rubi [A]  time = 0.091335, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{9} \sqrt{x^4+1} x^7-\frac{7}{45} \sqrt{x^4+1} x^3+\frac{7 \sqrt{x^4+1} x}{15 \left (x^2+1\right )}+\frac{7 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{30 \sqrt{x^4+1}}-\frac{7 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^10/Sqrt[1 + x^4],x]

[Out]

(-7*x^3*Sqrt[1 + x^4])/45 + (x^7*Sqrt[1 + x^4])/9 + (7*x*Sqrt[1 + x^4])/(15*(1 +
x^2)) - (7*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/2])/(
15*Sqrt[1 + x^4]) + (7*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[
x], 1/2])/(30*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 8.42862, size = 128, normalized size = 0.91 \[ \frac{x^{7} \sqrt{x^{4} + 1}}{9} - \frac{7 x^{3} \sqrt{x^{4} + 1}}{45} + \frac{7 x \sqrt{x^{4} + 1}}{15 \left (x^{2} + 1\right )} - \frac{7 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{15 \sqrt{x^{4} + 1}} + \frac{7 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{30 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(x**4+1)**(1/2),x)

[Out]

x**7*sqrt(x**4 + 1)/9 - 7*x**3*sqrt(x**4 + 1)/45 + 7*x*sqrt(x**4 + 1)/(15*(x**2
+ 1)) - 7*sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_e(2*atan(x), 1/2)/(
15*sqrt(x**4 + 1)) + 7*sqrt((x**4 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_f(2*at
an(x), 1/2)/(30*sqrt(x**4 + 1))

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Mathematica [C]  time = 0.0918306, size = 72, normalized size = 0.51 \[ \frac{1}{45} \left (\frac{\left (5 x^8-2 x^4-7\right ) x^3}{\sqrt{x^4+1}}+21 (-1)^{3/4} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-21 (-1)^{3/4} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/Sqrt[1 + x^4],x]

[Out]

((x^3*(-7 - 2*x^4 + 5*x^8))/Sqrt[1 + x^4] - 21*(-1)^(3/4)*EllipticE[I*ArcSinh[(-
1)^(1/4)*x], -1] + 21*(-1)^(3/4)*EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1])/45

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Maple [C]  time = 0.01, size = 107, normalized size = 0.8 \[{\frac{{x}^{7}}{9}\sqrt{{x}^{4}+1}}-{\frac{7\,{x}^{3}}{45}\sqrt{{x}^{4}+1}}+{\frac{{\frac{7\,i}{15}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(x^4+1)^(1/2),x)

[Out]

1/9*x^7*(x^4+1)^(1/2)-7/45*x^3*(x^4+1)^(1/2)+7/15*I/(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)-EllipticE(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. \[ \int \frac{x^{10}}{\sqrt{x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^10/sqrt(x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{10}}{\sqrt{x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(x^4 + 1),x, algorithm="fricas")

[Out]

integral(x^10/sqrt(x^4 + 1), x)

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Sympy [A]  time = 3.52164, size = 29, normalized size = 0.21 \[ \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(x**4+1)**(1/2),x)

[Out]

x**11*gamma(11/4)*hyper((1/2, 11/4), (15/4,), x**4*exp_polar(I*pi))/(4*gamma(15/
4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{\sqrt{x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^10/sqrt(x^4 + 1), x)