3.1428 \(\int \frac{x^3}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=378 \[ \frac{x^4}{6 \sqrt{x^6+2}}-\frac{\sqrt{x^6+2}}{6 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}+\frac{\sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{2\ 2^{5/6} 3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

[Out]

x^4/(6*Sqrt[2 + x^6]) - Sqrt[2 + x^6]/(6*(2^(1/3)*(1 + Sqrt[3]) + x^2)) + (Sqrt[
2 - Sqrt[3]]*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/(2^(1/3)*(1 + Sq
rt[3]) + x^2)^2]*EllipticE[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sq
rt[3]) + x^2)], -7 - 4*Sqrt[3]])/(2*2^(5/6)*3^(3/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3
)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6]) - ((2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1
/3)*x^2 + x^4)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*EllipticF[ArcSin[(2^(1/3)*(1 - S
qrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(3*2^(1/3)*3^(1/
4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6])

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Rubi [A]  time = 0.431473, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{x^4}{6 \sqrt{x^6+2}}-\frac{\sqrt{x^6+2}}{6 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{\left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [3]{2} \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}+\frac{\sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{2\ 2^{5/6} 3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(2 + x^6)^(3/2),x]

[Out]

x^4/(6*Sqrt[2 + x^6]) - Sqrt[2 + x^6]/(6*(2^(1/3)*(1 + Sqrt[3]) + x^2)) + (Sqrt[
2 - Sqrt[3]]*(2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1/3)*x^2 + x^4)/(2^(1/3)*(1 + Sq
rt[3]) + x^2)^2]*EllipticE[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1/3)*(1 + Sq
rt[3]) + x^2)], -7 - 4*Sqrt[3]])/(2*2^(5/6)*3^(3/4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3
)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6]) - ((2^(1/3) + x^2)*Sqrt[(2^(2/3) - 2^(1
/3)*x^2 + x^4)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*EllipticF[ArcSin[(2^(1/3)*(1 - S
qrt[3]) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(3*2^(1/3)*3^(1/
4)*Sqrt[(2^(1/3) + x^2)/(2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**6+2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [C]  time = 0.833944, size = 189, normalized size = 0.5 \[ \frac{x^4}{6 \sqrt{x^6+2}}+\frac{i \sqrt{(-1)^{5/6} \left (\sqrt [3]{-\frac{1}{2}} x^2-1\right )} \sqrt{\left (-\frac{1}{2}\right )^{2/3} x^4+\sqrt [3]{-\frac{1}{2}} x^2+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{(-1)^{5/6} x^2}{\sqrt [3]{2}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{(-1)^{5/6} x^2}{\sqrt [3]{2}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3 \sqrt [3]{2} \sqrt [4]{3} \sqrt{x^6+2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^3/(2 + x^6)^(3/2),x]

[Out]

x^4/(6*Sqrt[2 + x^6]) + ((I/3)*Sqrt[(-1)^(5/6)*(-1 + (-1/2)^(1/3)*x^2)]*Sqrt[1 +
(-1/2)^(1/3)*x^2 + (-1/2)^(2/3)*x^4]*((-I)*Sqrt[3]*EllipticE[ArcSin[Sqrt[-(-1)^
(5/6) - ((-1)^(5/6)*x^2)/2^(1/3)]/3^(1/4)], (-1)^(1/3)] + (-1)^(1/3)*EllipticF[A
rcSin[Sqrt[-(-1)^(5/6) - ((-1)^(5/6)*x^2)/2^(1/3)]/3^(1/4)], (-1)^(1/3)]))/(2^(1
/3)*3^(1/4)*Sqrt[2 + x^6])

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Maple [C]  time = 0.035, size = 33, normalized size = 0.1 \[{\frac{{x}^{4}}{6}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{{x}^{4}\sqrt{2}}{48}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{2}{3}};\,{\frac{5}{3}};\,-{\frac{{x}^{6}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^6+2)^(3/2),x)

[Out]

1/6*x^4/(x^6+2)^(1/2)-1/48*2^(1/2)*x^4*hypergeom([1/2,2/3],[5/3],-1/2*x^6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^3/(x^6 + 2)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{3}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

integral(x^3/(x^6 + 2)^(3/2), x)

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Sympy [A]  time = 1.97544, size = 36, normalized size = 0.1 \[ \frac{\sqrt{2} x^{4} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{3}{2} \\ \frac{5}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**6+2)**(3/2),x)

[Out]

sqrt(2)*x**4*gamma(2/3)*hyper((2/3, 3/2), (5/3,), x**6*exp_polar(I*pi)/2)/(24*ga
mma(5/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^3/(x^6 + 2)^(3/2), x)