Optimal. Leaf size=202 \[ -\frac{x^8}{3 \sqrt{x^6+2}}+\frac{8}{15} \sqrt{x^6+2} x^2-\frac{16\ 2^{5/6} \sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{15 \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.232819, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{x^8}{3 \sqrt{x^6+2}}+\frac{8}{15} \sqrt{x^6+2} x^2-\frac{16\ 2^{5/6} \sqrt{2+\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{15 \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
[In] Int[x^13/(2 + x^6)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.49002, size = 190, normalized size = 0.94 \[ - \frac{x^{8}}{3 \sqrt{x^{6} + 2}} + \frac{8 x^{2} \sqrt{x^{6} + 2}}{15} - \frac{16 \cdot 3^{\frac{3}{4}} \sqrt{\frac{2 \sqrt [3]{2} x^{4} - 2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (2 x^{2} + 2 \sqrt [3]{2}\right ) F\left (\operatorname{asin}{\left (\frac{2^{\frac{2}{3}} x^{2} - 2 \sqrt{3} + 2}{2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{45 \sqrt{\frac{2 \cdot 2^{\frac{2}{3}} x^{2} + 4}{\left (2^{\frac{2}{3}} x^{2} + 2 + 2 \sqrt{3}\right )^{2}}} \sqrt{x^{6} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**13/(x**6+2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.231441, size = 144, normalized size = 0.71 \[ \frac{9 x^8+48 x^2-16 \sqrt [6]{-1} \sqrt [3]{2} 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (2^{2/3} x^2+2 (-1)^{2/3}\right )} \sqrt{(-1)^{2/3} \sqrt [3]{2} x^4+\sqrt [3]{-1} 2^{2/3} x^2+2} F\left (\sin ^{-1}\left (\frac{\sqrt{\left (-i+\sqrt{3}\right ) \left (2^{2/3} x^2+2\right )}}{2 \sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{45 \sqrt{x^6+2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^13/(2 + x^6)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.036, size = 40, normalized size = 0.2 \[{\frac{{x}^{2} \left ( 3\,{x}^{6}+16 \right ) }{15}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\frac{8\,{x}^{2}\sqrt{2}}{15}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{2}};\,{\frac{4}{3}};\,-{\frac{{x}^{6}}{2}})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^13/(x^6+2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(x^6 + 2)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{13}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(x^6 + 2)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 7.36117, size = 36, normalized size = 0.18 \[ \frac{\sqrt{2} x^{14} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{10}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**13/(x**6+2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(x^6 + 2)^(3/2),x, algorithm="giac")
[Out]