Integrand size = 13, antiderivative size = 354 \[ \int \frac {x^3}{\sqrt {2+x^6}} \, dx=\frac {\sqrt {2+x^6}}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{2^{5/6} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}+\frac {2^{2/3} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}} \]
(x^6+2)^(1/2)/(x^2+2^(1/3)*(1+3^(1/2)))+1/3*2^(2/3)*(2^(1/3)+x^2)*Elliptic F((x^2+2^(1/3)*(1-3^(1/2)))/(x^2+2^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*((2^( 2/3)-2^(1/3)*x^2+x^4)/(x^2+2^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(3/4)/(x^6+2)^( 1/2)/((2^(1/3)+x^2)/(x^2+2^(1/3)*(1+3^(1/2)))^2)^(1/2)-1/2*2^(1/6)*3^(1/4) *(2^(1/3)+x^2)*EllipticE((x^2+2^(1/3)*(1-3^(1/2)))/(x^2+2^(1/3)*(1+3^(1/2) )),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((2^(2/3)-2^(1/3)*x^2+x^4)/(x^ 2+2^(1/3)*(1+3^(1/2)))^2)^(1/2)/(x^6+2)^(1/2)/((2^(1/3)+x^2)/(x^2+2^(1/3)* (1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.08 \[ \int \frac {x^3}{\sqrt {2+x^6}} \, dx=\frac {x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {x^6}{2}\right )}{4 \sqrt {2}} \]
Time = 0.42 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {807, 832, 759, 2416}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^3}{\sqrt {x^6+2}} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {x^2}{\sqrt {x^6+2}}dx^2\) |
\(\Big \downarrow \) 832 |
\(\displaystyle \frac {1}{2} \left (\int \frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{\sqrt {x^6+2}}dx^2-\sqrt [3]{2} \left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^6+2}}dx^2\right )\) |
\(\Big \downarrow \) 759 |
\(\displaystyle \frac {1}{2} \left (\int \frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{\sqrt {x^6+2}}dx^2-\frac {2 \sqrt [6]{2} \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}\right )\) |
\(\Big \downarrow \) 2416 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 \sqrt [6]{2} \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt [6]{2} \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {2 \sqrt {x^6+2}}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )\) |
((2*Sqrt[2 + x^6])/(2^(1/3)*(1 + Sqrt[3]) + x^2) - (2^(1/6)*3^(1/4)*Sqrt[2 - Sqrt[3]]*(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]*EllipticE[ArcSin[(2^(1/3)*(1 - Sqrt[3]) + x^2)/(2^(1 /3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(Sqrt[(2^(1/3) + x^2)/(2^(1/3) *(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6]) - (2*2^(1/6)*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*(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 - Sqrt[3]) + x^2)/(2^(1/ 3)*(1 + Sqrt[3]) + x^2)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(2^(1/3) + x^2)/( 2^(1/3)*(1 + Sqrt[3]) + x^2)^2]*Sqrt[2 + x^6]))/2
3.15.3.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 5.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.06
method | result | size |
meijerg | \(\frac {\sqrt {2}\, x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {x^{6}}{2}\right )}{8}\) | \(20\) |
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.03 \[ \int \frac {x^3}{\sqrt {2+x^6}} \, dx=-{\rm weierstrassZeta}\left (0, -8, {\rm weierstrassPInverse}\left (0, -8, x^{2}\right )\right ) \]
Time = 0.39 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.10 \[ \int \frac {x^3}{\sqrt {2+x^6}} \, dx=\frac {\sqrt {2} x^{4} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{12 \Gamma \left (\frac {5}{3}\right )} \]
\[ \int \frac {x^3}{\sqrt {2+x^6}} \, dx=\int { \frac {x^{3}}{\sqrt {x^{6} + 2}} \,d x } \]
\[ \int \frac {x^3}{\sqrt {2+x^6}} \, dx=\int { \frac {x^{3}}{\sqrt {x^{6} + 2}} \,d x } \]
Timed out. \[ \int \frac {x^3}{\sqrt {2+x^6}} \, dx=\int \frac {x^3}{\sqrt {x^6+2}} \,d x \]