Integrand size = 13, antiderivative size = 394 \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=-\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {10}{21} x^4 \sqrt {2+x^6}-\frac {80 \sqrt {2+x^6}}{21 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}+\frac {40 \sqrt [6]{2} \sqrt {2-\sqrt {3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right )|-7-4 \sqrt {3}\right )}{7\ 3^{3/4} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}}-\frac {80\ 2^{2/3} \left (\sqrt [3]{2}+x^2\right ) \sqrt {\frac {2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2}\right ),-7-4 \sqrt {3}\right )}{21 \sqrt [4]{3} \sqrt {\frac {\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )^2}} \sqrt {2+x^6}} \]
-1/3*x^10/(x^6+2)^(1/2)+10/21*x^4*(x^6+2)^(1/2)-80/21*(x^6+2)^(1/2)/(x^2+2 ^(1/3)*(1+3^(1/2)))-80/63*2^(2/3)*(2^(1/3)+x^2)*EllipticF((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)+40/21*2^(1/6)*3^(1/4)*(2^(1/3)+x^2)*E llipticE((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 = 8.99 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.14 \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\frac {x^4 \left (-20+x^6+10 \sqrt {2} \sqrt {2+x^6} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{2},\frac {5}{3},-\frac {x^6}{2}\right )\right )}{7 \sqrt {2+x^6}} \]
(x^4*(-20 + x^6 + 10*Sqrt[2]*Sqrt[2 + x^6]*Hypergeometric2F1[2/3, 3/2, 5/3 , -1/2*x^6]))/(7*Sqrt[2 + x^6])
Time = 0.44 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {807, 817, 843, 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.
| \(\displaystyle \int \frac {x^{15}}{\left (x^6+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {1}{2} \int \frac {x^{14}}{\left (x^6+2\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {1}{2} \left (\frac {10}{3} \int \frac {x^8}{\sqrt {x^6+2}}dx^2-\frac {2 x^{10}}{3 \sqrt {x^6+2}}\right )\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{2} \left (\frac {10}{3} \left (\frac {2}{7} x^4 \sqrt {x^6+2}-\frac {8}{7} \int \frac {x^2}{\sqrt {x^6+2}}dx^2\right )-\frac {2 x^{10}}{3 \sqrt {x^6+2}}\right )\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {1}{2} \left (\frac {10}{3} \left (\frac {2}{7} x^4 \sqrt {x^6+2}-\frac {8}{7} \left (\int \frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{\sqrt {x^6+2}}dx^2-\sqrt [3]{2} \left (1-\sqrt {3}\right ) \int \frac {1}{\sqrt {x^6+2}}dx^2\right )\right )-\frac {2 x^{10}}{3 \sqrt {x^6+2}}\right )\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {1}{2} \left (\frac {10}{3} \left (\frac {2}{7} x^4 \sqrt {x^6+2}-\frac {8}{7} \left (\int \frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{\sqrt {x^6+2}}dx^2-\frac {2 \sqrt [6]{2} \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}\right )\right )-\frac {2 x^{10}}{3 \sqrt {x^6+2}}\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {1}{2} \left (\frac {10}{3} \left (\frac {2}{7} x^4 \sqrt {x^6+2}-\frac {8}{7} \left (-\frac {2 \sqrt [6]{2} \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}-\frac {\sqrt [6]{2} \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\arcsin \left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {2 \sqrt {x^6+2}}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )\right )-\frac {2 x^{10}}{3 \sqrt {x^6+2}}\right )\) |
((-2*x^10)/(3*Sqrt[2 + x^6]) + (10*((2*x^4*Sqrt[2 + x^6])/7 - (8*((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 + Sqr t[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 + S qrt[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])))/7))/3)/2
3.15.28.3.1 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[((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*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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_.)*(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] - Simp[ 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]
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 = 7.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.05
| method | result | size |
| meijerg | \(\frac {\sqrt {2}\, x^{16} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{2},\frac {8}{3};\frac {11}{3};-\frac {x^{6}}{2}\right )}{64}\) | \(20\) |
| risch | \(\frac {x^{4} \left (3 x^{6}+20\right )}{21 \sqrt {x^{6}+2}}-\frac {10 \sqrt {2}\, x^{4} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {x^{6}}{2}\right )}{21}\) | \(40\) |
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.11 \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\frac {80 \, {\left (x^{6} + 2\right )} {\rm weierstrassZeta}\left (0, -8, {\rm weierstrassPInverse}\left (0, -8, x^{2}\right )\right ) + {\left (3 \, x^{10} + 20 \, x^{4}\right )} \sqrt {x^{6} + 2}}{21 \, {\left (x^{6} + 2\right )}} \]
1/21*(80*(x^6 + 2)*weierstrassZeta(0, -8, weierstrassPInverse(0, -8, x^2)) + (3*x^10 + 20*x^4)*sqrt(x^6 + 2))/(x^6 + 2)
Time = 0.76 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.09 \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\frac {\sqrt {2} x^{16} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac {11}{3}\right )} \]
sqrt(2)*x**16*gamma(8/3)*hyper((3/2, 8/3), (11/3,), x**6*exp_polar(I*pi)/2 )/(24*gamma(11/3))
\[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\int { \frac {x^{15}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\int { \frac {x^{15}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx=\int \frac {x^{15}}{{\left (x^6+2\right )}^{3/2}} \,d x \]