Integrand size = 15, antiderivative size = 149 \[ \int \frac {x^6}{\sqrt {-1-x^3}} \, dx=\frac {16}{55} x \sqrt {-1-x^3}-\frac {2}{11} x^4 \sqrt {-1-x^3}+\frac {32 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{55 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \] Output:
16/55*x*(-x^3-1)^(1/2)-2/11*x^4*(-x^3-1)^(1/2)+32/165*(1/2*6^(1/2)-1/2*2^( 1/2))*(1+x)*((x^2-x+1)/(1+x-3^(1/2))^2)^(1/2)*EllipticF((1+x+3^(1/2))/(1+x -3^(1/2)),2*I-I*3^(1/2))*3^(3/4)/(-(1+x)/(1+x-3^(1/2))^2)^(1/2)/(-x^3-1)^( 1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.36 \[ \int \frac {x^6}{\sqrt {-1-x^3}} \, dx=\frac {2 x \left (-8-3 x^3+5 x^6+8 \sqrt {1+x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-x^3\right )\right )}{55 \sqrt {-1-x^3}} \] Input:
Integrate[x^6/Sqrt[-1 - x^3],x]
Output:
(2*x*(-8 - 3*x^3 + 5*x^6 + 8*Sqrt[1 + x^3]*Hypergeometric2F1[1/3, 1/2, 4/3 , -x^3]))/(55*Sqrt[-1 - x^3])
Time = 0.40 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {843, 843, 760}
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.
\(\displaystyle \int \frac {x^6}{\sqrt {-x^3-1}} \, dx\) |
\(\Big \downarrow \) 843 |
\(\displaystyle -\frac {8}{11} \int \frac {x^3}{\sqrt {-x^3-1}}dx-\frac {2}{11} \sqrt {-x^3-1} x^4\) |
\(\Big \downarrow \) 843 |
\(\displaystyle -\frac {8}{11} \left (-\frac {2}{5} \int \frac {1}{\sqrt {-x^3-1}}dx-\frac {2}{5} \sqrt {-x^3-1} x\right )-\frac {2}{11} \sqrt {-x^3-1} x^4\) |
\(\Big \downarrow \) 760 |
\(\displaystyle -\frac {8}{11} \left (-\frac {4 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{5 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {2}{5} \sqrt {-x^3-1} x\right )-\frac {2}{11} \sqrt {-x^3-1} x^4\) |
Input:
Int[x^6/Sqrt[-1 - x^3],x]
Output:
(-2*x^4*Sqrt[-1 - x^3])/11 - (8*((-2*x*Sqrt[-1 - x^3])/5 - (4*Sqrt[2 - Sqr t[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(5*3^(1/4)*Sqrt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])))/11
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 ] && NegQ[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]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 0.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.12
method | result | size |
meijerg | \(-\frac {i x^{7} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {7}{3}\right ], \left [\frac {10}{3}\right ], -x^{3}\right )}{7}\) | \(18\) |
risch | \(\frac {2 x \left (5 x^{3}-8\right ) \left (x^{3}+1\right )}{55 \sqrt {-x^{3}-1}}-\frac {32 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{165 \sqrt {-x^{3}-1}}\) | \(132\) |
default | \(-\frac {2 x^{4} \sqrt {-x^{3}-1}}{11}+\frac {16 x \sqrt {-x^{3}-1}}{55}-\frac {32 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{165 \sqrt {-x^{3}-1}}\) | \(134\) |
elliptic | \(-\frac {2 x^{4} \sqrt {-x^{3}-1}}{11}+\frac {16 x \sqrt {-x^{3}-1}}{55}-\frac {32 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{165 \sqrt {-x^{3}-1}}\) | \(134\) |
Input:
int(x^6/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/7*I*x^7*hypergeom([1/2,7/3],[10/3],-x^3)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {x^6}{\sqrt {-1-x^3}} \, dx=-\frac {2}{55} \, {\left (5 \, x^{4} - 8 \, x\right )} \sqrt {-x^{3} - 1} - \frac {32}{55} i \, {\rm weierstrassPInverse}\left (0, -4, x\right ) \] Input:
integrate(x^6/(-x^3-1)^(1/2),x, algorithm="fricas")
Output:
-2/55*(5*x^4 - 8*x)*sqrt(-x^3 - 1) - 32/55*I*weierstrassPInverse(0, -4, x)
Time = 0.46 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.21 \[ \int \frac {x^6}{\sqrt {-1-x^3}} \, dx=- \frac {i x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \] Input:
integrate(x**6/(-x**3-1)**(1/2),x)
Output:
-I*x**7*gamma(7/3)*hyper((1/2, 7/3), (10/3,), x**3*exp_polar(I*pi))/(3*gam ma(10/3))
\[ \int \frac {x^6}{\sqrt {-1-x^3}} \, dx=\int { \frac {x^{6}}{\sqrt {-x^{3} - 1}} \,d x } \] Input:
integrate(x^6/(-x^3-1)^(1/2),x, algorithm="maxima")
Output:
integrate(x^6/sqrt(-x^3 - 1), x)
\[ \int \frac {x^6}{\sqrt {-1-x^3}} \, dx=\int { \frac {x^{6}}{\sqrt {-x^{3} - 1}} \,d x } \] Input:
integrate(x^6/(-x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate(x^6/sqrt(-x^3 - 1), x)
Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.34 \[ \int \frac {x^6}{\sqrt {-1-x^3}} \, dx=\frac {16\,x\,\sqrt {-x^3-1}}{55}-\frac {2\,x^4\,\sqrt {-x^3-1}}{11}+\frac {32\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{55\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(x^6/(- x^3 - 1)^(1/2),x)
Output:
(16*x*(- x^3 - 1)^(1/2))/55 - (2*x^4*(- x^3 - 1)^(1/2))/11 + (32*((3^(1/2) *1i)/2 + 3/2)*(x^3 + 1)^(1/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticF(asin(((x + 1)/((3^(1/2)*1i) /2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(55*(- x^3 - 1)^(1/2)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))
\[ \int \frac {x^6}{\sqrt {-1-x^3}} \, dx=\frac {2 i \left (-5 \sqrt {x^{3}+1}\, x^{4}+8 \sqrt {x^{3}+1}\, x -8 \left (\int \frac {\sqrt {x^{3}+1}}{x^{3}+1}d x \right )\right )}{55} \] Input:
int(x^6/(-x^3-1)^(1/2),x)
Output:
(2*i*( - 5*sqrt(x**3 + 1)*x**4 + 8*sqrt(x**3 + 1)*x - 8*int(sqrt(x**3 + 1) /(x**3 + 1),x)))/55