Integrand size = 15, antiderivative size = 264 \[ \int \frac {x^4}{\sqrt {-1-x^3}} \, dx=-\frac {2}{7} x^2 \sqrt {-1-x^3}+\frac {8 \sqrt {-1-x^3}}{7 \left (1-\sqrt {3}+x\right )}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right )|-7+4 \sqrt {3}\right )}{7 \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}}+\frac {8 \sqrt {2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{7 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \] Output:
-2/7*x^2*(-x^3-1)^(1/2)+8*(-x^3-1)^(1/2)/(7-7*3^(1/2)+7*x)-4/7*3^(1/4)*(1/ 2*6^(1/2)+1/2*2^(1/2))*(1+x)*((x^2-x+1)/(1+x-3^(1/2))^2)^(1/2)*EllipticE(( 1+x+3^(1/2))/(1+x-3^(1/2)),2*I-I*3^(1/2))/(-(1+x)/(1+x-3^(1/2))^2)^(1/2)/( -x^3-1)^(1/2)+8/21*2^(1/2)*(1+x)*((x^2-x+1)/(1+x-3^(1/2))^2)^(1/2)*Ellipti cF((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 = 10.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.19 \[ \int \frac {x^4}{\sqrt {-1-x^3}} \, dx=\frac {2 x^2 \left (1+x^3-\sqrt {1+x^3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-x^3\right )\right )}{7 \sqrt {-1-x^3}} \] Input:
Integrate[x^4/Sqrt[-1 - x^3],x]
Output:
(2*x^2*(1 + x^3 - Sqrt[1 + x^3]*Hypergeometric2F1[1/2, 2/3, 5/3, -x^3]))/( 7*Sqrt[-1 - x^3])
Time = 0.60 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {843, 833, 760, 2418}
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.
| \(\displaystyle \int \frac {x^4}{\sqrt {-x^3-1}} \, dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle -\frac {4}{7} \int \frac {x}{\sqrt {-x^3-1}}dx-\frac {2}{7} \sqrt {-x^3-1} x^2\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle -\frac {4}{7} \left (\int \frac {x+\sqrt {3}+1}{\sqrt {-x^3-1}}dx-\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {-x^3-1}}dx\right )-\frac {2}{7} \sqrt {-x^3-1} x^2\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle -\frac {4}{7} \left (\int \frac {x+\sqrt {3}+1}{\sqrt {-x^3-1}}dx-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\right )-\frac {2}{7} \sqrt {-x^3-1} x^2\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle -\frac {4}{7} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}-\frac {2 \sqrt {-x^3-1}}{x-\sqrt {3}+1}\right )-\frac {2}{7} \sqrt {-x^3-1} x^2\) |
Input:
Int[x^4/Sqrt[-1 - x^3],x]
Output:
(-2*x^2*Sqrt[-1 - x^3])/7 - (4*((-2*Sqrt[-1 - x^3])/(1 - Sqrt[3] + x) + (3 ^(1/4)*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*E llipticE[ArcSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(Sq rt[-((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3]) - (2*Sqrt[2 - Sqrt[3]]* (1 + Sqrt[3])*(1 + x)*Sqrt[(1 - x + x^2)/(1 - Sqrt[3] + x)^2]*EllipticF[Ar cSin[(1 + Sqrt[3] + x)/(1 - Sqrt[3] + x)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[ -((1 + x)/(1 - Sqrt[3] + x)^2)]*Sqrt[-1 - x^3])))/7
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[(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] && 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]
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 - S qrt[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] && NegQ[a] && EqQ[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 = 0.84 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.07
| method | result | size |
| meijerg | \(-\frac {i x^{5} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x^{3}\right )}{5}\) | \(18\) |
| default | \(-\frac {2 x^{2} \sqrt {-x^{3}-1}}{7}+\frac {8 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{21 \sqrt {-x^{3}-1}}\) | \(175\) |
| elliptic | \(-\frac {2 x^{2} \sqrt {-x^{3}-1}}{7}+\frac {8 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{21 \sqrt {-x^{3}-1}}\) | \(175\) |
| risch | \(\frac {2 x^{2} \left (x^{3}+1\right )}{7 \sqrt {-x^{3}-1}}+\frac {8 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {1+x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )-\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )\right )}{21 \sqrt {-x^{3}-1}}\) | \(180\) |
Input:
int(x^4/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/5*I*x^5*hypergeom([1/2,5/3],[8/3],-x^3)
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.09 \[ \int \frac {x^4}{\sqrt {-1-x^3}} \, dx=-\frac {2}{7} \, \sqrt {-x^{3} - 1} x^{2} - \frac {8}{7} i \, {\rm weierstrassZeta}\left (0, -4, {\rm weierstrassPInverse}\left (0, -4, x\right )\right ) \] Input:
integrate(x^4/(-x^3-1)^(1/2),x, algorithm="fricas")
Output:
-2/7*sqrt(-x^3 - 1)*x^2 - 8/7*I*weierstrassZeta(0, -4, weierstrassPInverse (0, -4, x))
Time = 0.45 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.12 \[ \int \frac {x^4}{\sqrt {-1-x^3}} \, dx=- \frac {i x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} \] Input:
integrate(x**4/(-x**3-1)**(1/2),x)
Output:
-I*x**5*gamma(5/3)*hyper((1/2, 5/3), (8/3,), x**3*exp_polar(I*pi))/(3*gamm a(8/3))
\[ \int \frac {x^4}{\sqrt {-1-x^3}} \, dx=\int { \frac {x^{4}}{\sqrt {-x^{3} - 1}} \,d x } \] Input:
integrate(x^4/(-x^3-1)^(1/2),x, algorithm="maxima")
Output:
integrate(x^4/sqrt(-x^3 - 1), x)
\[ \int \frac {x^4}{\sqrt {-1-x^3}} \, dx=\int { \frac {x^{4}}{\sqrt {-x^{3} - 1}} \,d x } \] Input:
integrate(x^4/(-x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate(x^4/sqrt(-x^3 - 1), x)
Time = 0.07 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.93 \[ \int \frac {x^4}{\sqrt {-1-x^3}} \, dx=-\frac {2\,x^2\,\sqrt {-x^3-1}}{7}+\frac {8\,\left (\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\left (-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{7\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(x^4/(- x^3 - 1)^(1/2),x)
Output:
(8*(((3^(1/2)*1i)/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)) - ((3^(1/2)*1i)/2 - 3/2)*ellipticE(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)))*((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))/(7*(- 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) ) - (2*x^2*(- x^3 - 1)^(1/2))/7
\[ \int \frac {x^4}{\sqrt {-1-x^3}} \, dx=\frac {2 i \left (-\sqrt {x^{3}+1}\, x^{2}+2 \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{3}+1}d x \right )\right )}{7} \] Input:
int(x^4/(-x^3-1)^(1/2),x)
Output:
(2*i*( - sqrt(x**3 + 1)*x**2 + 2*int((sqrt(x**3 + 1)*x)/(x**3 + 1),x)))/7