Integrand size = 19, antiderivative size = 22 \[ \int \frac {\sqrt {x}}{\sqrt {(4-x) (-2+x)}} \, dx=-4 E\left (\arcsin \left (\frac {\sqrt {4-x}}{\sqrt {2}}\right )|\frac {1}{2}\right ) \] Output:
-4*EllipticE(1/2*(4-x)^(1/2)*2^(1/2),1/2*2^(1/2))
Result contains complex when optimal does not.
Time = 3.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.50 \[ \int \frac {\sqrt {x}}{\sqrt {(4-x) (-2+x)}} \, dx=\frac {2 \sqrt {\frac {-4+x}{x}} \sqrt {\frac {x}{-4+x}} \left (\sqrt {-4+x} (-2+x) \sqrt {\frac {x}{-4+x}}+2 i (-4+x) \sqrt {\frac {-2+x}{-4+x}} E\left (i \text {arcsinh}\left (\frac {2}{\sqrt {-4+x}}\right )|\frac {1}{2}\right )\right )}{\sqrt {-8+6 x-x^2}} \] Input:
Integrate[Sqrt[x]/Sqrt[(4 - x)*(-2 + x)],x]
Output:
(2*Sqrt[(-4 + x)/x]*Sqrt[x/(-4 + x)]*(Sqrt[-4 + x]*(-2 + x)*Sqrt[x/(-4 + x )] + (2*I)*(-4 + x)*Sqrt[(-2 + x)/(-4 + x)]*EllipticE[I*ArcSinh[2/Sqrt[-4 + x]], 1/2]))/Sqrt[-8 + 6*x - x^2]
Time = 0.41 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2035, 2048, 1452, 27, 389, 322, 328}
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.
\(\displaystyle \int \frac {\sqrt {x}}{\sqrt {(4-x) (x-2)}} \, dx\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle 2 \int \frac {x}{\sqrt {-((2-x) (4-x))}}d\sqrt {x}\) |
\(\Big \downarrow \) 2048 |
\(\displaystyle 2 \int \frac {x}{\sqrt {-x^2+6 x-8}}d\sqrt {x}\) |
\(\Big \downarrow \) 1452 |
\(\displaystyle 4 \int \frac {x}{2 \sqrt {4-x} \sqrt {x-2}}d\sqrt {x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {x}{\sqrt {4-x} \sqrt {x-2}}d\sqrt {x}\) |
\(\Big \downarrow \) 389 |
\(\displaystyle 2 \left (2 \int \frac {1}{\sqrt {4-x} \sqrt {x-2}}d\sqrt {x}+\int \frac {\sqrt {x-2}}{\sqrt {4-x}}d\sqrt {x}\right )\) |
\(\Big \downarrow \) 322 |
\(\displaystyle 2 \left (\int \frac {\sqrt {x-2}}{\sqrt {4-x}}d\sqrt {x}-\sqrt {2} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt {x}}{2}\right ),2\right )\right )\) |
\(\Big \downarrow \) 328 |
\(\displaystyle 2 \left (-\sqrt {2} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt {x}}{2}\right ),2\right )-\sqrt {2} E\left (\left .\arccos \left (\frac {\sqrt {x}}{2}\right )\right |2\right )\right )\) |
Input:
Int[Sqrt[x]/Sqrt[(4 - x)*(-2 + x)],x]
Output:
2*(-(Sqrt[2]*EllipticE[ArcCos[Sqrt[x]/2], 2]) - Sqrt[2]*EllipticF[ArcCos[S qrt[x]/2], 2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(-(Sqrt[c]*Rt[-d/c, 2]*Sqrt[a - b*(c/d)])^(-1))*EllipticF[ArcCos[Rt[-d/ c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a - b*(c/d), 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (-Sqrt[a - b*(c/d)]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcCos[Rt[-d/c, 2]*x], b*(c/(b*c - a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a - b*(c/d), 0]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[1/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] - Simp[a/b Int [1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && N eQ[b*c - a*d, 0] && !SimplerSqrtQ[-b/a, -d/c]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*Sqrt[-c] Int[x^2/(Sqrt[b + q + 2*c*x^2]*Sqrt [-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) , x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F reeQ[{a, b, c, d, e, n, p}, x]
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55
method | result | size |
default | \(\frac {\left (\operatorname {EllipticE}\left (\frac {\sqrt {-2 x +4}}{2}, i\right )-2 \operatorname {EllipticF}\left (\frac {\sqrt {-2 x +4}}{2}, i\right )\right ) \sqrt {2}\, \sqrt {-2 x +8}\, \sqrt {-2 x +4}}{\sqrt {-\left (x -2\right ) \left (x -4\right )}}\) | \(56\) |
elliptic | \(-\frac {\sqrt {-\left (x -2\right ) \left (x -4\right ) x}\, \sqrt {-2 x +4}\, \sqrt {-2 x +8}\, \sqrt {2}\, \left (-2 \operatorname {EllipticE}\left (\frac {\sqrt {-2 x +4}}{2}, i\right )+4 \operatorname {EllipticF}\left (\frac {\sqrt {-2 x +4}}{2}, i\right )\right )}{2 \sqrt {-\left (x -2\right ) \left (x -4\right )}\, \sqrt {-x^{3}+6 x^{2}-8 x}}\) | \(86\) |
Input:
int(x^(1/2)/((4-x)*(x-2))^(1/2),x,method=_RETURNVERBOSE)
Output:
(EllipticE(1/2*(-2*x+4)^(1/2),I)-2*EllipticF(1/2*(-2*x+4)^(1/2),I))*2^(1/2 )*(-2*x+8)^(1/2)*(-2*x+4)^(1/2)/(-(x-2)*(x-4))^(1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x}}{\sqrt {(4-x) (-2+x)}} \, dx=-4 i \, {\rm weierstrassPInverse}\left (16, 0, x - 2\right ) + 2 i \, {\rm weierstrassZeta}\left (16, 0, {\rm weierstrassPInverse}\left (16, 0, x - 2\right )\right ) \] Input:
integrate(x^(1/2)/((4-x)*(-2+x))^(1/2),x, algorithm="fricas")
Output:
-4*I*weierstrassPInverse(16, 0, x - 2) + 2*I*weierstrassZeta(16, 0, weiers trassPInverse(16, 0, x - 2))
\[ \int \frac {\sqrt {x}}{\sqrt {(4-x) (-2+x)}} \, dx=\int \frac {\sqrt {x}}{\sqrt {- \left (x - 4\right ) \left (x - 2\right )}}\, dx \] Input:
integrate(x**(1/2)/((4-x)*(-2+x))**(1/2),x)
Output:
Integral(sqrt(x)/sqrt(-(x - 4)*(x - 2)), x)
\[ \int \frac {\sqrt {x}}{\sqrt {(4-x) (-2+x)}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {-{\left (x - 2\right )} {\left (x - 4\right )}}} \,d x } \] Input:
integrate(x^(1/2)/((4-x)*(-2+x))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x)/sqrt(-(x - 2)*(x - 4)), x)
\[ \int \frac {\sqrt {x}}{\sqrt {(4-x) (-2+x)}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {-{\left (x - 2\right )} {\left (x - 4\right )}}} \,d x } \] Input:
integrate(x^(1/2)/((4-x)*(-2+x))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(x)/sqrt(-(x - 2)*(x - 4)), x)
Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {(4-x) (-2+x)}} \, dx=\int \frac {\sqrt {x}}{\sqrt {-\left (x-2\right )\,\left (x-4\right )}} \,d x \] Input:
int(x^(1/2)/(-(x - 2)*(x - 4))^(1/2),x)
Output:
int(x^(1/2)/(-(x - 2)*(x - 4))^(1/2), x)
\[ \int \frac {\sqrt {x}}{\sqrt {(4-x) (-2+x)}} \, dx=-\left (\int \frac {\sqrt {x}\, \sqrt {-x^{2}+6 x -8}}{x^{2}-6 x +8}d x \right ) \] Input:
int(x^(1/2)/((4-x)*(-2+x))^(1/2),x)
Output:
- int((sqrt(x)*sqrt( - x**2 + 6*x - 8))/(x**2 - 6*x + 8),x)