\(\int \frac {\sqrt {x}}{\sqrt {-8+6 x-x^2}} \, dx\) [171]

Optimal result

Integrand size = 20, antiderivative size = 22 \[ \int \frac {\sqrt {x}}{\sqrt {-8+6 x-x^2}} \, dx=-4 E\left (\arcsin \left (\frac {\sqrt {4-x}}{\sqrt {2}}\right )|\frac {1}{2}\right ) \] Output:

-4*EllipticE(1/2*(4-x)^(1/2)*2^(1/2),1/2*2^(1/2))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(22)=44\).

Time = 20.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14 \[ \int \frac {\sqrt {x}}{\sqrt {-8+6 x-x^2}} \, dx=-\frac {4 \sqrt {2-x} \sqrt {4-x} \left (E\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {2}}\right )|\frac {1}{2}\right )-\operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {x}}{\sqrt {2}}\right ),\frac {1}{2}\right )\right )}{\sqrt {-8+6 x-x^2}} \] Input:

Integrate[Sqrt[x]/Sqrt[-8 + 6*x - x^2],x]
 

Output:

(-4*Sqrt[2 - x]*Sqrt[4 - x]*(EllipticE[ArcSin[Sqrt[x]/Sqrt[2]], 1/2] - Ell 
ipticF[ArcSin[Sqrt[x]/Sqrt[2]], 1/2]))/Sqrt[-8 + 6*x - x^2]
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1170, 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.

\(\displaystyle \int \frac {\sqrt {x}}{\sqrt {-x^2+6 x-8}} \, dx\)

\(\Big \downarrow \) 1170

\(\displaystyle 2 \int \frac {x}{\sqrt {-x^2+6 x-8}}d\sqrt {x}\)

\(\Big \downarrow \) 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[-8 + 6*x - x^2],x]
 

Output:

2*(-(Sqrt[2]*EllipticE[ArcCos[Sqrt[x]/2], 2]) - Sqrt[2]*EllipticF[ArcCos[S 
qrt[x]/2], 2])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 322

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]
 

rule 328

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]
 

rule 389

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]
 

rule 1170

Int[(x_)^(m_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 
 Subst[Int[x^(2*m + 1)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; Free 
Q[{a, b, c}, x] && EqQ[m^2, 1/4]
 

rule 1452

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.61 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64

method	result	size
default	\(\frac {\left (\operatorname {EllipticE}\left (\frac {\sqrt {-2 x +4}}{2}, i\right )-2 \operatorname {EllipticF}\left (\frac {\sqrt {-2 x +4}}{2}, i\right )\right ) \sqrt {-2 x +4}\, \sqrt {-2 x +8}\, \sqrt {2}}{\sqrt {-x^{2}+6 x -8}}\)	\(58\)
elliptic	\(-\frac {\sqrt {-\left (x^{2}-6 x +8\right ) x}\, \sqrt {-2 x +4}\, \sqrt {-2 x +8}\, \sqrt {2}\, \left (-2 \operatorname {EllipticE}\left (\frac {\sqrt {-2 x +4}}{2}, i\right )+4 \operatorname {EllipticF}\left (\frac {\sqrt {-2 x +4}}{2}, i\right )\right )}{2 \sqrt {-x^{2}+6 x -8}\, \sqrt {-x^{3}+6 x^{2}-8 x}}\)	\(90\)

Input:

int(x^(1/2)/(-x^2+6*x-8)^(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*x+ 
4)^(1/2)*(-2*x+8)^(1/2)*2^(1/2)/(-x^2+6*x-8)^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x}}{\sqrt {-8+6 x-x^2}} \, dx=-4 i \, {\rm weierstrassPInverse}\left (16, 0, x - 2\right ) + 2 i \, {\rm weierstrassZeta}\left (16, 0, {\rm weierstrassPInverse}\left (16, 0, x - 2\right )\right ) \] Input:

integrate(x^(1/2)/(-x^2+6*x-8)^(1/2),x, algorithm="fricas")
 

Output:

-4*I*weierstrassPInverse(16, 0, x - 2) + 2*I*weierstrassZeta(16, 0, weiers 
trassPInverse(16, 0, x - 2))
 

Sympy [F]

\[ \int \frac {\sqrt {x}}{\sqrt {-8+6 x-x^2}} \, dx=\int \frac {\sqrt {x}}{\sqrt {- \left (x - 4\right ) \left (x - 2\right )}}\, dx \] Input:

integrate(x**(1/2)/(-x**2+6*x-8)**(1/2),x)
 

Output:

Integral(sqrt(x)/sqrt(-(x - 4)*(x - 2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {x}}{\sqrt {-8+6 x-x^2}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {-x^{2} + 6 \, x - 8}} \,d x } \] Input:

integrate(x^(1/2)/(-x^2+6*x-8)^(1/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate(sqrt(x)/sqrt(-x^2 + 6*x - 8), x)
 

Giac [F]

\[ \int \frac {\sqrt {x}}{\sqrt {-8+6 x-x^2}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {-x^{2} + 6 \, x - 8}} \,d x } \] Input:

integrate(x^(1/2)/(-x^2+6*x-8)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(x)/sqrt(-x^2 + 6*x - 8), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {-8+6 x-x^2}} \, dx=\int \frac {\sqrt {x}}{\sqrt {-x^2+6\,x-8}} \,d x \] Input:

int(x^(1/2)/(6*x - x^2 - 8)^(1/2),x)
 

Output:

int(x^(1/2)/(6*x - x^2 - 8)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {x}}{\sqrt {-8+6 x-x^2}} \, dx=-\left (\int \frac {\sqrt {x}\, \sqrt {-x^{2}+6 x -8}}{x^{2}-6 x +8}d x \right ) \] Input:

int(x^(1/2)/(-x^2+6*x-8)^(1/2),x)
 

Output:

 - int((sqrt(x)*sqrt( - x**2 + 6*x - 8))/(x**2 - 6*x + 8),x)