\(\int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (1+x)}} \, dx\) [535]

Optimal result

Integrand size = 33, antiderivative size = 21 \[ \int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (1+x)}} \, dx=-2 E\left (\left .\arcsin \left (\sqrt {x}\right )\right |-1\right )+4 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {x}\right ),-1\right ) \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.79 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.48 \[ \int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (1+x)}} \, dx=\frac {2 (-x)^{3/2} \sqrt {1-x^2} \left (-3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},x^2\right )+x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},x^2\right )\right )}{3 \sqrt {-((-1+x) x)} \sqrt {-x (1+x)}} \] Input:

Integrate[-((Sqrt[1 - x]*Sqrt[x])/(Sqrt[-x]*Sqrt[-(x*(1 + x))])),x]
 

Output:

(2*(-x)^(3/2)*Sqrt[1 - x^2]*(-3*Hypergeometric2F1[1/4, 1/2, 5/4, x^2] + x* 
Hypergeometric2F1[1/2, 3/4, 7/4, x^2]))/(3*Sqrt[-((-1 + x)*x)]*Sqrt[-(x*(1 
 + x))])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {25, 30, 2048, 1168, 120}

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.

\(\displaystyle \int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (x+1)}} \, dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (x+1)}}dx\)

\(\Big \downarrow \) 30

\(\displaystyle -\frac {\sqrt {x} \int \frac {\sqrt {1-x}}{\sqrt {-x (x+1)}}dx}{\sqrt {-x}}\)

\(\Big \downarrow \) 2048

\(\displaystyle -\frac {\sqrt {x} \int \frac {\sqrt {1-x}}{\sqrt {-x^2-x}}dx}{\sqrt {-x}}\)

\(\Big \downarrow \) 1168

\(\displaystyle -\frac {\sqrt {x} \int \frac {\sqrt {1-x}}{\sqrt {-x} \sqrt {x+1}}dx}{\sqrt {-x}}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {2 \sqrt {x} E\left (\left .\arcsin \left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {-x}}\)

Input:

Int[-((Sqrt[1 - x]*Sqrt[x])/(Sqrt[-x]*Sqrt[-(x*(1 + x))])),x]
 

Output:

(2*Sqrt[x]*EllipticE[ArcSin[Sqrt[-x]], -1])/Sqrt[-x]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 120

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- 
b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt 
Q[e, 0] &&  !LtQ[-b/d, 0]
 

rule 1168

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> 
 Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1 + (c/b)*x]), x] /; FreeQ[{b, c, d, e}, x 
] && NeQ[c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[c, 0] && RationalQ[b]
 

rule 2048

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]
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57

method	result	size
default	\(-\frac {2 \operatorname {EllipticE}\left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {x +1}\, \sqrt {x}}{\sqrt {-x \left (x +1\right )}}\)	\(33\)

Input:

int(-(1-x)^(1/2)*x^(1/2)/(-x)^(1/2)/(-x*(x+1))^(1/2),x,method=_RETURNVERBO 
SE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (1+x)}} \, dx=2 i \, {\rm weierstrassPInverse}\left (4, 0, x\right ) + 2 i \, {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, x\right )\right ) \] Input:

integrate(-(1-x)^(1/2)*x^(1/2)/(-x)^(1/2)/(-x*(1+x))^(1/2),x, algorithm="f 
ricas")
 

Output:

2*I*weierstrassPInverse(4, 0, x) + 2*I*weierstrassZeta(4, 0, weierstrassPI 
nverse(4, 0, x))
 

Sympy [F(-1)]

Timed out. \[ \int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (1+x)}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (1+x)}} \, dx=\int { -\frac {\sqrt {x} \sqrt {-x + 1}}{\sqrt {-{\left (x + 1\right )} x} \sqrt {-x}} \,d x } \] Input:

integrate(-(1-x)^(1/2)*x^(1/2)/(-x)^(1/2)/(-x*(1+x))^(1/2),x, algorithm="m 
axima")
 

Output:

-integrate(sqrt(x)*sqrt(-x + 1)/(sqrt(-(x + 1)*x)*sqrt(-x)), x)
 

Giac [F]

\[ \int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (1+x)}} \, dx=\int { -\frac {\sqrt {x} \sqrt {-x + 1}}{\sqrt {-{\left (x + 1\right )} x} \sqrt {-x}} \,d x } \] Input:

integrate(-(1-x)^(1/2)*x^(1/2)/(-x)^(1/2)/(-x*(1+x))^(1/2),x, algorithm="g 
iac")
 

Output:

integrate(-sqrt(x)*sqrt(-x + 1)/(sqrt(-(x + 1)*x)*sqrt(-x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (1+x)}} \, dx=-\int \frac {\sqrt {x}\,\sqrt {1-x}}{\sqrt {-x}\,\sqrt {-x^2-x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int -\frac {\sqrt {1-x} \sqrt {x}}{\sqrt {-x} \sqrt {-x (1+x)}} \, dx=\int \frac {\sqrt {x}\, \sqrt {x +1}\, \sqrt {1-x}}{x^{2}+x}d x \] Input:

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

Output:

int((sqrt(x)*sqrt(x + 1)*sqrt( - x + 1))/(x**2 + x),x)