\(\int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx\) [458]

Optimal result

Integrand size = 24, antiderivative size = 86 \[ \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx=\frac {\sqrt {a-c} \sqrt {x} \sqrt {-\frac {c+2 x}{a-c}} E\left (\arcsin \left (\frac {\sqrt {a+2 x}}{\sqrt {a-c}}\right )|1-\frac {c}{a}\right )}{\sqrt {2} \sqrt {-\frac {x}{a}} \sqrt {c+2 x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.24 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx=-\frac {i c \sqrt {1+\frac {2 x}{a}} \sqrt {1+\frac {2 x}{c}} \left (E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {1}{a}} \sqrt {x}\right )|\frac {a}{c}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {1}{a}} \sqrt {x}\right ),\frac {a}{c}\right )\right )}{\sqrt {2} \sqrt {\frac {1}{a}} \sqrt {a+2 x} \sqrt {c+2 x}} \] Input:

Integrate[Sqrt[x]/(Sqrt[a + 2*x]*Sqrt[c + 2*x]),x]
 

Output:

((-I)*c*Sqrt[1 + (2*x)/a]*Sqrt[1 + (2*x)/c]*(EllipticE[I*ArcSinh[Sqrt[2]*S 
qrt[a^(-1)]*Sqrt[x]], a/c] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[a^(-1)]*Sqrt 
[x]], a/c]))/(Sqrt[2]*Sqrt[a^(-1)]*Sqrt[a + 2*x]*Sqrt[c + 2*x])
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {124, 27, 123}

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.

\(\displaystyle \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {\sqrt {x} \sqrt {-\frac {c+2 x}{a-c}} \int \frac {\sqrt {2} \sqrt {-\frac {x}{a}}}{\sqrt {a+2 x} \sqrt {-\frac {c}{a-c}-\frac {2 x}{a-c}}}dx}{\sqrt {2} \sqrt {-\frac {x}{a}} \sqrt {c+2 x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {x} \sqrt {-\frac {c+2 x}{a-c}} \int \frac {\sqrt {-\frac {x}{a}}}{\sqrt {a+2 x} \sqrt {-\frac {c}{a-c}-\frac {2 x}{a-c}}}dx}{\sqrt {-\frac {x}{a}} \sqrt {c+2 x}}\)

\(\Big \downarrow \) 123

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

Input:

Int[Sqrt[x]/(Sqrt[a + 2*x]*Sqrt[c + 2*x]),x]
 

Output:

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

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 123

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

rule 124

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(72)=144\).

Time = 0.77 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {\left (a \operatorname {EllipticF}\left (\sqrt {\frac {c +2 x}{c}}, \sqrt {-\frac {c}{a -c}}\right )-\operatorname {EllipticE}\left (\sqrt {\frac {c +2 x}{c}}, \sqrt {-\frac {c}{a -c}}\right ) a +\operatorname {EllipticE}\left (\sqrt {\frac {c +2 x}{c}}, \sqrt {-\frac {c}{a -c}}\right ) c \right ) \sqrt {2}\, \sqrt {-\frac {x}{c}}\, \sqrt {\frac {a +2 x}{a -c}}\, \sqrt {\frac {c +2 x}{c}}\, c \sqrt {a +2 x}\, \sqrt {c +2 x}}{2 \sqrt {x}\, \left (a c +2 a x +2 c x +4 x^{2}\right )}\)	\(157\)
elliptic	\(\frac {\sqrt {\left (a +2 x \right ) \left (c +2 x \right ) x}\, c \sqrt {2}\, \sqrt {\frac {x +\frac {c}{2}}{c}}\, \sqrt {\frac {x +\frac {a}{2}}{-\frac {c}{2}+\frac {a}{2}}}\, \sqrt {-\frac {2 x}{c}}\, \left (\left (-\frac {c}{2}+\frac {a}{2}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {x +\frac {c}{2}}{c}}, \frac {\sqrt {-\frac {2 c}{-\frac {c}{2}+\frac {a}{2}}}}{2}\right )-\frac {a \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {x +\frac {c}{2}}{c}}, \frac {\sqrt {-\frac {2 c}{-\frac {c}{2}+\frac {a}{2}}}}{2}\right )}{2}\right )}{\sqrt {a +2 x}\, \sqrt {c +2 x}\, \sqrt {x}\, \sqrt {a c x +2 a \,x^{2}+2 c \,x^{2}+4 x^{3}}}\)	\(173\)

Input:

int(x^(1/2)/(a+2*x)^(1/2)/(c+2*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx=-\frac {1}{6} \, {\left (a + c\right )} {\rm weierstrassPInverse}\left (\frac {1}{3} \, a^{2} - \frac {1}{3} \, a c + \frac {1}{3} \, c^{2}, -\frac {1}{27} \, a^{3} + \frac {1}{18} \, a^{2} c + \frac {1}{18} \, a c^{2} - \frac {1}{27} \, c^{3}, \frac {1}{6} \, a + \frac {1}{6} \, c + x\right ) - {\rm weierstrassZeta}\left (\frac {1}{3} \, a^{2} - \frac {1}{3} \, a c + \frac {1}{3} \, c^{2}, -\frac {1}{27} \, a^{3} + \frac {1}{18} \, a^{2} c + \frac {1}{18} \, a c^{2} - \frac {1}{27} \, c^{3}, {\rm weierstrassPInverse}\left (\frac {1}{3} \, a^{2} - \frac {1}{3} \, a c + \frac {1}{3} \, c^{2}, -\frac {1}{27} \, a^{3} + \frac {1}{18} \, a^{2} c + \frac {1}{18} \, a c^{2} - \frac {1}{27} \, c^{3}, \frac {1}{6} \, a + \frac {1}{6} \, c + x\right )\right ) \] Input:

integrate(x^(1/2)/(a+2*x)^(1/2)/(c+2*x)^(1/2),x, algorithm="fricas")
 

Output:

-1/6*(a + c)*weierstrassPInverse(1/3*a^2 - 1/3*a*c + 1/3*c^2, -1/27*a^3 + 
1/18*a^2*c + 1/18*a*c^2 - 1/27*c^3, 1/6*a + 1/6*c + x) - weierstrassZeta(1 
/3*a^2 - 1/3*a*c + 1/3*c^2, -1/27*a^3 + 1/18*a^2*c + 1/18*a*c^2 - 1/27*c^3 
, weierstrassPInverse(1/3*a^2 - 1/3*a*c + 1/3*c^2, -1/27*a^3 + 1/18*a^2*c 
+ 1/18*a*c^2 - 1/27*c^3, 1/6*a + 1/6*c + x))
 

Sympy [F]

\[ \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx=\int \frac {\sqrt {x}}{\sqrt {a + 2 x} \sqrt {c + 2 x}}\, dx \] Input:

integrate(x**(1/2)/(a+2*x)**(1/2)/(c+2*x)**(1/2),x)
 

Output:

Integral(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {a + 2 \, x} \sqrt {c + 2 \, x}} \,d x } \] Input:

integrate(x^(1/2)/(a+2*x)^(1/2)/(c+2*x)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)), x)
 

Giac [F]

\[ \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {a + 2 \, x} \sqrt {c + 2 \, x}} \,d x } \] Input:

integrate(x^(1/2)/(a+2*x)^(1/2)/(c+2*x)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(x)/(sqrt(a + 2*x)*sqrt(c + 2*x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx=\int \frac {\sqrt {x}}{\sqrt {a+2\,x}\,\sqrt {c+2\,x}} \,d x \] Input:

int(x^(1/2)/((a + 2*x)^(1/2)*(c + 2*x)^(1/2)),x)
 

Output:

int(x^(1/2)/((a + 2*x)^(1/2)*(c + 2*x)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {x}}{\sqrt {a+2 x} \sqrt {c+2 x}} \, dx=\int \frac {\sqrt {x}\, \sqrt {c +2 x}\, \sqrt {a +2 x}}{a c +2 a x +2 c x +4 x^{2}}d x \] Input:

int(x^(1/2)/(a+2*x)^(1/2)/(c+2*x)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int((sqrt(x)*sqrt(c + 2*x)*sqrt(a + 2*x))/(a*c + 2*a*x + 2*c*x + 4*x**2),x 
)