\(\int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\) [1246]

Optimal result

Integrand size = 40, antiderivative size = 58 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\frac {2 \sqrt {a} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right ),\frac {1-e}{1-c}\right )}{b \sqrt {1-c}} \] Output:

2*a^(1/2)*EllipticF((1-c)^(1/2)*(b*x+a)^(1/2)/a^(1/2),((1-e)/(1-c))^(1/2)) 
/b/(1-c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(58)=116\).

Time = 11.41 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.22 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 (a+b x) \sqrt {\frac {-1+c+\frac {a}{a+b x}}{-1+c}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-\frac {a}{-1+c}}}{\sqrt {a+b x}}\right ),\frac {-1+c}{-1+e}\right )}{b \sqrt {-\frac {a}{-1+c}} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \] Input:

Integrate[1/(Sqrt[a + b*x]*Sqrt[c + (b*(-1 + c)*x)/a]*Sqrt[e + (b*(-1 + e) 
*x)/a]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {129}

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.

\(\displaystyle \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b (c-1) x}{a}+c} \sqrt {\frac {b (e-1) x}{a}+e}} \, dx\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {2 \sqrt {a} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right ),\frac {1-e}{1-c}\right )}{b \sqrt {1-c}}\)

Input:

Int[1/(Sqrt[a + b*x]*Sqrt[c + (b*(-1 + c)*x)/a]*Sqrt[e + (b*(-1 + e)*x)/a] 
),x]
 

Output:

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

Defintions of rubi rules used

rule 129

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ 
Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - 
a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ 
[(b*e - a*f)/b, 0] && PosQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d 
*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(( 
-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ 
[((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f 
/b]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(49)=98\).

Time = 6.36 (sec) , antiderivative size = 179, normalized size of antiderivative = 3.09

method	result	size
default	\(\frac {2 a \sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {-\frac {\left (-1+e \right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) \left (c -e \right )}{\sqrt {b x +a}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \sqrt {\frac {b e x +a e -b x}{a}}\, b \left (-1+e \right ) \left (c -1\right )}\)	\(179\)
elliptic	\(\frac {2 \sqrt {\frac {\left (b x +a \right ) \left (b c x +a c -b x \right ) \left (b e x +a e -b x \right )}{a^{2}}}\, \left (\frac {a e}{\left (-1+e \right ) b}-\frac {a c}{b \left (c -1\right )}\right ) \sqrt {\frac {x +\frac {a e}{\left (-1+e \right ) b}}{\frac {a e}{\left (-1+e \right ) b}-\frac {a c}{b \left (c -1\right )}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a e}{\left (-1+e \right ) b}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {a c}{b \left (c -1\right )}}{-\frac {a e}{\left (-1+e \right ) b}+\frac {a c}{b \left (c -1\right )}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {a e}{\left (-1+e \right ) b}}{\frac {a e}{\left (-1+e \right ) b}-\frac {a c}{b \left (c -1\right )}}}, \sqrt {\frac {-\frac {a e}{\left (-1+e \right ) b}+\frac {a c}{b \left (c -1\right )}}{-\frac {a e}{\left (-1+e \right ) b}+\frac {a}{b}}}\right )}{\sqrt {b x +a}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \sqrt {\frac {b e x +a e -b x}{a}}\, \sqrt {\frac {b^{3} c e \,x^{3}}{a^{2}}+\frac {3 b^{2} c e \,x^{2}}{a}-\frac {b^{3} c \,x^{3}}{a^{2}}-\frac {b^{3} e \,x^{3}}{a^{2}}+3 b c e x -\frac {2 b^{2} c \,x^{2}}{a}-\frac {2 b^{2} e \,x^{2}}{a}+\frac {b^{3} x^{3}}{a^{2}}+a c e -b c x -b e x +\frac {b^{2} x^{2}}{a}}}\)	\(425\)

Input:

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

Output:

2/(b*x+a)^(1/2)*a*((c-1)*(b*e*x+a*e-b*x)/a/(c-e))^(1/2)*(-(b*x+a)*(-1+e)/a 
)^(1/2)*(-(-1+e)*(b*c*x+a*c-b*x)/(c-e)/a)^(1/2)*EllipticF(((c-1)*(b*e*x+a* 
e-b*x)/a/(c-e))^(1/2),((c-e)/(c-1))^(1/2))*(c-e)/((b*c*x+a*c-b*x)/a)^(1/2) 
/((b*e*x+a*e-b*x)/a)^(1/2)/b/(-1+e)/(c-1)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (45) = 90\).

Time = 0.09 (sec) , antiderivative size = 418, normalized size of antiderivative = 7.21 \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 \, a^{2} \sqrt {-\frac {b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e}{a^{2}}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} c^{2} + a^{2} e^{2} - a^{2} c + a^{2} - {\left (a^{2} c + a^{2}\right )} e\right )}}{3 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e^{2} + b^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e\right )}}, \frac {4 \, {\left (2 \, a^{3} c^{3} + 2 \, a^{3} e^{3} - 3 \, a^{3} c^{2} - 3 \, a^{3} c + 2 \, a^{3} - 3 \, {\left (a^{3} c + a^{3}\right )} e^{2} - 3 \, {\left (a^{3} c^{2} - 4 \, a^{3} c + a^{3}\right )} e\right )}}{27 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{3} - b^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{2} - 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e\right )}}, \frac {2 \, a c - {\left (3 \, a c - 2 \, a\right )} e + 3 \, {\left (b c - {\left (b c - b\right )} e - b\right )} x - a}{3 \, {\left (b c - {\left (b c - b\right )} e - b\right )}}\right )}{b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e} \] Input:

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

Output:

-2*a^2*sqrt(-(b^3*c - b^3 - (b^3*c - b^3)*e)/a^2)*weierstrassPInverse(4/3* 
(a^2*c^2 + a^2*e^2 - a^2*c + a^2 - (a^2*c + a^2)*e)/(b^2*c^2 - 2*b^2*c + ( 
b^2*c^2 - 2*b^2*c + b^2)*e^2 + b^2 - 2*(b^2*c^2 - 2*b^2*c + b^2)*e), 4/27* 
(2*a^3*c^3 + 2*a^3*e^3 - 3*a^3*c^2 - 3*a^3*c + 2*a^3 - 3*(a^3*c + a^3)*e^2 
 - 3*(a^3*c^2 - 4*a^3*c + a^3)*e)/(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - (b^3*c^ 
3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e^3 - b^3 + 3*(b^3*c^3 - 3*b^3*c^2 + 3*b^3* 
c - b^3)*e^2 - 3*(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e), 1/3*(2*a*c - (3 
*a*c - 2*a)*e + 3*(b*c - (b*c - b)*e - b)*x - a)/(b*c - (b*c - b)*e - b))/ 
(b^3*c - b^3 - (b^3*c - b^3)*e)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\int \frac {1}{\sqrt {a + b x} \sqrt {c + \frac {b c x}{a} - \frac {b x}{a}} \sqrt {e + \frac {b e x}{a} - \frac {b x}{a}}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(a + b*x)*sqrt(c + b*c*x/a - b*x/a)*sqrt(e + b*e*x/a - b*x 
/a)), x)
 

Maxima [F]

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

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

Output:

integrate(1/(sqrt(b*x + a)*sqrt(b*(c - 1)*x/a + c)*sqrt(b*(e - 1)*x/a + e) 
), x)
 

Giac [F]

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

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

Output:

integrate(1/(sqrt(b*x + a)*sqrt(b*(c - 1)*x/a + c)*sqrt(b*(e - 1)*x/a + e) 
), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\int \frac {1}{\sqrt {c+\frac {b\,x\,\left (c-1\right )}{a}}\,\sqrt {e+\frac {b\,x\,\left (e-1\right )}{a}}\,\sqrt {a+b\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\left (\int \frac {\sqrt {b x +a}\, \sqrt {b e x +a e -b x}\, \sqrt {b c x +a c -b x}}{b^{3} c e \,x^{3}+3 a \,b^{2} c e \,x^{2}-b^{3} c \,x^{3}-b^{3} e \,x^{3}+3 a^{2} b c e x -2 a \,b^{2} c \,x^{2}-2 a \,b^{2} e \,x^{2}+b^{3} x^{3}+a^{3} c e -a^{2} b c x -a^{2} b e x +a \,b^{2} x^{2}}d x \right ) a \] Input:

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

Output:

int((sqrt(a + b*x)*sqrt(a*e + b*e*x - b*x)*sqrt(a*c + b*c*x - b*x))/(a**3* 
c*e + 3*a**2*b*c*e*x - a**2*b*c*x - a**2*b*e*x + 3*a*b**2*c*e*x**2 - 2*a*b 
**2*c*x**2 - 2*a*b**2*e*x**2 + a*b**2*x**2 + b**3*c*e*x**3 - b**3*c*x**3 - 
 b**3*e*x**3 + b**3*x**3),x)*a