\(\int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [1253]

Optimal result

Integrand size = 30, antiderivative size = 100 \[ \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 \sqrt {c} \sqrt {1-\frac {d x}{c}} \sqrt {\frac {d (e+f x)}{d e-c f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d x}}{\sqrt {2} \sqrt {c}}\right ),-\frac {2 c f}{d e-c f}\right )}{d \sqrt {-c+d x} \sqrt {e+f x}} \] Output:

2*c^(1/2)*(1-d*x/c)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*EllipticF(1/2*(d*x+ 
c)^(1/2)*2^(1/2)/c^(1/2),(-2*c*f/(-c*f+d*e))^(1/2))/d/(d*x-c)^(1/2)/(f*x+e 
)^(1/2)
 

Mathematica [A] (verified)

Time = 5.55 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {\sqrt {2} (c-d x) \sqrt {\frac {c+d x}{-c+d x}} \sqrt {\frac {d (e+f x)}{f (-c+d x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {-c}}{\sqrt {-c+d x}}\right ),\frac {1}{2} \left (1+\frac {d e}{c f}\right )\right )}{\sqrt {-c} d \sqrt {c+d x} \sqrt {e+f x}} \] Input:

Integrate[1/(Sqrt[-c + d*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {131, 27, 131, 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.

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

\(\Big \downarrow \) 131

\(\displaystyle \frac {\sqrt {\frac {c-d x}{c}} \int \frac {\sqrt {2}}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x}}dx}{\sqrt {2} \sqrt {d x-c}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {\frac {c-d x}{c}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {e+f x}}dx}{\sqrt {d x-c}}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\sqrt {\frac {c-d x}{c}} \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {d x}{c}} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {d x-c} \sqrt {e+f x}}\)

\(\Big \downarrow \) 129

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

Input:

Int[1/(Sqrt[-c + d*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
 

Output:

(2*Sqrt[c]*Sqrt[(c - d*x)/c]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*EllipticF[Arc 
Sin[Sqrt[c + d*x]/(Sqrt[2]*Sqrt[c])], (-2*c*f)/(d*e - c*f)])/(d*Sqrt[-c + 
d*x]*Sqrt[e + f*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 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]))
 

rule 131

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

Maple [A] (verified)

Time = 2.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.72

method	result	size
default	\(\frac {2 \sqrt {f x +e}\, \sqrt {x d +c}\, \sqrt {x d -c}\, \sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {\frac {\left (-x d +c \right ) f}{c f +d e}}\, \sqrt {\frac {f \left (x d +c \right )}{c f -d e}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) \left (c f -d e \right )}{d f \left (-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +c^{2} e \right )}\)	\(172\)
elliptic	\(\frac {2 \sqrt {-\left (f x +e \right ) \left (-d^{2} x^{2}+c^{2}\right )}\, \left (-\frac {c}{d}+\frac {e}{f}\right ) \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}, \sqrt {\frac {\frac {c}{d}-\frac {e}{f}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{\sqrt {x d -c}\, \sqrt {x d +c}\, \sqrt {f x +e}\, \sqrt {d^{2} f \,x^{3}+d^{2} e \,x^{2}-c^{2} f x -c^{2} e}}\)	\(220\)

Input:

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

Output:

2*(f*x+e)^(1/2)*(d*x+c)^(1/2)*(d*x-c)^(1/2)*(-d*(f*x+e)/(c*f-d*e))^(1/2)*( 
(-d*x+c)*f/(c*f+d*e))^(1/2)*(f/(c*f-d*e)*(d*x+c))^(1/2)*EllipticF((-d*(f*x 
+e)/(c*f-d*e))^(1/2),(-(c*f-d*e)/(c*f+d*e))^(1/2))*(c*f-d*e)/d/f/(-d^2*f*x 
^3-d^2*e*x^2+c^2*f*x+c^2*e)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 \, \sqrt {d^{2} f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )}{d^{2} f} \] Input:

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

Output:

2*sqrt(d^2*f)*weierstrassPInverse(4/3*(d^2*e^2 + 3*c^2*f^2)/(d^2*f^2), -8/ 
27*(d^2*e^3 - 9*c^2*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f)/(d^2*f)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {1}{\sqrt {- c + d x} \sqrt {c + d x} \sqrt {e + f x}}\, dx \] Input:

integrate(1/(d*x-c)**(1/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
 

Output:

Integral(1/(sqrt(-c + d*x)*sqrt(c + d*x)*sqrt(e + f*x)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {1}{\sqrt {d x + c} \sqrt {d x - c} \sqrt {f x + e}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(d*x + c)*sqrt(d*x - c)*sqrt(f*x + e)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {1}{\sqrt {d x + c} \sqrt {d x - c} \sqrt {f x + e}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(d*x + c)*sqrt(d*x - c)*sqrt(f*x + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-c+d x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {c+d\,x}\,\sqrt {d\,x-c}} \,d x \] Input:

int(1/((e + f*x)^(1/2)*(c + d*x)^(1/2)*(d*x - c)^(1/2)),x)
 

Output:

int(1/((e + f*x)^(1/2)*(c + d*x)^(1/2)*(d*x - c)^(1/2)), x)
 

Reduce [F]

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

int(1/(d*x-c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

 - int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt( - c + d*x))/(c**2*e + c**2*f*x - 
 d**2*e*x**2 - d**2*f*x**3),x)