\(\int \frac {\sqrt {c+d x}}{\sqrt {c-d x} \sqrt {e+f x}} \, dx\) [294]

Optimal result

Integrand size = 29, antiderivative size = 120 \[ \int \frac {\sqrt {c+d x}}{\sqrt {c-d x} \sqrt {e+f x}} \, dx=-\frac {2 \sqrt {2} \sqrt {d e+c f} \sqrt {c+d x} \sqrt {\frac {d (e+f x)}{d e+c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c-d x}}{\sqrt {d e+c f}}\right )|\frac {1}{2} \left (1+\frac {d e}{c f}\right )\right )}{d \sqrt {f} \sqrt {1+\frac {d x}{c}} \sqrt {e+f x}} \] Output:

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

Mathematica [A] (verified)

Time = 8.68 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {c+d x}}{\sqrt {c-d x} \sqrt {e+f x}} \, dx=\frac {2 \sqrt {e+f x} \left (\frac {c+d x}{\sqrt {c-d x}}-\frac {\sqrt {2} \sqrt {c} \sqrt {\frac {c+d x}{-c+d x}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {c-d x}}\right )|\frac {1}{2} \left (1+\frac {d e}{c f}\right )\right )}{\sqrt {\frac {d (e+f x)}{f (-c+d x)}}}\right )}{f \sqrt {c+d x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, 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.

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 123

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

Input:

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

Output:

(-2*Sqrt[2]*Sqrt[d*e + c*f]*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(d*e + c*f)]* 
EllipticE[ArcSin[(Sqrt[f]*Sqrt[c - d*x])/Sqrt[d*e + c*f]], (1 + (d*e)/(c*f 
))/2])/(d*Sqrt[f]*Sqrt[(c + d*x)/c]*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 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. \(318\) vs. \(2(100)=200\).

Time = 1.99 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.66

method	result	size
default	\(-\frac {2 \left (2 \operatorname {EllipticF}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) c^{2} f^{2}-2 \operatorname {EllipticF}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) c d e f -\operatorname {EllipticE}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) c^{2} f^{2}+\operatorname {EllipticE}\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {-\frac {c f -d e}{c f +d e}}\right ) d^{2} e^{2}\right ) \sqrt {\frac {f \left (d x +c \right )}{c f -d e}}\, \sqrt {\frac {f \left (-d x +c \right )}{c f +d e}}\, \sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {d x +c}\, \sqrt {-d x +c}\, \sqrt {f x +e}}{f^{2} d \left (-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}\right )}\)	\(319\)
elliptic	\(\frac {\sqrt {\left (f x +e \right ) \left (-d^{2} x^{2}+c^{2}\right )}\, \left (\frac {2 c \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}+\frac {2 d \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x -\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}-\frac {c}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )+\frac {c \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}-\frac {c}{d}}}\right )}{d}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+c^{2} f x +e \,c^{2}}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {-d x +c}}\)	\(472\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (100) = 200\).

Time = 0.08 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {c+d x}}{\sqrt {c-d x} \sqrt {e+f x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-d^{2} f} d f {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right ) + \sqrt {-d^{2} f} {\left (d e - 3 \, c f\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 3 \, c^{2} f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 9 \, c^{2} e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right )}}{3 \, d^{2} f^{2}} \] Input:

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

Output:

2/3*(3*sqrt(-d^2*f)*d*f*weierstrassZeta(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), 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)) + sqrt(-d^2*f)*(d*e - 3*c*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^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {c+d x}}{\sqrt {c-d x} \sqrt {e+f x}} \, dx=\int \frac {\sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {-d x +c}}{-d f \,x^{2}+c f x -d e x +c e}d x \] Input:

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