\(\int \frac {\sqrt {2+d x}}{\sqrt {2-d x} \sqrt {e+f x}} \, dx\) [291]

Optimal result

Integrand size = 29, antiderivative size = 91 \[ \int \frac {\sqrt {2+d x}}{\sqrt {2-d x} \sqrt {e+f x}} \, dx=-\frac {4 \sqrt {d e+2 f} \sqrt {\frac {d (e+f x)}{d e+2 f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {2-d x}}{\sqrt {d e+2 f}}\right )|\frac {1}{4} \left (2+\frac {d e}{f}\right )\right )}{d \sqrt {f} \sqrt {e+f x}} \] Output:

-4*(d*e+2*f)^(1/2)*(d*(f*x+e)/(d*e+2*f))^(1/2)*EllipticE(f^(1/2)*(-d*x+2)^ 
(1/2)/(d*e+2*f)^(1/2),1/2*(2+d*e/f)^(1/2))/d/f^(1/2)/(f*x+e)^(1/2)
 

Mathematica [A] (verified)

Time = 7.90 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {2+d x}}{\sqrt {2-d x} \sqrt {e+f x}} \, dx=\frac {2 \sqrt {e+f x} \left (\frac {2+d x}{\sqrt {2-d x}}-\frac {2 \sqrt {\frac {2+d x}{-2+d x}} E\left (\arcsin \left (\frac {2}{\sqrt {2-d x}}\right )|\frac {1}{4} \left (2+\frac {d e}{f}\right )\right )}{\sqrt {\frac {d (e+f x)}{f (-2+d x)}}}\right )}{f \sqrt {2+d x}} \] Input:

Integrate[Sqrt[2 + d*x]/(Sqrt[2 - d*x]*Sqrt[e + f*x]),x]
 

Output:

(2*Sqrt[e + f*x]*((2 + d*x)/Sqrt[2 - d*x] - (2*Sqrt[(2 + d*x)/(-2 + d*x)]* 
EllipticE[ArcSin[2/Sqrt[2 - d*x]], (2 + (d*e)/f)/4])/Sqrt[(d*(e + f*x))/(f 
*(-2 + d*x))]))/(f*Sqrt[2 + d*x])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, 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.

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 123

\(\displaystyle -\frac {4 \sqrt {d e+2 f} \sqrt {\frac {d (e+f x)}{d e+2 f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {2-d x}}{\sqrt {d e+2 f}}\right )|\frac {1}{4} \left (\frac {d e}{f}+2\right )\right )}{d \sqrt {f} \sqrt {e+f x}}\)

Input:

Int[Sqrt[2 + d*x]/(Sqrt[2 - d*x]*Sqrt[e + f*x]),x]
 

Output:

(-4*Sqrt[d*e + 2*f]*Sqrt[(d*(e + f*x))/(d*e + 2*f)]*EllipticE[ArcSin[(Sqrt 
[f]*Sqrt[2 - d*x])/Sqrt[d*e + 2*f]], (2 + (d*e)/f)/4])/(d*Sqrt[f]*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. \(163\) vs. \(2(76)=152\).

Time = 1.90 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.80

method	result	size
default	\(\frac {2 \left (d^{2} e^{2}-4 f^{2}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}, \sqrt {\frac {d e +2 f}{d e -2 f}}\right ) \sqrt {-\frac {f \left (d x -2\right )}{d e +2 f}}\, \sqrt {-\frac {\left (d x +2\right ) f}{d e -2 f}}\, \sqrt {\frac {d \left (f x +e \right )}{d e +2 f}}\, \sqrt {d x +2}\, \sqrt {-d x +2}\, \sqrt {f x +e}}{f^{2} d \left (d^{2} f \,x^{3}+d^{2} e \,x^{2}-4 f x -4 e \right )}\)	\(164\)
elliptic	\(\frac {\sqrt {-\left (f x +e \right ) \left (d^{2} x^{2}-4\right )}\, \left (\frac {4 \left (\frac {e}{f}+\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}+\frac {2 d \left (\frac {e}{f}+\frac {2}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x +\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\, \sqrt {\frac {x -\frac {2}{d}}{-\frac {e}{f}-\frac {2}{d}}}\, \left (\left (-\frac {e}{f}+\frac {2}{d}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )-\frac {2 \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}+\frac {2}{d}}}, \sqrt {\frac {-\frac {e}{f}-\frac {2}{d}}{-\frac {e}{f}+\frac {2}{d}}}\right )}{d}\right )}{\sqrt {-d^{2} f \,x^{3}-d^{2} e \,x^{2}+4 f x +4 e}}\right )}{\sqrt {d x +2}\, \sqrt {-d x +2}\, \sqrt {f x +e}}\)	\(446\)

Input:

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

Output:

2*(d^2*e^2-4*f^2)*EllipticE((d*(f*x+e)/(d*e+2*f))^(1/2),((d*e+2*f)/(d*e-2* 
f))^(1/2))*(-f*(d*x-2)/(d*e+2*f))^(1/2)*(-(d*x+2)*f/(d*e-2*f))^(1/2)*(d*(f 
*x+e)/(d*e+2*f))^(1/2)*(d*x+2)^(1/2)*(-d*x+2)^(1/2)*(f*x+e)^(1/2)/f^2/d/(d 
^2*f*x^3+d^2*e*x^2-4*f*x-4*e)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (76) = 152\).

Time = 0.07 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.10 \[ \int \frac {\sqrt {2+d x}}{\sqrt {2-d x} \sqrt {e+f x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-d^{2} f} d f {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right ) + \sqrt {-d^{2} f} {\left (d e - 6 \, f\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} e^{2} + 12 \, f^{2}\right )}}{3 \, d^{2} f^{2}}, -\frac {8 \, {\left (d^{2} e^{3} - 36 \, e f^{2}\right )}}{27 \, d^{2} f^{3}}, \frac {3 \, f x + e}{3 \, f}\right )\right )}}{3 \, d^{2} f^{2}} \] Input:

integrate((d*x+2)^(1/2)/(-d*x+2)^(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 + 12*f^2)/(d^2*f^2), 
-8/27*(d^2*e^3 - 36*e*f^2)/(d^2*f^3), weierstrassPInverse(4/3*(d^2*e^2 + 1 
2*f^2)/(d^2*f^2), -8/27*(d^2*e^3 - 36*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f) 
) + sqrt(-d^2*f)*(d*e - 6*f)*weierstrassPInverse(4/3*(d^2*e^2 + 12*f^2)/(d 
^2*f^2), -8/27*(d^2*e^3 - 36*e*f^2)/(d^2*f^3), 1/3*(3*f*x + e)/f))/(d^2*f^ 
2)
 

Sympy [F]

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

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

Output:

Integral(sqrt(d*x + 2)/(sqrt(e + f*x)*sqrt(-d*x + 2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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