\(\int \frac {\sqrt {1+c x}}{\sqrt {1-c x} \sqrt {d+e x}} \, dx\) [3]

Optimal result

Integrand size = 29, antiderivative size = 90 \[ \int \frac {\sqrt {1+c x}}{\sqrt {1-c x} \sqrt {d+e x}} \, dx=-\frac {2 \sqrt {2} \sqrt {c d+e} \sqrt {\frac {c (d+e x)}{c d+e}} E\left (\arcsin \left (\frac {\sqrt {e} \sqrt {1-c x}}{\sqrt {c d+e}}\right )|\frac {c d+e}{2 e}\right )}{c \sqrt {e} \sqrt {d+e x}} \] Output:

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

Mathematica [A] (verified)

Time = 7.75 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {1+c x}}{\sqrt {1-c x} \sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (\frac {1+c x}{\sqrt {1-c x}}-\frac {\sqrt {2} \sqrt {\frac {1+c x}{-1+c x}} E\left (\arcsin \left (\frac {\sqrt {2}}{\sqrt {1-c x}}\right )|\frac {c d+e}{2 e}\right )}{\sqrt {\frac {c (d+e x)}{e (-1+c x)}}}\right )}{e \sqrt {1+c x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 90, 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.

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 123

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

Input:

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

Output:

(-2*Sqrt[2]*Sqrt[c*d + e]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticE[ArcSin[( 
Sqrt[e]*Sqrt[1 - c*x])/Sqrt[c*d + e]], (c*d + e)/(2*e)])/(c*Sqrt[e]*Sqrt[d 
 + e*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. \(277\) vs. \(2(77)=154\).

Time = 2.20 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.09

method	result	size
default	\(\frac {2 \left (\operatorname {EllipticE}\left (\sqrt {\frac {c \left (e x +d \right )}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}-2 \operatorname {EllipticF}\left (\sqrt {\frac {c \left (e x +d \right )}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e -\operatorname {EllipticE}\left (\sqrt {\frac {c \left (e x +d \right )}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}+2 \operatorname {EllipticF}\left (\sqrt {\frac {c \left (e x +d \right )}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}\right ) \sqrt {-\frac {e \left (c x +1\right )}{c d -e}}\, \sqrt {-\frac {e \left (c x -1\right )}{c d +e}}\, \sqrt {\frac {c \left (e x +d \right )}{c d -e}}\, \sqrt {c x +1}\, \sqrt {-c x +1}\, \sqrt {e x +d}}{e^{2} c \left (e \,x^{3} c^{2}+c^{2} d \,x^{2}-e x -d \right )}\)	\(278\)
elliptic	\(\frac {\sqrt {-\left (e x +d \right ) \left (c^{2} x^{2}-1\right )}\, \left (\frac {2 \left (\frac {d}{e}-\frac {1}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\, \sqrt {\frac {x +\frac {1}{c}}{-\frac {d}{e}+\frac {1}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\right )}{\sqrt {-e \,x^{3} c^{2}-c^{2} d \,x^{2}+e x +d}}+\frac {2 c \left (\frac {d}{e}-\frac {1}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\, \sqrt {\frac {x +\frac {1}{c}}{-\frac {d}{e}+\frac {1}{c}}}\, \left (\left (-\frac {d}{e}-\frac {1}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\right )+\frac {\operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\right )}{c}\right )}{\sqrt {-e \,x^{3} c^{2}-c^{2} d \,x^{2}+e x +d}}\right )}{\sqrt {c x +1}\, \sqrt {-c x +1}\, \sqrt {e x +d}}\)	\(425\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (77) = 154\).

Time = 0.08 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {1+c x}}{\sqrt {1-c x} \sqrt {d+e x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-c^{2} e} c e {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} + 3 \, e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {8 \, {\left (c^{2} d^{3} - 9 \, d e^{2}\right )}}{27 \, c^{2} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + 3 \, e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {8 \, {\left (c^{2} d^{3} - 9 \, d e^{2}\right )}}{27 \, c^{2} e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + \sqrt {-c^{2} e} {\left (c d - 3 \, e\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + 3 \, e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {8 \, {\left (c^{2} d^{3} - 9 \, d e^{2}\right )}}{27 \, c^{2} e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right )}}{3 \, c^{2} e^{2}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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