\(\int \frac {\sqrt {a+b x}}{\sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\) [1251]

Optimal result

Integrand size = 40, antiderivative size = 147 \[ \int \frac {\sqrt {a+b x}}{\sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 a \sqrt {a+b x} E\left (\arccos \left (\sqrt {c-\frac {b (1-c) x}{a}}\right )|\frac {1-e}{1-c}\right )}{b (1-e) \sqrt {\frac {(1-c) (a+b x)}{a}}}+\frac {2 a \sqrt {a+b x} \operatorname {EllipticF}\left (\arccos \left (\sqrt {c-\frac {b (1-c) x}{a}}\right ),\frac {1-e}{1-c}\right )}{b (1-e) \sqrt {\frac {(1-c) (a+b x)}{a}}} \] Output:

-2*a*(b*x+a)^(1/2)*EllipticE((1-c+b*(1-c)*x/a)^(1/2),((1-e)/(1-c))^(1/2))/ 
b/(1-e)/((1-c)*(b*x+a)/a)^(1/2)+2*a*(b*x+a)^(1/2)*InverseJacobiAM(arccos(( 
c-b*(1-c)*x/a)^(1/2)),((1-e)/(1-c))^(1/2))/b/(1-e)/((1-c)*(b*x+a)/a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.39 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+b x}}{\sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 i a \sqrt {a+b x} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {(-1+c) (a+b x)}{a}}\right )|\frac {-1+e}{-1+c}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {(-1+c) (a+b x)}{a}}\right ),\frac {-1+e}{-1+c}\right )\right )}{b (-1+e) \sqrt {\frac {(-1+c) (a+b x)}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {124, 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.

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

Input:

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

Output:

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

Defintions of rubi rules used

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 [A] (verified)

Time = 4.53 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.23

method	result	size
default	\(-\frac {2 a^{2} \left (c -e \right ) \sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) \sqrt {-\frac {\left (-1+e \right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}}{\sqrt {b x +a}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \sqrt {\frac {b e x +a e -b x}{a}}\, \left (-1+e \right )^{2} b \left (c -1\right )}\)	\(181\)
elliptic	\(\frac {\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 {2 a \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 {\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}}}+\frac {2 b \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 )}}}\, \left (\left (-\frac {a e}{\left (-1+e \right ) b}+\frac {a}{b}\right ) \operatorname {EllipticE}\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 )-\frac {a \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 )}{b}\right )}{\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}}}\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}}}\)	\(891\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1177 vs. \(2 (122) = 244\).

Time = 0.09 (sec) , antiderivative size = 1177, normalized size of antiderivative = 8.01 \[ \int \frac {\sqrt {a+b x}}{\sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/3*((a^3*c + a^3*e - 2*a^3)*sqrt(-(b^3*c - b^3 - (b^3*c - b^3)*e)/a^2)*w 
eierstrassPInverse(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)) - 3*(a^2*b*c - a^2*b - (a^2*b*c - a^2*b)*e)*sqrt(-( 
b^3*c - b^3 - (b^3*c - b^3)*e)/a^2)*weierstrassZeta(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), 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*...
 

Sympy [F]

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

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

Output:

Integral(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]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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