\(\int \frac {\sqrt {e+\frac {b (-1+e) x}{a}}}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}}} \, dx\) [1249]

Optimal result

Integrand size = 40, antiderivative size = 58 \[ \int \frac {\sqrt {e+\frac {b (-1+e) x}{a}}}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}}} \, dx=\frac {2 \sqrt {a} E\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b \sqrt {1-c}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(191\) vs. \(2(58)=116\).

Time = 12.49 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.29 \[ \int \frac {\sqrt {e+\frac {b (-1+e) x}{a}}}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}}} \, dx=-\frac {2 (a+b x)^{3/2} \left (-\frac {\sqrt {-\frac {a}{-1+e}} \left (-1+c+\frac {a}{a+b x}\right ) \left (-1+e+\frac {a}{a+b x}\right )}{-1+c}+\frac {a \sqrt {\frac {-1+c+\frac {a}{a+b x}}{-1+c}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} E\left (\arcsin \left (\frac {\sqrt {-\frac {a}{-1+e}}}{\sqrt {a+b x}}\right )|\frac {-1+e}{-1+c}\right )}{\sqrt {a+b x}}\right )}{a b \sqrt {-\frac {a}{-1+e}} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \] Input:

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

Output:

(-2*(a + b*x)^(3/2)*(-((Sqrt[-(a/(-1 + e))]*(-1 + c + a/(a + b*x))*(-1 + e 
 + a/(a + b*x)))/(-1 + c)) + (a*Sqrt[(-1 + c + a/(a + b*x))/(-1 + c)]*Sqrt 
[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticE[ArcSin[Sqrt[-(a/(-1 + e))]/Sqr 
t[a + b*x]], (-1 + e)/(-1 + c)])/Sqrt[a + b*x]))/(a*b*Sqrt[-(a/(-1 + e))]* 
Sqrt[c + (b*(-1 + c)*x)/a]*Sqrt[e + (b*(-1 + e)*x)/a])
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {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.

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

\(\Big \downarrow \) 123

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

Input:

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

Output:

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

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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(347\) vs. \(2(49)=98\).

Time = 4.27 (sec) , antiderivative size = 348, normalized size of antiderivative = 6.00

method	result	size
default	\(\frac {2 a^{2} \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) c -\operatorname {EllipticF}\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) e -\operatorname {EllipticE}\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right ) c +e \operatorname {EllipticE}\left (\sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{c -1}}\right )\right ) \sqrt {-\frac {\left (-1+e \right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}\, \sqrt {b x +a}\, \sqrt {\frac {b e x +a e -b x}{a}}}{\sqrt {\frac {b c x +a c -b x}{a}}\, \left (b^{2} e \,x^{2}+2 a b e x -b^{2} x^{2}+a^{2} e -a b x \right ) b \left (-1+e \right ) \left (c -1\right )}\)	\(348\)
elliptic	\(\frac {\sqrt {\frac {b e x +a e -b x}{a}}\, a \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 e \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 (-1+e \right ) \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 )}{a \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}}\, \left (b e x +a e -b x \right )}\)	\(912\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1150 vs. \(2 (45) = 90\).

Time = 0.09 (sec) , antiderivative size = 1150, normalized size of antiderivative = 19.83 \[ \int \frac {\sqrt {e+\frac {b (-1+e) x}{a}}}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/3*((2*a^2*c - a^2*e - a^2)*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*b*c - a*b - (a*b*c - a*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*b^3*c^2 ...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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