\(\int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\) [211]

Optimal result

Integrand size = 45, antiderivative size = 145 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 a^{3/2} B E\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)}+\frac {2 \sqrt {a} (a B e+A (b-b e)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right ),\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)} \] Output:

-2*a^(3/2)*B*EllipticE((1-c)^(1/2)*(b*x+a)^(1/2)/a^(1/2),((1-e)/(1-c))^(1/ 
2))/b^2/(1-c)^(1/2)/(1-e)+2*a^(1/2)*(a*B*e+A*(-b*e+b))*EllipticF((1-c)^(1/ 
2)*(b*x+a)^(1/2)/a^(1/2),((1-e)/(1-c))^(1/2))/b^2/(1-c)^(1/2)/(1-e)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 16.27 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.13 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 \sqrt {\frac {a}{-1+c}} (a+b x)^{3/2} \left (-B \sqrt {\frac {a}{-1+c}} \left (-1+c+\frac {a}{a+b x}\right ) \left (-1+e+\frac {a}{a+b x}\right )-\frac {i a B (-1+e) \sqrt {\frac {-1+c+\frac {a}{a+b x}}{-1+c}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{-1+c}}}{\sqrt {a+b x}}\right )|\frac {-1+c}{-1+e}\right )}{\sqrt {a+b x}}+\frac {i (a B c+A (b-b c)) (-1+e) \sqrt {\frac {-1+c+\frac {a}{a+b x}}{-1+c}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{-1+c}}}{\sqrt {a+b x}}\right ),\frac {-1+c}{-1+e}\right )}{\sqrt {a+b x}}\right )}{a b^2 (-1+e) \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \] Input:

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

Output:

(-2*Sqrt[a/(-1 + c)]*(a + b*x)^(3/2)*(-(B*Sqrt[a/(-1 + c)]*(-1 + c + a/(a 
+ b*x))*(-1 + e + a/(a + b*x))) - (I*a*B*(-1 + e)*Sqrt[(-1 + c + a/(a + b* 
x))/(-1 + c)]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticE[I*ArcSinh[Sq 
rt[a/(-1 + c)]/Sqrt[a + b*x]], (-1 + c)/(-1 + e)])/Sqrt[a + b*x] + (I*(a*B 
*c + A*(b - b*c))*(-1 + e)*Sqrt[(-1 + c + a/(a + b*x))/(-1 + c)]*Sqrt[(-1 
+ e + a/(a + b*x))/(-1 + e)]*EllipticF[I*ArcSinh[Sqrt[a/(-1 + c)]/Sqrt[a + 
 b*x]], (-1 + c)/(-1 + e)])/Sqrt[a + b*x]))/(a*b^2*(-1 + e)*Sqrt[c + (b*(- 
1 + c)*x)/a]*Sqrt[e + (b*(-1 + e)*x)/a])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {176, 123, 129}

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.

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 123

\(\displaystyle \left (\frac {a B e}{b-b e}+A\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c-\frac {b (1-c) x}{a}} \sqrt {e-\frac {b (1-e) x}{a}}}dx-\frac {2 a^{3/2} B E\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {2 \sqrt {a} \left (\frac {a B e}{b-b e}+A\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right ),\frac {1-e}{1-c}\right )}{b \sqrt {1-c}}-\frac {2 a^{3/2} B E\left (\arcsin \left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)}\)

Input:

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

Output:

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

rule 129

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ 
Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - 
a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ 
[(b*e - a*f)/b, 0] && PosQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d 
*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(( 
-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ 
[((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f 
/b]))
 

rule 176

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* 
Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f   Int[Sqrt[e + f*x]/(Sqrt[a + b*x 
]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f   Int[1/(Sqrt[a + b*x]*Sqrt[c 
 + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim 
plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(603\) vs. \(2(127)=254\).

Time = 12.84 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.17

method	result	size
default	\(\frac {2 \left (A \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 ) b c e -A \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 ) b \,e^{2}-B \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 ) a c e +B \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 ) a \,e^{2}-A \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 ) b c +A \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 ) b e +B \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 ) a c -B \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 ) a e -B \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 ) a c +B \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 ) a e \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 )}}\, 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} \left (c -1\right ) b^{2}}\)	\(604\)
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)/(b*x+a)^(1/2)/(c+b*(c-1)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x,m 
ethod=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1228 vs. \(2 (117) = 234\).

Time = 0.10 (sec) , antiderivative size = 1228, normalized size of antiderivative = 8.47 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/3*((B*a^3 - 3*A*a^2*b - (2*B*a^3 - 3*A*a^2*b)*c - (2*B*a^3 - 3*A*a^2*b 
- 3*(B*a^3 - A*a^2*b)*c)*e)*sqrt(-(b^3*c - b^3 - (b^3*c - b^3)*e)/a^2)*wei 
erstrassPInverse(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*(B*a^2*b*c - B*a^2*b - (B*a^2*b*c - B*a^2*b)*e)*s 
qrt(-(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...
 

Sympy [F]

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

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

Output:

Integral((A + B*x)/(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]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((B*x+A)/(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