\(\int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\) [212]

Optimal result

Integrand size = 39, antiderivative size = 208 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\frac {2 \sqrt {a} B \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b d \sqrt {1-e} \sqrt {\frac {b (c+d x)}{b c-a d}}}-\frac {2 \sqrt {a} (B c-A d) \sqrt {\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right ),-\frac {a d}{(b c-a d) (1-e)}\right )}{b d \sqrt {1-e} \sqrt {c+d x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 18.37 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.50 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=-\frac {2 \sqrt {\frac {a}{-1+e}} (a+b x)^{3/2} \left (-\frac {b B \sqrt {\frac {a}{-1+e}} (c+d x) (a e+b (-1+e) x)}{(a+b x)^2}-\frac {i a B d \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{-1+e}}}{\sqrt {a+b x}}\right )|\frac {(b c-a d) (-1+e)}{a d}\right )}{\sqrt {a+b x}}+\frac {i d (a B e+A (b-b e)) \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {a}{-1+e}}}{\sqrt {a+b x}}\right ),\frac {(b c-a d) (-1+e)}{a d}\right )}{\sqrt {a+b x}}\right )}{a b^2 d \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {176, 124, 123, 131, 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.

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 129

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

Input:

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

Output:

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

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]
 

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 131

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

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. \(728\) vs. \(2(182)=364\).

Time = 4.54 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.50

method	result	size
elliptic	\(\frac {\sqrt {\frac {\left (b x +a \right ) \left (x d +c \right ) \left (b e x +a e -b x \right )}{a}}\, \left (\frac {2 A \left (\frac {a e}{\left (-1+e \right ) b}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {a e}{\left (-1+e \right ) b}}{\frac {a e}{\left (-1+e \right ) b}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a e}{\left (-1+e \right ) b}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {a e}{\left (-1+e \right ) b}+\frac {c}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {a e}{\left (-1+e \right ) b}}{\frac {a e}{\left (-1+e \right ) b}-\frac {c}{d}}}, \sqrt {\frac {-\frac {a e}{\left (-1+e \right ) b}+\frac {c}{d}}{-\frac {a e}{\left (-1+e \right ) b}+\frac {a}{b}}}\right )}{\sqrt {\frac {b^{2} d e \,x^{3}}{a}+2 b d e \,x^{2}+\frac {b^{2} c e \,x^{2}}{a}-\frac {x^{3} d \,b^{2}}{a}+a d e x +2 b c e x -b d \,x^{2}-\frac {b^{2} c \,x^{2}}{a}+a c e -b c x}}+\frac {2 B \left (\frac {a e}{\left (-1+e \right ) b}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {a e}{\left (-1+e \right ) b}}{\frac {a e}{\left (-1+e \right ) b}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a e}{\left (-1+e \right ) b}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {a e}{\left (-1+e \right ) b}+\frac {c}{d}}}\, \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 {c}{d}}}, \sqrt {\frac {-\frac {a e}{\left (-1+e \right ) b}+\frac {c}{d}}{-\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 {c}{d}}}, \sqrt {\frac {-\frac {a e}{\left (-1+e \right ) b}+\frac {c}{d}}{-\frac {a e}{\left (-1+e \right ) b}+\frac {a}{b}}}\right )}{b}\right )}{\sqrt {\frac {b^{2} d e \,x^{3}}{a}+2 b d e \,x^{2}+\frac {b^{2} c e \,x^{2}}{a}-\frac {x^{3} d \,b^{2}}{a}+a d e x +2 b c e x -b d \,x^{2}-\frac {b^{2} c \,x^{2}}{a}+a c e -b c x}}\right )}{\sqrt {b x +a}\, \sqrt {x d +c}\, \sqrt {\frac {b e x +a e -b x}{a}}}\)	\(729\)
default	\(\frac {2 \sqrt {b x +a}\, \sqrt {x d +c}\, \sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {-\frac {\left (x d +c \right ) b \left (-1+e \right )}{a d e -b c e +b c}}\, \left (A \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b d \,e^{2}-A \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) b^{2} c \,e^{2}-B \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a^{2} d \,e^{2}+B \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c \,e^{2}-A \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b d e +2 A \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) b^{2} c e +B \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a^{2} d e -2 B \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c e -B \operatorname {EllipticE}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a^{2} d e +B \operatorname {EllipticE}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c e -A \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) b^{2} c +B \operatorname {EllipticF}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c -B \operatorname {EllipticE}\left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c \right )}{\sqrt {\frac {b e x +a e -b x}{a}}\, \left (b d \,x^{2}+a d x +b c x +a c \right ) \left (-1+e \right )^{2} d \,b^{2}}\)	\(940\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1126 vs. \(2 (176) = 352\).

Time = 0.11 (sec) , antiderivative size = 1126, normalized size of antiderivative = 5.41 \[ \int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/3*((B*a*b*c + (B*a^2 - 3*A*a*b)*d - (B*a*b*c + (2*B*a^2 - 3*A*a*b)*d)*e) 
*sqrt((b^2*d*e - b^2*d)/a)*weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^ 
2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2 - (2*b^2*c^2 - 3*a*b*c*d + a^2 
*d^2)*e)/(b^2*d^2*e^2 - 2*b^2*d^2*e + b^2*d^2), 4/27*(2*b^3*c^3 - 3*a*b^2* 
c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3 - 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c 
*d^2 - a^3*d^3)*e^3 + 3*(2*b^3*c^3 - 5*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d 
^3)*e^2 - 3*(2*b^3*c^3 - 4*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*e)/(b^3*d^ 
3*e^3 - 3*b^3*d^3*e^2 + 3*b^3*d^3*e - b^3*d^3), -1/3*(b*c + a*d - (b*c + 2 
*a*d)*e - 3*(b*d*e - b*d)*x)/(b*d*e - b*d)) - 3*(B*a*b*d*e - B*a*b*d)*sqrt 
((b^2*d*e - b^2*d)/a)*weierstrassZeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2 + ( 
b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2 - (2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*e)/ 
(b^2*d^2*e^2 - 2*b^2*d^2*e + b^2*d^2), 4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3 
*a^2*b*c*d^2 + 2*a^3*d^3 - 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^ 
3*d^3)*e^3 + 3*(2*b^3*c^3 - 5*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*e^2 - 
 3*(2*b^3*c^3 - 4*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*e)/(b^3*d^3*e^3 - 3 
*b^3*d^3*e^2 + 3*b^3*d^3*e - b^3*d^3), weierstrassPInverse(4/3*(b^2*c^2 - 
a*b*c*d + a^2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2 - (2*b^2*c^2 - 3*a 
*b*c*d + a^2*d^2)*e)/(b^2*d^2*e^2 - 2*b^2*d^2*e + b^2*d^2), 4/27*(2*b^3*c^ 
3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3 - 2*(b^3*c^3 - 3*a*b^2*c^2*d 
 + 3*a^2*b*c*d^2 - a^3*d^3)*e^3 + 3*(2*b^3*c^3 - 5*a*b^2*c^2*d + 4*a^2*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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