\(\int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [4]

Optimal result

Integrand size = 40, antiderivative size = 313 \[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 B \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (A-\frac {a B}{b}\right ) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d) \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \] Output:

2*B*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1 
/2)*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f/(-c*h+ 
d*g))^(1/2))/b/d/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*(A-a*B/b)*(c*f-d*e) 
^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)*EllipticP 
i(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),-b*(-c*f+d*e)/(-a*d+b*c)/f,((-c*f+ 
d*e)*h/f/(-c*h+d*g))^(1/2))/(-a*d+b*c)/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.07 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 i \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left (b (-B c+A d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )+(-A b+a B) d \operatorname {EllipticPi}\left (-\frac {b c f-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{b (-b c+a d) \sqrt {-c+\frac {d e}{f}} f \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {g+h x}} \] Input:

Integrate[(A + B*x)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), 
x]
 

Output:

((2*I)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*(b*(-(B*c) + A*d)*E 
llipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e 
*h - c*f*h)] + (-(A*b) + a*B)*d*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c* 
f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - 
 c*f*h)]))/(b*(-(b*c) + a*d)*Sqrt[-c + (d*e)/f]*f*Sqrt[(d*(e + f*x))/(f*(c 
 + d*x))]*Sqrt[g + h*x])
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {2110, 27, 131, 131, 130, 187, 413, 413, 412}

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.

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

\(\Big \downarrow \) 2110

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 130

\(\displaystyle \left (A-\frac {a B}{b}\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}\)

\(\Big \downarrow \) 187

\(\displaystyle \frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-2 \left (A-\frac {a B}{b}\right ) \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {e-\frac {c f}{d}+\frac {f (c+d x)}{d}} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (A-\frac {a B}{b}\right ) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {g-\frac {c h}{d}+\frac {h (c+d x)}{d}}}d\sqrt {c+d x}}{\sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (A-\frac {a B}{b}\right ) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \int \frac {1}{(b c-a d-b (c+d x)) \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1}}d\sqrt {c+d x}}{\sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {2 B \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b d \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \left (A-\frac {a B}{b}\right ) \sqrt {c f-d e} \sqrt {\frac {f (c+d x)}{d e-c f}+1} \sqrt {\frac {h (c+d x)}{d g-c h}+1} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} (b c-a d) \sqrt {\frac {f (c+d x)}{d}-\frac {c f}{d}+e} \sqrt {\frac {h (c+d x)}{d}-\frac {c h}{d}+g}}\)

Input:

Int[(A + B*x)/((a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
 

Output:

(2*B*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x)) 
/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]] 
, ((d*e - c*f)*h)/(f*(d*g - c*h))])/(b*d*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h* 
x]) - (2*(A - (a*B)/b)*Sqrt[-(d*e) + c*f]*Sqrt[1 + (f*(c + d*x))/(d*e - c* 
f)]*Sqrt[1 + (h*(c + d*x))/(d*g - c*h)]*EllipticPi[-((b*(d*e - c*f))/((b*c 
 - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c 
*f)*h)/(f*(d*g - c*h))])/((b*c - a*d)*Sqrt[f]*Sqrt[e - (c*f)/d + (f*(c + d 
*x))/d]*Sqrt[g - (c*h)/d + (h*(c + d*x))/d])
 

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 130

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/(b*c - a*d), 0] && GtQ 
[b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f 
*x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])
 

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 187

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] &&  !SimplerQ[e + f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]
 

rule 412

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

rule 413

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

rule 2110

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_))^(q_.), x_Symbol] :> Simp[PolynomialRem 
ainder[Px, a + b*x, x]   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^ 
q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d 
*x)^n*(e + f*x)^p*(g + h*x)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p 
, q}, x] && PolyQ[Px, x] && EqQ[m, -1]
 

Maple [A] (verified)

Time = 7.74 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.53

method	result	size
elliptic	\(\frac {\sqrt {\left (h x +g \right ) \left (f x +e \right ) \left (x d +c \right )}\, \left (\frac {2 B \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{b \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (A b -B a \right ) \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {a}{b}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{b^{2} \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}\, \left (-\frac {c}{d}+\frac {a}{b}\right )}\right )}{\sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {x d +c}}\)	\(478\)
default	\(\frac {2 \sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {x d +c}\, \sqrt {\frac {f \left (x d +c \right )}{c f -d e}}\, \sqrt {-\frac {\left (h x +g \right ) d}{c h -d g}}\, \sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \left (A \operatorname {EllipticPi}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, -\frac {b \left (c f -d e \right )}{f \left (a d -b c \right )}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) b c d f -A \operatorname {EllipticPi}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, -\frac {b \left (c f -d e \right )}{f \left (a d -b c \right )}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) b \,d^{2} e +B \operatorname {EllipticF}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) a c d f -B \operatorname {EllipticF}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) a \,d^{2} e -B \operatorname {EllipticF}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) b \,c^{2} f +B \operatorname {EllipticF}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) b c d e -B \operatorname {EllipticPi}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, -\frac {b \left (c f -d e \right )}{f \left (a d -b c \right )}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) a c d f +B \operatorname {EllipticPi}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, -\frac {b \left (c f -d e \right )}{f \left (a d -b c \right )}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) a \,d^{2} e \right )}{b d f \left (a d -b c \right ) \left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right )}\)	\(665\)

Input:

int((B*x+A)/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RE 
TURNVERBOSE)
 

Output:

((h*x+g)*(f*x+e)*(d*x+c))^(1/2)/(h*x+g)^(1/2)/(f*x+e)^(1/2)/(d*x+c)^(1/2)* 
(2*B/b*(c/d-e/f)*((x+c/d)/(c/d-e/f))^(1/2)*((x+g/h)/(-c/d+g/h))^(1/2)*((x+ 
e/f)/(-c/d+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c* 
f*g*x+d*e*g*x+c*e*g)^(1/2)*EllipticF(((x+c/d)/(c/d-e/f))^(1/2),((-c/d+e/f) 
/(-c/d+g/h))^(1/2))+2*(A*b-B*a)/b^2*(c/d-e/f)*((x+c/d)/(c/d-e/f))^(1/2)*(( 
x+g/h)/(-c/d+g/h))^(1/2)*((x+e/f)/(-c/d+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d 
*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)/(-c/d+a/b)*Ellipti 
cPi(((x+c/d)/(c/d-e/f))^(1/2),(-c/d+e/f)/(-c/d+a/b),((-c/d+e/f)/(-c/d+g/h) 
)^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((B*x+A)/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, alg 
orithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((B*x + A)/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), 
 x)
 

Giac [F]

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

integrate((B*x+A)/(b*x+a)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, alg 
orithm="giac")
 

Output:

integrate((B*x + A)/((b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), 
 x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {d x +c}}{d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}d x \] Input:

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

Output:

int((sqrt(g + h*x)*sqrt(e + f*x)*sqrt(c + d*x))/(c*e*g + c*e*h*x + c*f*g*x 
 + c*f*h*x**2 + d*e*g*x + d*e*h*x**2 + d*f*g*x**2 + d*f*h*x**3),x)