\(\int \frac {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [3]

Optimal result

Integrand size = 33, antiderivative size = 284 \[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 B \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {-d e+c f} (B g-A h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}} \] Output:

2*B*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2)*EllipticE(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))/d/ 
f^(1/2)/h/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)-2*(c*f-d*e)^(1/2)*(-A 
*h+B*g)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)*Elliptic 
F(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)) 
/d/f^(1/2)/h/(f*x+e)^(1/2)/(h*x+g)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 19.32 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \left (-B d^2 \sqrt {-c+\frac {d e}{f}} (e+f x) (g+h x)-i B (d e-c f) h (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+i d (B e-A f) h (c+d x)^{3/2} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{d^2 \sqrt {-c+\frac {d e}{f}} f h \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \] Input:

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

Output:

(-2*(-(B*d^2*Sqrt[-c + (d*e)/f]*(e + f*x)*(g + h*x)) - I*B*(d*e - c*f)*h*( 
c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c 
+ d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c 
*f*h)/(d*e*h - c*f*h)] + I*d*(B*e - A*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f* 
x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[S 
qrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(d^2* 
Sqrt[-c + (d*e)/f]*f*h*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {176, 124, 123, 131, 131, 130}

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.

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 130

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

Input:

Int[(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[g + h*x]*Elli 
pticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/ 
(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h 
)]) - (2*Sqrt[-(d*e) + c*f]*(B*g - A*h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sq 
rt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqr 
t[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[e + 
f*x]*Sqrt[g + h*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 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 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 [A] (verified)

Time = 4.99 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.75

method	result	size
elliptic	\(\frac {\sqrt {\left (h x +g \right ) \left (f x +e \right ) \left (x d +c \right )}\, \left (\frac {2 A \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 B \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \left (\left (-\frac {c}{d}+\frac {g}{h}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )-\frac {g \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{h}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {x d +c}}\)	\(498\)
default	\(\frac {2 \left (A \operatorname {EllipticF}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) c d f h -A \operatorname {EllipticF}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) d^{2} e h -B \operatorname {EllipticF}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) c d f g +B \operatorname {EllipticF}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) d^{2} e g -B \operatorname {EllipticE}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) c^{2} f h +B \operatorname {EllipticE}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) c d e h +B \operatorname {EllipticE}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) c d f g -B \operatorname {EllipticE}\left (\sqrt {\frac {f \left (x d +c \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{f \left (c h -d g \right )}}\right ) d^{2} e g \right ) \sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {-\frac {\left (h x +g \right ) d}{c h -d g}}\, \sqrt {\frac {f \left (x d +c \right )}{c f -d e}}\, \sqrt {x d +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}{h f \,d^{2} \left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right )}\)	\(559\)

Input:

int((B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVERB 
OSE)
 

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*A*(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*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)*((-c/d+g/h)*EllipticE(((x+c/d)/( 
c/d-e/f))^(1/2),((-c/d+e/f)/(-c/d+g/h))^(1/2))-g/h*EllipticF(((x+c/d)/(c/d 
-e/f))^(1/2),((-c/d+e/f)/(-c/d+g/h))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (250) = 500\).

Time = 0.11 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.36 \[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {d f h} B d f h {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + {\left (d e + c f\right )} h}{3 \, d f h}\right )\right ) + {\left (B d f g + {\left (B d e + {\left (B c - 3 \, A d\right )} f\right )} h\right )} \sqrt {d f h} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + {\left (d e + c f\right )} h}{3 \, d f h}\right )\right )}}{3 \, d^{2} f^{2} h^{2}} \] Input:

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

Output:

-2/3*(3*sqrt(d*f*h)*B*d*f*h*weierstrassZeta(4/3*(d^2*f^2*g^2 - (d^2*e*f + 
c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2* 
d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e*f 
^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3 
*f^3)*h^3)/(d^3*f^3*h^3), weierstrassPInverse(4/3*(d^2*f^2*g^2 - (d^2*e*f 
+ c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2*f^2)*h^2)/(d^2*f^2*h^2), -4/27*( 
2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)*g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e 
*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c 
^3*f^3)*h^3)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x + d*f*g + (d*e + c*f)*h)/(d*f*h 
))) + (B*d*f*g + (B*d*e + (B*c - 3*A*d)*f)*h)*sqrt(d*f*h)*weierstrassPInve 
rse(4/3*(d^2*f^2*g^2 - (d^2*e*f + c*d*f^2)*g*h + (d^2*e^2 - c*d*e*f + c^2* 
f^2)*h^2)/(d^2*f^2*h^2), -4/27*(2*d^3*f^3*g^3 - 3*(d^3*e*f^2 + c*d^2*f^3)* 
g^2*h - 3*(d^3*e^2*f - 4*c*d^2*e*f^2 + c^2*d*f^3)*g*h^2 + (2*d^3*e^3 - 3*c 
*d^2*e^2*f - 3*c^2*d*e*f^2 + 2*c^3*f^3)*h^3)/(d^3*f^3*h^3), 1/3*(3*d*f*h*x 
 + d*f*g + (d*e + c*f)*h)/(d*f*h)))/(d^2*f^2*h^2)
 

Sympy [F]

\[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {A + B x}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {B x + A}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((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)*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)*b + i 
nt((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)*a