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

Optimal result

Integrand size = 38, antiderivative size = 404 \[ \int \frac {(a+b x) (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 b B \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}+\frac {2 \sqrt {-d e+c f} (3 A b d f h+3 a B d f h-2 b B (d f g+d e h+c f h)) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {-d e+c f} (3 a d f h (B g-A h)+b (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {g+h x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.28 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x) (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {\sqrt {c+d x} \left (2 b B d^2 f h (e+f x) (g+h x)-\frac {2 d^2 (-3 A b d f h-3 a B d f h+2 b B (d f g+d e h+c f h)) (e+f x) (g+h x)}{c+d x}+\frac {2 i (d e-c f) h (3 A b d f h+3 a B d f h-2 b B (d f g+d e h+c f h)) \sqrt {c+d x} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )}{\sqrt {-c+\frac {d e}{f}}}+\frac {2 i d h (3 a d f (-B e+A f) h+b (-3 A d e f h+B c f (-f g+e h)+B d e (f g+2 e h))) \sqrt {c+d x} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )}{\sqrt {-c+\frac {d e}{f}}}\right )}{3 d^3 f^2 h^2 \sqrt {e+f x} \sqrt {g+h x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {2097, 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.

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

\(\Big \downarrow \) 2097

\(\displaystyle \frac {\int \frac {3 a A d f h-b B (d e g+c f g+c e h)+(3 A b d f h+3 a B d f h-2 b B (d f g+d e h+c f h)) x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{3 d f h}+\frac {2 b B \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {\frac {(3 a B d f h+3 A b d f h-2 b B (c f h+d e h+d f g)) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}}dx}{h}-\frac {(3 a d f h (B g-A h)+b (3 A d f g h-B c h (f g-e h)-B d g (e h+2 f g))) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}}{3 d f h}+\frac {2 b B \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {\frac {\sqrt {g+h x} \sqrt {\frac {d (e+f x)}{d e-c f}} (3 a B d f h+3 A b d f h-2 b B (c f h+d e h+d f g)) \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {(3 a d f h (B g-A h)+b (3 A d f g h-B c h (f g-e h)-B d g (e h+2 f g))) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}}{3 d f h}+\frac {2 b B \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

\(\Big \downarrow \) 123

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 130

\(\displaystyle \frac {\frac {2 \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right ) (3 a B d f h+3 A b d f h-2 b B (c f h+d e h+d f g))}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right ) (3 a d f h (B g-A h)+b (3 A d f g h-B c h (f g-e h)-B d g (e h+2 f g)))}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}}{3 d f h}+\frac {2 b B \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

Input:

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

Output:

(2*b*B*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) + ((2*Sqrt[-(d 
*e) + c*f]*(3*A*b*d*f*h + 3*a*B*d*f*h - 2*b*B*(d*f*g + d*e*h + c*f*h))*Sqr 
t[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[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] 
*(3*a*d*f*h*(B*g - A*h) + b*(3*A*d*f*g*h - B*c*h*(f*g - e*h) - B*d*g*(2*f* 
g + e*h)))*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))])/(d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[g + h*x]))/(3*d*f*h 
)
 

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]
 

rule 2097

Int[(((a_.) + (b_.)*(x_))*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x_)]*S 
qrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[2*b*B* 
Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(3*d*f*h)), x] + Simp[1/(3*d*f*h 
)   Int[(1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[3*a*A*d*f*h - 
b*B*(d*e*g + c*f*g + c*e*h) + (3*A*b*d*f*h + B*(3*a*d*f*h - 2*b*(d*f*g + d* 
e*h + c*f*h)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x]
 

Maple [A] (verified)

Time = 5.82 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.55

method	result	size
elliptic	\(\frac {\sqrt {\left (h x +g \right ) \left (f x +e \right ) \left (x d +c \right )}\, \left (\frac {2 B b \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{3 d f h}+\frac {2 \left (A a -\frac {2 B b \left (\frac {1}{2} c e h +\frac {1}{2} c f g +\frac {1}{2} d e g \right )}{3 d f h}\right ) \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (A b +B a -\frac {2 B b \left (c f h +d e h +d f g \right )}{3 d f h}\right ) \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \left (\left (-\frac {c}{d}+\frac {g}{h}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )-\frac {g \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{h}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {x d +c}}\)	\(625\)
default	\(\text {Expression too large to display}\)	\(3224\)

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)* 
(2/3*B*b/d/f/h*(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)+2*(A*a-2/3*B*b/d/f/h*(1/2*c*e*h+1/2*c*f*g+1/2*d*e*g))*( 
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-2/3*B*b/d/f/h*(c*f*h+d*e*h+d*f*g))*(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. 842 vs. \(2 (358) = 716\).

Time = 0.16 (sec) , antiderivative size = 842, normalized size of antiderivative = 2.08 \[ \int \frac {(a+b x) (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx =\text {Too large to display} \] 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:

2/9*(3*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)*B*b*d^2*f^2*h^2 + (2*B*b* 
d^2*f^2*g^2 + (B*b*d^2*e*f + (B*b*c*d - 3*(B*a + A*b)*d^2)*f^2)*g*h + (2*B 
*b*d^2*e^2 + (B*b*c*d - 3*(B*a + A*b)*d^2)*e*f + (2*B*b*c^2 + 9*A*a*d^2 - 
3*(B*a + A*b)*c*d)*f^2)*h^2)*sqrt(d*f*h)*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)) + 3*(2*B*b*d^2*f^2*g*h + (2*B*b*d^2*e*f + (2*B*b*c*d - 3 
*(B*a + A*b)*d^2)*f^2)*h^2)*sqrt(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))))/(d^3*f^3*h^3)
 

Sympy [F]

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

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b x) (A+B x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {too large to display} \] 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:

(2*sqrt(g + h*x)*sqrt(e + f*x)*sqrt(c + d*x)*a*b - 3*int((sqrt(g + h*x)*sq 
rt(e + f*x)*sqrt(c + d*x)*x**2)/(c**2*e*f*g*h + c**2*e*f*h**2*x + c**2*f** 
2*g*h*x + c**2*f**2*h**2*x**2 + c*d*e**2*g*h + c*d*e**2*h**2*x + c*d*e*f*g 
**2 + 3*c*d*e*f*g*h*x + 2*c*d*e*f*h**2*x**2 + c*d*f**2*g**2*x + 2*c*d*f**2 
*g*h*x**2 + c*d*f**2*h**2*x**3 + d**2*e**2*g*h*x + d**2*e**2*h**2*x**2 + d 
**2*e*f*g**2*x + 2*d**2*e*f*g*h*x**2 + d**2*e*f*h**2*x**3 + d**2*f**2*g**2 
*x**2 + d**2*f**2*g*h*x**3),x)*a*b*c*d*f**2*h**2 - 3*int((sqrt(g + h*x)*sq 
rt(e + f*x)*sqrt(c + d*x)*x**2)/(c**2*e*f*g*h + c**2*e*f*h**2*x + c**2*f** 
2*g*h*x + c**2*f**2*h**2*x**2 + c*d*e**2*g*h + c*d*e**2*h**2*x + c*d*e*f*g 
**2 + 3*c*d*e*f*g*h*x + 2*c*d*e*f*h**2*x**2 + c*d*f**2*g**2*x + 2*c*d*f**2 
*g*h*x**2 + c*d*f**2*h**2*x**3 + d**2*e**2*g*h*x + d**2*e**2*h**2*x**2 + d 
**2*e*f*g**2*x + 2*d**2*e*f*g*h*x**2 + d**2*e*f*h**2*x**3 + d**2*f**2*g**2 
*x**2 + d**2*f**2*g*h*x**3),x)*a*b*d**2*e*f*h**2 - 3*int((sqrt(g + h*x)*sq 
rt(e + f*x)*sqrt(c + d*x)*x**2)/(c**2*e*f*g*h + c**2*e*f*h**2*x + c**2*f** 
2*g*h*x + c**2*f**2*h**2*x**2 + c*d*e**2*g*h + c*d*e**2*h**2*x + c*d*e*f*g 
**2 + 3*c*d*e*f*g*h*x + 2*c*d*e*f*h**2*x**2 + c*d*f**2*g**2*x + 2*c*d*f**2 
*g*h*x**2 + c*d*f**2*h**2*x**3 + d**2*e**2*g*h*x + d**2*e**2*h**2*x**2 + d 
**2*e*f*g**2*x + 2*d**2*e*f*g*h*x**2 + d**2*e*f*h**2*x**3 + d**2*f**2*g**2 
*x**2 + d**2*f**2*g*h*x**3),x)*a*b*d**2*f**2*g*h + int((sqrt(g + h*x)*sqrt 
(e + f*x)*sqrt(c + d*x)*x**2)/(c**2*e*f*g*h + c**2*e*f*h**2*x + c**2*f*...