\(\int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [29]

Optimal result

Integrand size = 35, antiderivative size = 367 \[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}-\frac {4 C \sqrt {-d e+c f} (d f g+d e h+c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}+\frac {2 \sqrt {-d e+c f} \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right ) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt {e+f x} \sqrt {g+h x}} \] Output:

2/3*C*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/d/f/h-4/3*C*(c*f-d*e)^(1/2 
)*(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^2+c*C*h*(-e*h+f*g)+C*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 = 23.87 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.06 \[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {\sqrt {c+d x} \left (2 C d^2 f h (e+f x) (g+h x)-\frac {4 C d^2 (d f g+d e h+c f h) (e+f x) (g+h x)}{c+d x}-4 i C \sqrt {-c+\frac {d e}{f}} f h (d f g+d e h+c f h) \sqrt {c+d x} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+\frac {2 i d h \left (3 A d f^2 h+c C f (-f g+e h)+C d e (f g+2 e h)\right ) \sqrt {c+d x} \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )}{\sqrt {-c+\frac {d e}{f}}}\right )}{3 d^3 f^2 h^2 \sqrt {e+f x} \sqrt {g+h x}} \] Input:

Integrate[(A + C*x^2)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
 

Output:

(Sqrt[c + d*x]*(2*C*d^2*f*h*(e + f*x)*(g + h*x) - (4*C*d^2*(d*f*g + d*e*h 
+ c*f*h)*(e + f*x)*(g + h*x))/(c + d*x) - (4*I)*C*Sqrt[-c + (d*e)/f]*f*h*( 
d*f*g + d*e*h + c*f*h)*Sqrt[c + d*x]*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqr 
t[(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)] + ((2*I)*d*h*(3*A*d*f^2*h + c 
*C*f*(-(f*g) + e*h) + C*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[Sqr 
t[-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 = 0.65 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {2118, 27, 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.

\(\displaystyle \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\)

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

\(\displaystyle \frac {\frac {\left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}-\frac {2 C (c f h+d e h+d f g) \int \frac {\sqrt {g+h x}}{\sqrt {c+d x} \sqrt {e+f x}}dx}{h}}{3 d f h}+\frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {\frac {\left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}-\frac {2 C \sqrt {g+h x} \sqrt {\frac {d (e+f x)}{d e-c f}} (c f h+d e h+d f g) \int \frac {\sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}+\frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {\frac {\left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{h}-\frac {4 C \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} (c f h+d e h+d f g) E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}+\frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\frac {\sqrt {\frac {d (e+f x)}{d e-c f}} \left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}}dx}{h \sqrt {e+f x}}-\frac {4 C \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} (c f h+d e h+d f g) E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}+\frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {\frac {\sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}}dx}{h \sqrt {e+f x} \sqrt {g+h x}}-\frac {4 C \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} (c f h+d e h+d f g) E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}+\frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

\(\Big \downarrow \) 130

\(\displaystyle \frac {\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}}-\frac {4 C \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} (c f h+d e h+d f g) E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}}{3 d f h}+\frac {2 C \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 d f h}\)

Input:

Int[(A + C*x^2)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
 

Output:

(2*C*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) + ((-4*C*Sqrt[-( 
d*e) + c*f]*(d*f*g + d*e*h + c*f*h)*Sqrt[(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^2 + c*C*h*(f*g - e*h) + 
 C*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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2118

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo 
n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 
1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + 
q + 1))   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + 
n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q 
- 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + 
 c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m 
 + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] && PolyQ[Px, x]
 

Maple [A] (verified)

Time = 4.80 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.66

method	result	size
elliptic	\(\frac {\sqrt {\left (h x +g \right ) \left (f x +e \right ) \left (x d +c \right )}\, \left (\frac {2 C \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}{3 d f h}+\frac {2 \left (A -\frac {2 C \left (\frac {1}{2} c e h +\frac {1}{2} c f g +\frac {1}{2} d e g \right )}{3 d f h}\right ) \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}-\frac {4 C \left (c f h +d e h +d f g \right ) \left (\frac {c}{d}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {g}{h}}{-\frac {c}{d}+\frac {g}{h}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \left (\left (-\frac {c}{d}+\frac {g}{h}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )-\frac {g \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {e}{f}}}, \sqrt {\frac {-\frac {c}{d}+\frac {e}{f}}{-\frac {c}{d}+\frac {g}{h}}}\right )}{h}\right )}{3 d f h \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {h x +g}\, \sqrt {f x +e}\, \sqrt {x d +c}}\)	\(611\)
default	\(\text {Expression too large to display}\)	\(1804\)

Input:

int((C*x^2+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVE 
RBOSE)
 

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*C/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-2/3*C/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))-4/3*C/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)*Elli 
pticE(((x+c/d)/(c/d-e/f))^(1/2),((-c/d+e/f)/(-c/d+g/h))^(1/2))-g/h*Ellipti 
cF(((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. 775 vs. \(2 (321) = 642\).

Time = 0.11 (sec) , antiderivative size = 775, normalized size of antiderivative = 2.11 \[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/9*(3*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)*C*d^2*f^2*h^2 + (2*C*d^2* 
f^2*g^2 + (C*d^2*e*f + C*c*d*f^2)*g*h + (2*C*d^2*e^2 + C*c*d*e*f + (2*C*c^ 
2 + 9*A*d^2)*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)) + 6*(C*d^2*f^2*g*h + (C*d^2*e*f + C*c*d*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), we 
ierstrassPInverse(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]

\[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {A + C x^{2}}{\sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \] Input:

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

Output:

Integral((A + C*x**2)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x)
 

Maxima [F]

\[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

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

Output:

integrate((C*x^2 + A)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
 

Giac [F]

\[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C x^{2} + A}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \] Input:

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

Output:

integrate((C*x^2 + A)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+C x^2}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {C\,x^2+A}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

int((C*x^2+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**2)/(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)*c 
+ 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)*a