\(\int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [97]

Optimal result

Integrand size = 38, antiderivative size = 384 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 C \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b d f}+\frac {2 \sqrt {-b c+a d} (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 d^{3/2} f^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (3 b d f (B e-A f)-a C f (d e-c f)-b C e (2 d e+c f)) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 d^{3/2} f^2 \sqrt {c+d x} \sqrt {e+f x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.58 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {\sqrt {a+b x} \left (2 b^2 C d f (c+d x) (e+f x)-\frac {2 b^2 (-3 b B d f+2 a C d f+2 b C (d e+c f)) (c+d x) (e+f x)}{a+b x}+2 i \sqrt {-a+\frac {b c}{d}} d f (3 b B d f-2 a C d f-2 b C (d e+c f)) \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )+\frac {2 i b f \left (a C d (-d e+c f)+b \left (2 c^2 C f+3 A d^2 f+c d (C e-3 B f)\right )\right ) \sqrt {a+b x} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )}{\sqrt {-a+\frac {b c}{d}}}\right )}{3 b^3 d^2 f^2 \sqrt {c+d x} \sqrt {e+f x}} \] Input:

Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, 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.

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

\(\Big \downarrow \) 2118

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 130

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

Input:

Int[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
 

Output:

(2*C*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*d*f) - ((-2*Sqrt[-(b* 
c) + a*d]*(3*b*B*d*f - 2*a*C*d*f - 2*b*C*(d*e + c*f))*Sqrt[(b*(c + d*x))/( 
b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-( 
b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*f*Sqrt[c + d*x] 
*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(B*e - 
A*f) - a*C*f*(d*e - c*f) - b*C*e*(2*d*e + c*f))*Sqrt[(b*(c + d*x))/(b*c - 
a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b 
*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*f*S 
qrt[c + d*x]*Sqrt[e + f*x]))/(3*b*d*f)
 

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 = 5.25 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.60

method	result	size
elliptic	\(\frac {\sqrt {\left (f x +e \right ) \left (b x +a \right ) \left (x d +c \right )}\, \left (\frac {2 C \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}{3 b d f}+\frac {2 \left (A -\frac {2 C \left (\frac {1}{2} a c f +\frac {1}{2} a d e +\frac {1}{2} b c e \right )}{3 b d f}\right ) \left (\frac {c}{d}-\frac {a}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}+\frac {2 \left (B -\frac {2 C \left (a d f +b c f +b d e \right )}{3 b d f}\right ) \left (\frac {c}{d}-\frac {a}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {c}{d}+\frac {e}{f}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {c}{d}+\frac {a}{b}}}\, \left (\left (-\frac {c}{d}+\frac {e}{f}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )-\frac {e \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {a}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {a}{b}}{-\frac {c}{d}+\frac {e}{f}}}\right )}{f}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\right )}{\sqrt {b x +a}\, \sqrt {x d +c}\, \sqrt {f x +e}}\)	\(615\)
default	\(\text {Expression too large to display}\)	\(2545\)

Input:

int((C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x,method=_RETU 
RNVERBOSE)
 

Output:

((f*x+e)*(b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)* 
(2/3*C/b/d/f*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c* 
e*x+a*c*e)^(1/2)+2*(A-2/3*C/b/d/f*(1/2*a*c*f+1/2*a*d*e+1/2*b*c*e))*(c/d-a/ 
b)*((x+c/d)/(c/d-a/b))^(1/2)*((x+e/f)/(-c/d+e/f))^(1/2)*((x+a/b)/(-c/d+a/b 
))^(1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+ 
a*c*e)^(1/2)*EllipticF(((x+c/d)/(c/d-a/b))^(1/2),((-c/d+a/b)/(-c/d+e/f))^( 
1/2))+2*(B-2/3*C/b/d/f*(a*d*f+b*c*f+b*d*e))*(c/d-a/b)*((x+c/d)/(c/d-a/b))^ 
(1/2)*((x+e/f)/(-c/d+e/f))^(1/2)*((x+a/b)/(-c/d+a/b))^(1/2)/(b*d*f*x^3+a*d 
*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)*((-c/d+e/f 
)*EllipticE(((x+c/d)/(c/d-a/b))^(1/2),((-c/d+a/b)/(-c/d+e/f))^(1/2))-e/f*E 
llipticF(((x+c/d)/(c/d-a/b))^(1/2),((-c/d+a/b)/(-c/d+e/f))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (336) = 672\).

Time = 0.13 (sec) , antiderivative size = 807, normalized size of antiderivative = 2.10 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor 
ithm="fricas")
 

Output:

2/9*(3*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*C*b^2*d^2*f^2 + (2*C*b^2* 
d^2*e^2 + (C*b^2*c*d + (C*a*b - 3*B*b^2)*d^2)*e*f + (2*C*b^2*c^2 + (C*a*b 
- 3*B*b^2)*c*d + (2*C*a^2 - 3*B*a*b + 9*A*b^2)*d^2)*f^2)*sqrt(b*d*f)*weier 
strassPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b 
*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + 
a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^ 
3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3 
*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f)) + 3*(2*C*b^2*d^2*e*f + (2*C* 
b^2*c*d + (2*C*a*b - 3*B*b^2)*d^2)*f^2)*sqrt(b*d*f)*weierstrassZeta(4/3*(b 
^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/ 
(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3* 
(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d 
 - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), weierstrassPInverse(4/3* 
(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2 
)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 
3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2 
*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e 
 + (b*c + a*d)*f)/(b*d*f))))/(b^3*d^3*f^3)
 

Sympy [F]

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

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

Output:

Integral((A + B*x + C*x**2)/(sqrt(a + b*x)*sqrt(c + d*x)*sqrt(e + f*x)), x 
)
 

Maxima [F]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)), x 
)
 

Giac [F]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor 
ithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)), x 
)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {too large to display} \] Input:

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

Output:

(2*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*b + 2*int((sqrt(e + f*x)*sqrt 
(c + d*x)*sqrt(a + b*x)*x**2)/(a**2*c*d*e*f + a**2*c*d*f**2*x + a**2*d**2* 
e*f*x + a**2*d**2*f**2*x**2 + a*b*c**2*e*f + a*b*c**2*f**2*x + a*b*c*d*e** 
2 + 3*a*b*c*d*e*f*x + 2*a*b*c*d*f**2*x**2 + a*b*d**2*e**2*x + 2*a*b*d**2*e 
*f*x**2 + a*b*d**2*f**2*x**3 + b**2*c**2*e*f*x + b**2*c**2*f**2*x**2 + b** 
2*c*d*e**2*x + 2*b**2*c*d*e*f*x**2 + b**2*c*d*f**2*x**3 + b**2*d**2*e**2*x 
**2 + b**2*d**2*e*f*x**3),x)*a**2*c*d**2*f**2 - 3*int((sqrt(e + f*x)*sqrt( 
c + d*x)*sqrt(a + b*x)*x**2)/(a**2*c*d*e*f + a**2*c*d*f**2*x + a**2*d**2*e 
*f*x + a**2*d**2*f**2*x**2 + a*b*c**2*e*f + a*b*c**2*f**2*x + a*b*c*d*e**2 
 + 3*a*b*c*d*e*f*x + 2*a*b*c*d*f**2*x**2 + a*b*d**2*e**2*x + 2*a*b*d**2*e* 
f*x**2 + a*b*d**2*f**2*x**3 + b**2*c**2*e*f*x + b**2*c**2*f**2*x**2 + b**2 
*c*d*e**2*x + 2*b**2*c*d*e*f*x**2 + b**2*c*d*f**2*x**3 + b**2*d**2*e**2*x* 
*2 + b**2*d**2*e*f*x**3),x)*a*b**2*d**2*f**2 + 4*int((sqrt(e + f*x)*sqrt(c 
 + d*x)*sqrt(a + b*x)*x**2)/(a**2*c*d*e*f + a**2*c*d*f**2*x + a**2*d**2*e* 
f*x + a**2*d**2*f**2*x**2 + a*b*c**2*e*f + a*b*c**2*f**2*x + a*b*c*d*e**2 
+ 3*a*b*c*d*e*f*x + 2*a*b*c*d*f**2*x**2 + a*b*d**2*e**2*x + 2*a*b*d**2*e*f 
*x**2 + a*b*d**2*f**2*x**3 + b**2*c**2*e*f*x + b**2*c**2*f**2*x**2 + b**2* 
c*d*e**2*x + 2*b**2*c*d*e*f*x**2 + b**2*c*d*f**2*x**3 + b**2*d**2*e**2*x** 
2 + b**2*d**2*e*f*x**3),x)*a*b*c**2*d*f**2 + 4*int((sqrt(e + f*x)*sqrt(c + 
 d*x)*sqrt(a + b*x)*x**2)/(a**2*c*d*e*f + a**2*c*d*f**2*x + a**2*d**2*e...