\(\int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [98]

Optimal result

Integrand size = 38, antiderivative size = 422 \[ \int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \left (2 a^2 C d f+b^2 (c C e+A d f)-a b (C d e+c C f+B d f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt {d} \sqrt {-b c+a d} f (b e-a f) \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 (a C (d e-c f)-b (c C e-B c f+A d f)) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt {d} \sqrt {-b c+a d} f \sqrt {c+d x} \sqrt {e+f x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.18 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 \left (-b^2 \left (A b^2+a (-b B+a C)\right ) (c+d x) (e+f x)+\frac {b^2 \left (2 a^2 C d f+b^2 (c C e+A d f)-a b (C d e+c C f+B d f)\right ) (c+d x) (e+f x)}{d f}+\frac {i (b c-a d) \left (2 a^2 C d f+b^2 (c C e+A d f)-a b (C d e+c C f+B d f)\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )}{\sqrt {-a+\frac {b c}{d}} d}+\frac {i b (-b c+a d) (a C (d e-c f)+b (c C e-B d e+A d f)) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )}{\sqrt {-a+\frac {b c}{d}} d}\right )}{b^3 (b c-a d) (b e-a f) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \] Input:

Integrate[(A + B*x + C*x^2)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]), 
x]
 

Output:

(2*(-(b^2*(A*b^2 + a*(-(b*B) + a*C))*(c + d*x)*(e + f*x)) + (b^2*(2*a^2*C* 
d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d*e + c*C*f + B*d*f))*(c + d*x)*(e + f* 
x))/(d*f) + (I*(b*c - a*d)*(2*a^2*C*d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d*e 
 + c*C*f + B*d*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[ 
(b*(e + f*x))/(f*(a + b*x))]*EllipticE[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]*d) + (I*b* 
(-(b*c) + a*d)*(a*C*(d*e - c*f) + b*(c*C*e - B*d*e + A*d*f))*(a + b*x)^(3/ 
2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*Ell 
ipticF[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]*d)))/(b^3*(b*c - a*d)*(b*e - a*f)*Sqrt[a + 
 b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2117, 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.

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

\(\Big \downarrow \) 2117

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 176

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

\(\Big \downarrow \) 124

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

\(\Big \downarrow \) 123

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 131

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

\(\Big \downarrow \) 130

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

Input:

Int[(A + B*x + C*x^2)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
 

Output:

(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)*(b 
*e - a*f)*Sqrt[a + b*x]) + ((2*Sqrt[-(b*c) + a*d]*(2*a^2*C*d*f + b^2*(c*C* 
e + A*d*f) - a*b*(C*d*e + c*C*f + B*d*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]*(b*e - a*f)*(a*C*(d*e - c*f) 
 - b*(c*C*e - B*c*f + A*d*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*Sqrt[c + d*x]*Sqrt[ 
e + f*x]))/(b*(b*c - a*d)*(b*e - a*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 2117

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_ 
.)*(x_))^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px, a + b*x, x], 
 R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*(c + 
d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Si 
mp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n* 
(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1 
) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x] 
, x], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, - 
1] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(783\) vs. \(2(382)=764\).

Time = 7.75 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.86

method	result	size
elliptic	\(\frac {\sqrt {\left (f x +e \right ) \left (b x +a \right ) \left (x d +c \right )}\, \left (-\frac {2 \left (b d f \,x^{2}+b c f x +b d e x +b c e \right ) \left (b^{2} A -a b B +a^{2} C \right )}{\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) b^{2} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}}+\frac {2 \left (\frac {B b -C a}{b^{2}}+\frac {\left (a d f -b c f -b d e \right ) \left (b^{2} A -a b B +a^{2} C \right )}{b^{2} \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right )}+\frac {\left (b c f +b d e \right ) \left (b^{2} A -a b B +a^{2} C \right )}{\left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right ) b^{2}}\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 (\frac {C}{b}+\frac {d f \left (b^{2} A -a b B +a^{2} C \right )}{b \left (a^{2} d f -a b c f -a b d e +c e \,b^{2}\right )}\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}}\)	\(784\)
default	\(\text {Expression too large to display}\)	\(4175\)

Input:

int((C*x^2+B*x+A)/(b*x+a)^(3/2)/(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*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^ 
2*(A*b^2-B*a*b+C*a^2)/((x+a/b)*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e))^(1/2)+2* 
((B*b-C*a)/b^2+1/b^2*(a*d*f-b*c*f-b*d*e)*(A*b^2-B*a*b+C*a^2)/(a^2*d*f-a*b* 
c*f-a*b*d*e+b^2*c*e)+(b*c*f+b*d*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2*( 
A*b^2-B*a*b+C*a^2))*(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*(C/b+1/b*d*f*(A*b^2-B*a*b+C*a^2)/(a^2*d 
*f-a*b*c*f-a*b*d*e+b^2*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)*((-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*EllipticF(((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. 1240 vs. \(2 (381) = 762\).

Time = 0.14 (sec) , antiderivative size = 1240, normalized size of antiderivative = 2.94 \[ \int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor 
ithm="fricas")
 

Output:

-2/3*(3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x 
 + e)*d^2*f^2 + ((C*a*b^3*c*d - C*a^2*b^2*d^2)*e^2 + (C*a*b^3*c^2 + (2*C*a 
^2*b^2 - 3*B*a*b^3)*c*d - (2*C*a^3*b - 2*B*a^2*b^2 - A*a*b^3)*d^2)*e*f - ( 
C*a^2*b^2*c^2 + (2*C*a^3*b - 2*B*a^2*b^2 - A*a*b^3)*c*d - (2*C*a^4 - B*a^3 
*b - 2*A*a^2*b^2)*d^2)*f^2 + ((C*b^4*c*d - C*a*b^3*d^2)*e^2 + (C*b^4*c^2 + 
 (2*C*a*b^3 - 3*B*b^4)*c*d - (2*C*a^2*b^2 - 2*B*a*b^3 - A*b^4)*d^2)*e*f - 
(C*a*b^3*c^2 + (2*C*a^2*b^2 - 2*B*a*b^3 - A*b^4)*c*d - (2*C*a^3*b - B*a^2* 
b^2 - 2*A*a*b^3)*d^2)*f^2)*x)*sqrt(b*d*f)*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)) + 3*sqrt(b*d*f)*((C*a*b^3*c*d - C*a^2*b^2*d^2)*e*f - (C 
*a^2*b^2*c*d - (2*C*a^3*b - B*a^2*b^2 + A*a*b^3)*d^2)*f^2 + ((C*b^4*c*d - 
C*a*b^3*d^2)*e*f - (C*a*b^3*c*d - (2*C*a^2*b^2 - B*a*b^3 + A*b^4)*d^2)*f^2 
)*x)*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^...
 

Sympy [F]

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

integrate((C*x**2+B*x+A)/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
 

Output:

Integral((A + B*x + C*x**2)/((a + b*x)**(3/2)*sqrt(c + d*x)*sqrt(e + f*x)) 
, x)
 

Maxima [F]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)), 
 x)
 

Giac [F]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor 
ithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)), 
 x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {too large to display} \] Input:

int((C*x^2+B*x+A)/(b*x+a)^(3/2)/(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*e + a**2*c*f*x + a**2*d*e*x + a**2*d 
*f*x**2 + 2*a*b*c*e*x + 2*a*b*c*f*x**2 + 2*a*b*d*e*x**2 + 2*a*b*d*f*x**3 + 
 b**2*c*e*x**2 + b**2*c*f*x**3 + b**2*d*e*x**3 + b**2*d*f*x**4),x)*a**2*c* 
d*f - int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2)/(a**2*c*e + a** 
2*c*f*x + a**2*d*e*x + a**2*d*f*x**2 + 2*a*b*c*e*x + 2*a*b*c*f*x**2 + 2*a* 
b*d*e*x**2 + 2*a*b*d*f*x**3 + b**2*c*e*x**2 + b**2*c*f*x**3 + b**2*d*e*x** 
3 + b**2*d*f*x**4),x)*a*b**2*d*f + 2*int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt 
(a + b*x)*x**2)/(a**2*c*e + a**2*c*f*x + a**2*d*e*x + a**2*d*f*x**2 + 2*a* 
b*c*e*x + 2*a*b*c*f*x**2 + 2*a*b*d*e*x**2 + 2*a*b*d*f*x**3 + b**2*c*e*x**2 
 + b**2*c*f*x**3 + b**2*d*e*x**3 + b**2*d*f*x**4),x)*a*b*c*d*f*x - int((sq 
rt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2)/(a**2*c*e + a**2*c*f*x + a** 
2*d*e*x + a**2*d*f*x**2 + 2*a*b*c*e*x + 2*a*b*c*f*x**2 + 2*a*b*d*e*x**2 + 
2*a*b*d*f*x**3 + b**2*c*e*x**2 + b**2*c*f*x**3 + b**2*d*e*x**3 + b**2*d*f* 
x**4),x)*b**3*d*f*x + 2*int((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x))/(a 
**2*c*e + a**2*c*f*x + a**2*d*e*x + a**2*d*f*x**2 + 2*a*b*c*e*x + 2*a*b*c* 
f*x**2 + 2*a*b*d*e*x**2 + 2*a*b*d*f*x**3 + b**2*c*e*x**2 + b**2*c*f*x**3 + 
 b**2*d*e*x**3 + b**2*d*f*x**4),x)*a**3*d*f - int((sqrt(e + f*x)*sqrt(c + 
d*x)*sqrt(a + b*x))/(a**2*c*e + a**2*c*f*x + a**2*d*e*x + a**2*d*f*x**2 + 
2*a*b*c*e*x + 2*a*b*c*f*x**2 + 2*a*b*d*e*x**2 + 2*a*b*d*f*x**3 + b**2*c...