\(\int (A+B x+C x^2) \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx\) [115]

Optimal result

Integrand size = 59, antiderivative size = 1393 \[ \int \left (A+B x+C x^2\right ) \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx =\text {Too large to display} \] Output:

2/315*(8*a^3*C*d^3*f^3+3*a^2*b*d^2*f^2*(-4*B*d*f-C*c*f+C*d*e)-3*a*b^2*d*f^ 
2*((-7*A*d^2+C*c^2)*f+B*d*(-2*c*f+d*e))-b^3*(C*(-8*c^3*f^3-3*c^2*d*e*f^2+1 
6*d^3*e^3)+3*d*f*(7*A*d*f*(-c*f+2*d*e)-B*(-4*c^2*f^2-c*d*e*f+8*d^2*e^2)))) 
*(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2)/b^3 
/d^3/f^3-2/105*(7*d*f*(-3*A*b*d*f+C*a*c*f+C*a*d*e+C*b*c*e)+(a*d*f-4*b*(c*f 
+d*e))*(2*a*C*d*f-b*(3*B*d*f-2*C*(c*f+d*e)))/b)*(f*x+e)*(a*c*e+(a*c*f+a*d* 
e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2)/b/d^2/f^3+2/21*(3*B*d* 
f-2*C*(a*d*f+b*c*f+b*d*e)/b)*(d*x+c)*(f*x+e)*(a*c*e+(a*c*f+a*d*e+b*c*e)*x+ 
(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2)/d^2/f^2+2/9*C*(b*x+a)*(d*x+c)*(f* 
x+e)*(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(1/2) 
/b/d/f-2/315*(a*d-b*c)^(1/2)*(16*a^4*C*d^4*f^4-8*a^3*b*d^3*f^3*(3*B*d*f+C* 
c*f+C*d*e)+3*a^2*b^2*d^2*f^2*(d*f*(14*A*d*f+5*B*c*f+5*B*d*e)-2*C*(c^2*f^2- 
c*d*e*f+d^2*e^2))-a*b^3*d*f*(C*(8*c^3*f^3-6*c^2*d*e*f^2-6*c*d^2*e^2*f+8*d^ 
3*e^3)+3*d*f*(14*A*d*f*(c*f+d*e)-B*(5*c^2*f^2-6*c*d*e*f+5*d^2*e^2)))+b^4*( 
2*C*(8*c^4*f^4-4*c^3*d*e*f^3-3*c^2*d^2*e^2*f^2-4*c*d^3*e^3*f+8*d^4*e^4)+3* 
d*f*(14*A*d*f*(c^2*f^2-c*d*e*f+d^2*e^2)-B*(8*c^3*f^3-5*c^2*d*e*f^2-5*c*d^2 
*e^2*f+8*d^3*e^3))))*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(a*c*e+(a*c*f+a*d*e+b*c* 
e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^(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^4/d^(7/2)/f^4/(b* 
x+a)^(1/2)/(d*x+c)/(b*(f*x+e)/(-a*f+b*e))^(1/2)-2/315*(a*d-b*c)^(1/2)*(...
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 39.04 (sec) , antiderivative size = 12161, normalized size of antiderivative = 8.73 \[ \int \left (A+B x+C x^2\right ) \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2526

\(\displaystyle \frac {\int -\left ((a C (d e+c f)+b (c C e-3 A d f)-(3 b B d f-2 a C d f-2 b C (d e+c f)) x) \sqrt {b d f x^3+(b d e+b c f+a d f) x^2+(b c e+a d e+a c f) x+a c e}\right )dx}{3 b d f}+\frac {2 C \left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^{3/2}}{9 b d f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 C \left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^{3/2}}{9 b d f}-\frac {\int (b c C e+a C d e+a c C f-3 A b d f-(3 b B d f-2 a C d f-2 b C (d e+c f)) x) \sqrt {b d f x^3+(b d e+b c f+a d f) x^2+(b c e+a d e+a c f) x+a c e}dx}{3 b d f}\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {2 C \left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^{3/2}}{9 b d f}-\frac {\int \left (\frac {3 b d f (b c C e+a C d e+a c C f-3 A b d f)-(b d e+b c f+a d f) (-3 b B d f+2 a C d f+2 b C (d e+c f))}{3 b d f}+(-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )\right ) \sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {\left (3 b d f (b c e+a d e+a c f)-(b d e+b c f+a d f)^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}}d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {2 C \left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^{3/2}}{9 b d f}-\frac {\int \left (\frac {3 b d f (b c C e+a C d e+a c C f-3 A b d f)-(b d e+b c f+a d f) (-3 b B d f+2 a C d f+2 b C (d e+c f))}{3 b d f}+(-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )\right ) \sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (3 (b c e+a d e+a c f)-\frac {(b d e+b c f+a d f)^2}{b d f}\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}}d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {2 C \left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^{3/2}}{9 b d f}-\frac {\int \left (\frac {\sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (3 (b c e+a d e+a c f)-\frac {(b d e+b c f+a d f)^2}{b d f}\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}} \left (-\left (\left (C \left (2 d^2 e^2+c d f e+2 c^2 f^2\right )+3 d f (3 A d f-B (d e+c f))\right ) b^2\right )-a d f (C d e+c C f-3 B d f) b-2 a^2 C d^2 f^2\right )}{3 b d f}+(-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right ) \sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (3 (b c e+a d e+a c f)-\frac {(b d e+b c f+a d f)^2}{b d f}\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}}\right )d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {2 C \left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^{3/2}}{9 b d f}-\frac {\int \left (\frac {\sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (-\frac {(b d e+b c f+a d f)^2}{b d f}+3 b c e+3 a d e+3 a c f\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}} \left (-\left (\left (C \left (2 d^2 e^2+c d f e+2 c^2 f^2\right )+3 d f (3 A d f-B (d e+c f))\right ) b^2\right )-a d f (C d e+c C f-3 B d f) b-2 a^2 C d^2 f^2\right )}{3 b d f}+(-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right ) \sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (-\frac {(b d e+b c f+a d f)^2}{b d f}+3 b c e+3 a d e+3 a c f\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}}\right )d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\)

\(\Big \downarrow \) 7299

\(\displaystyle \frac {2 C \left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^{3/2}}{9 b d f}-\frac {\int \left (\frac {\sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (-\frac {(b d e+b c f+a d f)^2}{b d f}+3 b c e+3 a d e+3 a c f\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}} \left (-\left (\left (C \left (2 d^2 e^2+c d f e+2 c^2 f^2\right )+3 d f (3 A d f-B (d e+c f))\right ) b^2\right )-a d f (C d e+c C f-3 B d f) b-2 a^2 C d^2 f^2\right )}{3 b d f}+(-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right ) \sqrt {b d f \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3+\frac {1}{3} \left (-\frac {(b d e+b c f+a d f)^2}{b d f}+3 b c e+3 a d e+3 a c f\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+\frac {(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{27 b^2 d^2 f^2}}\right )d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp 
[1/(n*Coeff[Qn, x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x 
, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [A] (verified)

Time = 4.02 (sec) , antiderivative size = 2037, normalized size of antiderivative = 1.46

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2037\)
default	\(\text {Expression too large to display}\)	\(3074\)

Input:

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

Output:

2/9*C*x^3*(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)+2/7*(b*d*f*B+a*C*d*f+C*b*c*f+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4* 
b*d*e)*C)/b/d/f*x^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)+2/5*(b*d*f*A+B*a*d*f+B*b*c*f+B*b*d*e+C*a*c*f+C*a*d 
*e+C*b*c*e-2/9*C*(7/2*a*c*f+7/2*a*d*e+7/2*b*c*e)-2/7*(b*d*f*B+a*C*d*f+C*b* 
c*f+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4*b*d*e)*C)/b/d/f*(3*a*d*f+3*b*c*f+3*b*d* 
e))/b/d/f*x*(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)+2/3*(A*a*d*f+A*b*c*f+A*b*d*e+B*a*c*f+a*B*d*e+B*b*c*e+1/3*C 
*a*e*c-2/7*(b*d*f*B+a*C*d*f+C*b*c*f+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4*b*d*e)* 
C)/b/d/f*(5/2*a*c*f+5/2*a*d*e+5/2*b*c*e)-2/5*(b*d*f*A+B*a*d*f+B*b*c*f+B*b* 
d*e+C*a*c*f+C*a*d*e+C*b*c*e-2/9*C*(7/2*a*c*f+7/2*a*d*e+7/2*b*c*e)-2/7*(b*d 
*f*B+a*C*d*f+C*b*c*f+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4*b*d*e)*C)/b/d/f*(3*a*d 
*f+3*b*c*f+3*b*d*e))/b/d/f*(2*a*d*f+2*b*c*f+2*b*d*e))/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*A*c*e 
-2/5*(b*d*f*A+B*a*d*f+B*b*c*f+B*b*d*e+C*a*c*f+C*a*d*e+C*b*c*e-2/9*C*(7/2*a 
*c*f+7/2*a*d*e+7/2*b*c*e)-2/7*(b*d*f*B+a*C*d*f+C*b*c*f+C*b*d*e-2/9*(4*a*d* 
f+4*b*c*f+4*b*d*e)*C)/b/d/f*(3*a*d*f+3*b*c*f+3*b*d*e))/b/d/f*a*c*e-2/3*(A* 
a*d*f+A*b*c*f+A*b*d*e+B*a*c*f+a*B*d*e+B*b*c*e+1/3*C*a*e*c-2/7*(b*d*f*B+a*C 
*d*f+C*b*c*f+C*b*d*e-2/9*(4*a*d*f+4*b*c*f+4*b*d*e)*C)/b/d/f*(5/2*a*c*f+5/2 
*a*d*e+5/2*b*c*e)-2/5*(b*d*f*A+B*a*d*f+B*b*c*f+B*b*d*e+C*a*c*f+C*a*d*e+...
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1943, normalized size of antiderivative = 1.39 \[ \int \left (A+B x+C x^2\right ) \sqrt {a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3} \, dx=\text {Too large to display} \] Input:

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

Output:

2/945*((16*C*b^5*d^5*e^5 - 8*(2*C*b^5*c*d^4 + (2*C*a*b^4 + 3*B*b^5)*d^5)*e 
^4*f - (5*C*b^5*c^2*d^3 - (20*C*a*b^4 + 27*B*b^5)*c*d^4 + (5*C*a^2*b^3 - 2 
7*B*a*b^4 - 42*A*b^5)*d^5)*e^3*f^2 - (5*C*b^5*c^3*d^2 - 6*(C*a*b^4 + 2*B*b 
^5)*c^2*d^3 - 3*(2*C*a^2*b^3 - 14*B*a*b^4 - 21*A*b^5)*c*d^4 + (5*C*a^3*b^2 
 - 12*B*a^2*b^3 + 63*A*a*b^4)*d^5)*e^2*f^3 - (16*C*b^5*c^4*d - (20*C*a*b^4 
 + 27*B*b^5)*c^3*d^2 - 3*(2*C*a^2*b^3 - 14*B*a*b^4 - 21*A*b^5)*c^2*d^3 - 2 
*(10*C*a^3*b^2 - 21*B*a^2*b^3 + 126*A*a*b^4)*c*d^4 + (16*C*a^4*b - 27*B*a^ 
3*b^2 + 63*A*a^2*b^3)*d^5)*e*f^4 + (16*C*b^5*c^5 - 8*(2*C*a*b^4 + 3*B*b^5) 
*c^4*d - (5*C*a^2*b^3 - 27*B*a*b^4 - 42*A*b^5)*c^3*d^2 - (5*C*a^3*b^2 - 12 
*B*a^2*b^3 + 63*A*a*b^4)*c^2*d^3 - (16*C*a^4*b - 27*B*a^3*b^2 + 63*A*a^2*b 
^3)*c*d^4 + 2*(8*C*a^5 - 12*B*a^4*b + 21*A*a^3*b^2)*d^5)*f^5)*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*(16*C*b^5*d^5*e^4*f 
 - 8*(C*b^5*c*d^4 + (C*a*b^4 + 3*B*b^5)*d^5)*e^3*f^2 - 3*(2*C*b^5*c^2*d^3 
- (2*C*a*b^4 + 5*B*b^5)*c*d^4 + (2*C*a^2*b^3 - 5*B*a*b^4 - 14*A*b^5)*d^5)* 
e^2*f^3 - (8*C*b^5*c^3*d^2 - 3*(2*C*a*b^4 + 5*B*b^5)*c^2*d^3 - 6*(C*a^2*b^ 
3 - 3*B*a*b^4 - 7*A*b^5)*c*d^4 + (8*C*a^3*b^2 - 15*B*a^2*b^3 + 42*A*a*b...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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