\(\int \frac {A+B x+C x^2}{(a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3)^{3/2}} \, dx\) [117]

Optimal result

Integrand size = 59, antiderivative size = 1151 \[ \int \frac {A+B x+C x^2}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^{3/2}} \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 36.97 (sec) , antiderivative size = 9971, normalized size of antiderivative = 8.66 \[ \int \frac {A+B x+C x^2}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x + C*x^2)/(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)^(3/2),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.

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

\(\Big \downarrow \) 2526

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2490

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (C \left (2 d^2 e^2+c d f e+2 c^2 f^2\right )+3 d f (3 A d f-B (d e+c f))\right ) b^2\right )-a d f (C d e+c C f-3 B d f) b-2 a^2 C d^2 f^2\right )}{b d f \left (\frac {27 b^3 d^3 f^3 \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3-9 b d f \left (\left (d^2 e^2-c d f e+c^2 f^2\right ) b^2-a d f (d e+c f) b+a^2 d^2 f^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{b^2 d^2 f^2}\right )^{3/2}}+\frac {81 \sqrt {3} (-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{\left (\frac {27 b^3 d^3 f^3 \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3-9 b d f \left (\left (d^2 e^2-c d f e+c^2 f^2\right ) b^2-a d f (d e+c f) b+a^2 d^2 f^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{b^2 d^2 f^2}\right )^{3/2}}\right )d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}-\frac {2 C}{3 b d f \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}}\)

\(\Big \downarrow \) 7299

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (C \left (2 d^2 e^2+c d f e+2 c^2 f^2\right )+3 d f (3 A d f-B (d e+c f))\right ) b^2\right )-a d f (C d e+c C f-3 B d f) b-2 a^2 C d^2 f^2\right )}{b d f \left (\frac {27 b^3 d^3 f^3 \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3-9 b d f \left (\left (d^2 e^2-c d f e+c^2 f^2\right ) b^2-a d f (d e+c f) b+a^2 d^2 f^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{b^2 d^2 f^2}\right )^{3/2}}+\frac {81 \sqrt {3} (-3 b B d f+2 a C d f+2 b C (d e+c f)) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{\left (\frac {27 b^3 d^3 f^3 \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )^3-9 b d f \left (\left (d^2 e^2-c d f e+c^2 f^2\right ) b^2-a d f (d e+c f) b+a^2 d^2 f^2\right ) \left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )+(b d e+b c f-2 a d f) (2 b d e-b c f-a d f) (b d e-2 b c f+a d f)}{b^2 d^2 f^2}\right )^{3/2}}\right )d\left (\frac {b d e+b c f+a d f}{3 b d f}+x\right )}{3 b d f}-\frac {2 C}{3 b d f \sqrt {x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3}}\)

Input:

Int[(A + B*x + C*x^2)/(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)^(3/2),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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4110\) vs. \(2(1111)=2222\).

Time = 4.33 (sec) , antiderivative size = 4111, normalized size of antiderivative = 3.57

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(4111\)
default	\(\text {Expression too large to display}\)	\(8188\)

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)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2*b*d*f*((2*A*a^2*d^2*f^2-2*A*a*b*c*d*f^2-2*A*a*b*d^2*e*f+2*A*b^2*c^2*f^2 
-2*A*b^2*c*d*e*f+2*A*b^2*d^2*e^2-B*a^2*c*d*f^2-B*a^2*d^2*e*f-B*a*b*c^2*f^2 
+6*B*a*b*c*d*e*f-B*a*b*d^2*e^2-B*b^2*c^2*e*f-B*b^2*c*d*e^2+2*C*a^2*c^2*f^2 
-2*C*a^2*c*d*e*f+2*C*a^2*d^2*e^2-2*C*a*b*c^2*e*f-2*C*a*b*c*d*e^2+2*C*b^2*c 
^2*e^2)/(a^4*c^2*d^2*f^4-2*a^4*c*d^3*e*f^3+a^4*d^4*e^2*f^2-2*a^3*b*c^3*d*f 
^4+2*a^3*b*c^2*d^2*e*f^3+2*a^3*b*c*d^3*e^2*f^2-2*a^3*b*d^4*e^3*f+a^2*b^2*c 
^4*f^4+2*a^2*b^2*c^3*d*e*f^3-6*a^2*b^2*c^2*d^2*e^2*f^2+2*a^2*b^2*c*d^3*e^3 
*f+a^2*b^2*d^4*e^4-2*a*b^3*c^4*e*f^3+2*a*b^3*c^3*d*e^2*f^2+2*a*b^3*c^2*d^2 
*e^3*f-2*a*b^3*c*d^3*e^4+b^4*c^4*e^2*f^2-2*b^4*c^3*d*e^3*f+b^4*c^2*d^2*e^4 
)*x^2+(2*A*a^3*d^3*f^3-A*a^2*b*c*d^2*f^3-A*a^2*b*d^3*e*f^2-A*a*b^2*c^2*d*f 
^3-A*a*b^2*d^3*e^2*f+2*A*b^3*c^3*f^3-A*b^3*c^2*d*e*f^2-A*b^3*c*d^2*e^2*f+2 
*A*b^3*d^3*e^3-B*a^3*c*d^2*f^3-B*a^3*d^3*e*f^2+2*B*a^2*b*c*d^2*e*f^2-B*a*b 
^2*c^3*f^3+2*B*a*b^2*c^2*d*e*f^2+2*B*a*b^2*c*d^2*e^2*f-B*a*b^2*d^3*e^3-B*b 
^3*c^3*e*f^2-B*b^3*c*d^2*e^3+C*a^3*c^2*d*f^3+C*a^3*d^3*e^2*f+C*a^2*b*c^3*f 
^3-2*C*a^2*b*c^2*d*e*f^2-2*C*a^2*b*c*d^2*e^2*f+C*a^2*b*d^3*e^3-2*C*a*b^2*c 
^2*d*e^2*f+C*b^3*c^3*e^2*f+C*b^3*c^2*d*e^3)/b/d/f/(a^4*c^2*d^2*f^4-2*a^4*c 
*d^3*e*f^3+a^4*d^4*e^2*f^2-2*a^3*b*c^3*d*f^4+2*a^3*b*c^2*d^2*e*f^3+2*a^3*b 
*c*d^3*e^2*f^2-2*a^3*b*d^4*e^3*f+a^2*b^2*c^4*f^4+2*a^2*b^2*c^3*d*e*f^3-6*a 
^2*b^2*c^2*d^2*e^2*f^2+2*a^2*b^2*c*d^3*e^3*f+a^2*b^2*d^4*e^4-2*a*b^3*c^4*e 
*f^3+2*a*b^3*c^3*d*e^2*f^2+2*a*b^3*c^2*d^2*e^3*f-2*a*b^3*c*d^3*e^4+b^4*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5532 vs. \(2 (1110) = 2220\).

Time = 0.37 (sec) , antiderivative size = 5532, normalized size of antiderivative = 4.81 \[ \int \frac {A+B x+C x^2}{\left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^{3/2}} \, dx=\text {Too large to display} \] 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)^(3/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

Output:

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

Maxima [F]

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

Output:

integrate((C*x^2 + B*x + A)/(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)^(3/2), x)
 

Giac [F]

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

Output:

integrate((C*x^2 + B*x + A)/(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)^(3/2), x)
 

Mupad [F(-1)]

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

Reduce [F]

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