\(\int \frac {x^4}{(a+b x+c x^2)^{3/2} (d+e x+f x^2)} \, dx\) [148]

Optimal result

Integrand size = 30, antiderivative size = 773 \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=-\frac {2 \left (a \left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right )+\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) x\right )}{c \left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {a+b x+c x^2}}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} f}+\frac {\left (2 d f \left (c d^2-b d e+a \left (e^2-d f\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (c d^2 e+a e \left (e^2-2 d f\right )-b d \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {\left (2 d f \left (c d^2-b d e+a \left (e^2-d f\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (c d^2 e+a e \left (e^2-2 d f\right )-b d \left (e^2-d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} f \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \] Output:

(-2*a*(b^3*d-a*b^2*e+2*a^2*c*e-a*b*(-a*f+3*c*d))-2*(b^4*d-a*b^3*e+3*a^2*b* 
c*e+2*a^2*c*(-a*f+c*d)-a*b^2*(-a*f+4*c*d))*x)/c/(-4*a*c+b^2)/((-a*f+c*d)^2 
-(-a*e+b*d)*(-b*f+c*e))/(c*x^2+b*x+a)^(1/2)+arctanh(1/2*(2*c*x+b)/c^(1/2)/ 
(c*x^2+b*x+a)^(1/2))/c^(3/2)/f+1/2*(2*d*f*(c*d^2-b*d*e+a*(-d*f+e^2))-(e-(- 
4*d*f+e^2)^(1/2))*(c*d^2*e+a*e*(-2*d*f+e^2)-b*d*(-d*f+e^2)))*arctanh(1/4*( 
4*a*f-b*(e-(-4*d*f+e^2)^(1/2))+2*(b*f-c*(e-(-4*d*f+e^2)^(1/2)))*x)*2^(1/2) 
/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+ 
b*x+a)^(1/2))*2^(1/2)/f/(-4*d*f+e^2)^(1/2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+ 
c*e))/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)-1/ 
2*(2*d*f*(c*d^2-b*d*e+a*(-d*f+e^2))-(e+(-4*d*f+e^2)^(1/2))*(c*d^2*e+a*e*(- 
2*d*f+e^2)-b*d*(-d*f+e^2)))*arctanh(1/4*(4*a*f-b*(e+(-4*d*f+e^2)^(1/2))+2* 
(b*f-c*(e+(-4*d*f+e^2)^(1/2)))*x)*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b 
*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+b*x+a)^(1/2))*2^(1/2)/f/(-4*d*f+e 
^2)^(1/2)/((-a*f+c*d)^2-(-a*e+b*d)*(-b*f+c*e))/(c*e^2-2*c*d*f-b*e*f+2*a*f^ 
2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 13.78 (sec) , antiderivative size = 1471, normalized size of antiderivative = 1.90 \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx =\text {Too large to display} \] Input:

Integrate[x^4/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]
 

Output:

(2*(b^4*d*x + a*b^2*(-4*c*d*x + b*(d - e*x)) + a^3*(b*f + 2*c*(e - f*x)) + 
 a^2*(2*c^2*d*x - 3*b*c*(d - e*x) + b^2*(-e + f*x))))/(c*(-b^2 + 4*a*c)*(c 
^2*d^2 - b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))*Sqrt[a + 
 x*(b + c*x)]) - ((c*d^2*(-e^2 + 2*d*f + e*Sqrt[e^2 - 4*d*f]) + b*d*(e^3 - 
 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*f]) + a*(-e^4 + 4*d* 
e^2*f - 2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*d*e*f*Sqrt[e^2 - 4*d*f]))*Lo 
g[-e + Sqrt[e^2 - 4*d*f] - 2*f*x])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f]*(c^2*d^2 - 
 b*c*d*e + f*(b^2*d - a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))*Sqrt[c*(e^2 - 2* 
d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f + b*(-e + Sqrt[e^2 - 4*d*f]))]) - (( 
c*d^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + b*d*(-e^3 + 3*d*e*f - e^2*Sqrt 
[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4*d*f]) + a*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + 
e^3*Sqrt[e^2 - 4*d*f] - 2*d*e*f*Sqrt[e^2 - 4*d*f]))*Log[e + Sqrt[e^2 - 4*d 
*f] + 2*f*x])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f]*(c^2*d^2 - b*c*d*e + f*(b^2*d - 
 a*b*e + a^2*f) + a*c*(e^2 - 2*d*f))*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^2 - 4* 
d*f]) + f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f]))]) + Log[b + 2*c*x + 2*Sqrt[c 
]*Sqrt[a + x*(b + c*x)]]/(c^(3/2)*f) + ((c*d^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 
 4*d*f]) + b*d*(-e^3 + 3*d*e*f - e^2*Sqrt[e^2 - 4*d*f] + d*f*Sqrt[e^2 - 4* 
d*f]) + a*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*Sqrt[e^2 - 4*d*f] - 2*d*e*f*S 
qrt[e^2 - 4*d*f]))*Log[-4*a*f + 2*c*e*x + 2*c*Sqrt[e^2 - 4*d*f]*x + b*(e + 
 Sqrt[e^2 - 4*d*f] - 2*f*x) - 2*Sqrt[2]*Sqrt[c*(e^2 - 2*d*f + e*Sqrt[e^...
 

Rubi [A] (warning: unable to verify)

Time = 9.19 (sec) , antiderivative size = 971, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7279, 2009}

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.

\(\displaystyle \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {c x^2+b x+a} b}{c \left (b^2-4 a c\right ) f}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{c^{3/2} f}+\frac {\left (2 d f \left (c d^2-b e d+a \left (e^2-d f\right )\right )-\left (e-\sqrt {e^2-4 d f}\right ) \left (c e d^2-b \left (e^2-d f\right ) d+a e \left (e^2-2 d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} f \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {\left (2 d f \left (c d^2-b e d+a \left (e^2-d f\right )\right )-\left (e+\sqrt {e^2-4 d f}\right ) \left (c e d^2-b \left (e^2-d f\right ) d+a e \left (e^2-2 d f\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} f \sqrt {e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}}}+\frac {2 x (2 a+b x)}{\left (b^2-4 a c\right ) f \sqrt {c x^2+b x+a}}-\frac {2 e (2 a+b x)}{\left (b^2-4 a c\right ) f^2 \sqrt {c x^2+b x+a}}-\frac {2 \left (e^2-d f\right ) (b+2 c x)}{\left (b^2-4 a c\right ) f^3 \sqrt {c x^2+b x+a}}-\frac {2 \left (c d (b c d-2 a c e+a b f) \left (e^2-d f\right )+\left (f b^2+2 c^2 d-c (b e+2 a f)\right ) \left (a e \left (e^2-2 d f\right )-b d \left (e^2-d f\right )\right )+c \left (e (b c d-2 a c e+a b f) \left (e^2-2 d f\right )-d \left (e^2-d f\right ) \left (f b^2+2 c^2 d-c (b e+2 a f)\right )\right ) x\right )}{\left (b^2-4 a c\right ) f^3 \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt {c x^2+b x+a}}\)

Input:

Int[x^4/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]
 

Output:

(-2*e*(2*a + b*x))/((b^2 - 4*a*c)*f^2*Sqrt[a + b*x + c*x^2]) + (2*x*(2*a + 
 b*x))/((b^2 - 4*a*c)*f*Sqrt[a + b*x + c*x^2]) - (2*(e^2 - d*f)*(b + 2*c*x 
))/((b^2 - 4*a*c)*f^3*Sqrt[a + b*x + c*x^2]) - (2*(c*d*(b*c*d - 2*a*c*e + 
a*b*f)*(e^2 - d*f) + (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*e*(e^2 - 2*d*f 
) - b*d*(e^2 - d*f)) + c*(e*(b*c*d - 2*a*c*e + a*b*f)*(e^2 - 2*d*f) - d*(e 
^2 - d*f)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*x))/((b^2 - 4*a*c)*f^3*((c* 
d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[a + b*x + c*x^2]) - (2*b*Sqrt[a 
 + b*x + c*x^2])/(c*(b^2 - 4*a*c)*f) + ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt 
[a + b*x + c*x^2])]/(c^(3/2)*f) + ((2*d*f*(c*d^2 - b*d*e + a*(e^2 - d*f)) 
- (e - Sqrt[e^2 - 4*d*f])*(c*d^2*e + a*e*(e^2 - 2*d*f) - b*d*(e^2 - d*f))) 
*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 - 4 
*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f) 
*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 - 4*d*f]* 
((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2 
*a*f^2 - (c*e - b*f)*Sqrt[e^2 - 4*d*f]]) - ((2*d*f*(c*d^2 - b*d*e + a*(e^2 
 - d*f)) - (e + Sqrt[e^2 - 4*d*f])*(c*d^2*e + a*e*(e^2 - 2*d*f) - b*d*(e^2 
 - d*f)))*ArcTanh[(4*a*f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqr 
t[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c 
*e - b*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*f*Sqrt[e^2 
- 4*d*f]*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*Sqrt[c*e^2 - 2*c*d*f...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2257\) vs. \(2(717)=1434\).

Time = 2.87 (sec) , antiderivative size = 2258, normalized size of antiderivative = 2.92

method	result	size
default	\(\text {Expression too large to display}\)	\(2258\)

Input:

int(x^4/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/2/f^4*(2*d*e*f*(-4*d*f+e^2)^(1/2)-e^3*(-4*d*f+e^2)^(1/2)-2*d^2*f^2+4*d*e 
^2*f-e^4)/(-4*d*f+e^2)^(1/2)*(2/(-f*b*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2 
)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)*f^2/(c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f) 
^2+1/f*(-c*(-4*d*f+e^2)^(1/2)+f*b-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/ 
2*(-f*b*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c* 
e^2)/f^2)^(1/2)-2*f*(-c*(-4*d*f+e^2)^(1/2)+f*b-c*e)/(-f*b*(-4*d*f+e^2)^(1/ 
2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)*(2*c*(x+1/2*(e+(-4* 
d*f+e^2)^(1/2))/f)+1/f*(-c*(-4*d*f+e^2)^(1/2)+f*b-c*e))/(2*c*(-f*b*(-4*d*f 
+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2-1/f^2* 
(-c*(-4*d*f+e^2)^(1/2)+f*b-c*e)^2)/(c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2+1 
/f*(-c*(-4*d*f+e^2)^(1/2)+f*b-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*(- 
f*b*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2) 
/f^2)^(1/2)-2/(-f*b*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e* 
f-2*d*f*c+c*e^2)*f^2*2^(1/2)/((-f*b*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)* 
c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2)^(1/2)*ln(((-f*b*(-4*d*f+e^2)^(1/2)+( 
-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2+1/f*(-c*(-4*d*f+e^2 
)^(1/2)+f*b-c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*((-f*b*(-4*d 
*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2)^(1/ 
2)*(4*c*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2+4/f*(-c*(-4*d*f+e^2)^(1/2)+f*b- 
c*e)*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*(-f*b*(-4*d*f+e^2)^(1/2)+(-4*d*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(x^4/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {x^{4}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}\, dx \] Input:

integrate(x**4/(c*x**2+b*x+a)**(3/2)/(f*x**2+e*x+d),x)
 

Output:

Integral(x**4/((a + b*x + c*x**2)**(3/2)*(d + e*x + f*x**2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^4/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*d*f-e^2>0)', see `assume?` for 
 more deta
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4/(c*x^2+b*x+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {x^4}{{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^4}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx=\int \frac {x^{4}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (f \,x^{2}+e x +d \right )}d x \] Input:

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

Output:

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