\(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x+f x^2)^2} \, dx\) [138]

Optimal result

Integrand size = 27, antiderivative size = 488 \[ \int \frac {\sqrt {a+b x+c x^2}}{\left (d+e x+f x^2\right )^2} \, dx=-\frac {(e+2 f x) \sqrt {a+b x+c x^2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}-\frac {\left (f (b e-4 a f)-(c e-b f) \left (e-\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \left (e^2-4 d f\right )^{3/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {\left (f (b e-4 a f)-(c e-b f) \left (e+\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \left (e^2-4 d f\right )^{3/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 3.32 (sec) , antiderivative size = 1691, normalized size of antiderivative = 3.47 \[ \int \frac {\sqrt {a+b x+c x^2}}{\left (d+e x+f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(2*d^3*e*Sqrt[a + x*(b + c*x)] + 4*d^3*f*x*Sqrt[a + x*(b + c*x)] - 2* 
(e^2 - 4*d*f)*(d + x*(e + f*x))*RootSum[c^2*d - b*c*e + b^2*f + 2*Sqrt[a]* 
c*e*#1 - 4*Sqrt[a]*b*f*#1 - 2*c*d*#1^2 + b*e*#1^2 + 4*a*f*#1^2 - 2*Sqrt[a] 
*e*#1^3 + d*#1^4 & , (-(b^2*d^2*Log[x]) - 3*a*c*d^2*Log[x] + 5*a*b*d*e*Log 
[x] - 4*a^2*e^2*Log[x] + 4*a^2*d*f*Log[x] + b^2*d^2*Log[-Sqrt[a] + Sqrt[a 
+ b*x + c*x^2] - x*#1] + 3*a*c*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - 
x*#1] - 5*a*b*d*e*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] + 4*a^2*e^2 
*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1] - 4*a^2*d*f*Log[-Sqrt[a] + S 
qrt[a + b*x + c*x^2] - x*#1] + 2*Sqrt[a]*b*d^2*Log[x]*#1 - 2*a^(3/2)*d*e*L 
og[x]*#1 - 2*Sqrt[a]*b*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 
 + 2*a^(3/2)*d*e*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1 - a*d^2*L 
og[x]*#1^2 + a*d^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#1]*#1^2)/(-(S 
qrt[a]*c*e) + 2*Sqrt[a]*b*f + 2*c*d*#1 - b*e*#1 - 4*a*f*#1 + 3*Sqrt[a]*e*# 
1^2 - 2*d*#1^3) & ] + (d + x*(e + f*x))*RootSum[c^2*d - b*c*e + b^2*f + 2* 
Sqrt[a]*c*e*#1 - 4*Sqrt[a]*b*f*#1 - 2*c*d*#1^2 + b*e*#1^2 + 4*a*f*#1^2 - 2 
*Sqrt[a]*e*#1^3 + d*#1^4 & , (-(b*c*d^3*e*Log[x]) + 2*b^2*d^2*e^2*Log[x] + 
 6*a*c*d^2*e^2*Log[x] - 10*a*b*d*e^3*Log[x] + 8*a^2*e^4*Log[x] - 6*b^2*d^3 
*f*Log[x] - 28*a*c*d^3*f*Log[x] + 40*a*b*d^2*e*f*Log[x] - 40*a^2*d*e^2*f*L 
og[x] + 32*a^2*d^2*f^2*Log[x] + b*c*d^3*e*Log[-Sqrt[a] + Sqrt[a + b*x + c* 
x^2] - x*#1] - 2*b^2*d^2*e^2*Log[-Sqrt[a] + Sqrt[a + b*x + c*x^2] - x*#...
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1302, 27, 1365, 1154, 219}

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.

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

\(\Big \downarrow \) 1302

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1365

\(\displaystyle \frac {\frac {2 \left (f (b e-4 a f)-\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f)\right ) \int \frac {1}{\left (e+2 f x-\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}-\frac {2 \left (f (b e-4 a f)-\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f)\right ) \int \frac {1}{\left (e+2 f x+\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \sqrt {a+b x+c x^2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {4 \left (f (b e-4 a f)-\left (\sqrt {e^2-4 d f}+e\right ) (c e-b f)\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}-\frac {4 \left (f (b e-4 a f)-\left (e-\sqrt {e^2-4 d f}\right ) (c e-b f)\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}}{2 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \sqrt {a+b x+c x^2}}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

-(((e + 2*f*x)*Sqrt[a + b*x + c*x^2])/((e^2 - 4*d*f)*(d + e*x + f*x^2))) + 
 (-((Sqrt[2]*(f*(b*e - 4*a*f) - (c*e - b*f)*(e - Sqrt[e^2 - 4*d*f]))*ArcTa 
nh[(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[e^2 - 4*d*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]])) + (Sqrt[2]*(f*(b 
*e - 4*a*f) - (c*e - b*f)*(e + Sqrt[e^2 - 4*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]*Sq 
rt[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[e^2 - 4*d*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*(e^2 - 4*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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1302

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x 
_)^2)^(q_), x_Symbol] :> Simp[(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e 
*x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) 
   Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p 
+ 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ 
[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !IGtQ[q, 0]
 

rule 1365

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( 
e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(2*c*g - h*(b - q))/q   Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] 
, x] - Simp[(2*c*g - h*(b + q))/q   Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f 
*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 
] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4690\) vs. \(2(439)=878\).

Time = 2.88 (sec) , antiderivative size = 4691, normalized size of antiderivative = 9.61

method	result	size
default	\(\text {Expression too large to display}\)	\(4691\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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