\(\int \frac {1}{\sqrt {d+e x+f x^2} (a e+b e x+b f x^2)^2} \, dx\) [40]

Optimal result

Integrand size = 31, antiderivative size = 162 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}-\frac {\left (8 a e f-b \left (e^2+4 d f\right )\right ) \text {arctanh}\left (\frac {\sqrt {b d-a e} (e+2 f x)}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{e^{3/2} (b d-a e)^{3/2} (b e-4 a f)^{3/2}} \] Output:

-b*(2*f*x+e)*(f*x^2+e*x+d)^(1/2)/e/(-a*e+b*d)/(-4*a*f+b*e)/(b*f*x^2+b*e*x+ 
a*e)-(8*a*e*f-b*(4*d*f+e^2))*arctanh((-a*e+b*d)^(1/2)*(2*f*x+e)/e^(1/2)/(- 
4*a*f+b*e)^(1/2)/(f*x^2+e*x+d)^(1/2))/e^(3/2)/(-a*e+b*d)^(3/2)/(-4*a*f+b*e 
)^(3/2)
 

Mathematica [C] (verified)

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

Time = 3.25 (sec) , antiderivative size = 1420, normalized size of antiderivative = 8.77 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

Integrate[1/(Sqrt[d + e*x + f*x^2]*(a*e + b*e*x + b*f*x^2)^2),x]
 

Output:

((-2*RootSum[a*e*f^2 - 2*b*Sqrt[d]*e*f*#1 + b*e^2*#1^2 + 4*b*d*f*#1^2 - 2* 
a*e*f*#1^2 - 2*b*Sqrt[d]*e*#1^3 + a*e*#1^4 & , (-4*b^2*d*e*Log[x] + a*b*e^ 
2*Log[x] + 4*a*b*d*f*Log[x] + a^2*e*f*Log[x] + 4*b^2*d*e*Log[-Sqrt[d] + Sq 
rt[d + e*x + f*x^2] - x*#1] - a*b*e^2*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] 
 - x*#1] - 4*a*b*d*f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - a^2*e* 
f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 2*a*b*Sqrt[d]*e*Log[x]*#1 
 + 2*a*b*Sqrt[d]*e*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1 - a^2*e 
*Log[x]*#1^2 + a^2*e*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1]*#1^2)/(- 
(b*Sqrt[d]*e*f) + b*e^2*#1 + 4*b*d*f*#1 - 2*a*e*f*#1 - 3*b*Sqrt[d]*e*#1^2 
+ 2*a*e*#1^3) & ])/a^3 + (b*((-2*e*(e + 2*f*x)*Sqrt[d + x*(e + f*x)])/(a*e 
 + b*x*(e + f*x)) + RootSum[a*e*f^2 - 2*b*Sqrt[d]*e*f*#1 + b*e^2*#1^2 + 4* 
b*d*f*#1^2 - 2*a*e*f*#1^2 - 2*b*Sqrt[d]*e*#1^3 + a*e*#1^4 & , (-8*b^3*d^2* 
e^2*Log[x] + 10*a*b^2*d*e^3*Log[x] - 2*a^2*b*e^4*Log[x] + 40*a*b^2*d^2*e*f 
*Log[x] - 46*a^2*b*d*e^2*f*Log[x] + 7*a^3*e^3*f*Log[x] - 32*a^2*b*d^2*f^2* 
Log[x] + 28*a^3*d*e*f^2*Log[x] + 8*b^3*d^2*e^2*Log[-Sqrt[d] + Sqrt[d + e*x 
 + f*x^2] - x*#1] - 10*a*b^2*d*e^3*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - 
x*#1] + 2*a^2*b*e^4*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 40*a*b^ 
2*d^2*e*f*Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] + 46*a^2*b*d*e^2*f* 
Log[-Sqrt[d] + Sqrt[d + e*x + f*x^2] - x*#1] - 7*a^3*e^3*f*Log[-Sqrt[d] + 
Sqrt[d + e*x + f*x^2] - x*#1] + 32*a^2*b*d^2*f^2*Log[-Sqrt[d] + Sqrt[d ...
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1305, 27, 1313, 221}

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.

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

\(\Big \downarrow \) 1305

\(\displaystyle \frac {\int \frac {b (b d-a e) f^2 \left (8 a e f-b \left (e^2+4 d f\right )\right )}{2 \sqrt {f x^2+e x+d} \left (b f x^2+b e x+a e\right )}dx}{b e f^2 (b d-a e)^2 (b e-4 a f)}-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (8 a e f-b \left (4 d f+e^2\right )\right ) \int \frac {1}{\sqrt {f x^2+e x+d} \left (b f x^2+b e x+a e\right )}dx}{2 e (b d-a e) (b e-4 a f)}-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\)

\(\Big \downarrow \) 1313

\(\displaystyle -\frac {\left (8 a e f-b \left (4 d f+e^2\right )\right ) \int \frac {1}{e^2 (b e-4 a f)-\frac {e (b d-a e) (e+2 f x)^2}{f x^2+e x+d}}d\frac {e+2 f x}{\sqrt {f x^2+e x+d}}}{(b d-a e) (b e-4 a f)}-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\left (8 a e f-b \left (4 d f+e^2\right )\right ) \text {arctanh}\left (\frac {(e+2 f x) \sqrt {b d-a e}}{\sqrt {e} \sqrt {b e-4 a f} \sqrt {d+e x+f x^2}}\right )}{e^{3/2} (b d-a e)^{3/2} (b e-4 a f)^{3/2}}-\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{e (b d-a e) (b e-4 a f) \left (a e+b e x+b f x^2\right )}\)

Input:

Int[1/(Sqrt[d + e*x + f*x^2]*(a*e + b*e*x + b*f*x^2)^2),x]
 

Output:

-((b*(e + 2*f*x)*Sqrt[d + e*x + f*x^2])/(e*(b*d - a*e)*(b*e - 4*a*f)*(a*e 
+ b*e*x + b*f*x^2))) - ((8*a*e*f - b*(e^2 + 4*d*f))*ArcTanh[(Sqrt[b*d - a* 
e]*(e + 2*f*x))/(Sqrt[e]*Sqrt[b*e - 4*a*f]*Sqrt[d + e*x + f*x^2])])/(e^(3/ 
2)*(b*d - a*e)^(3/2)*(b*e - 4*a*f)^(3/2))
 

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 221

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

rule 1305

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x 
_)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a 
*f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( 
d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - 
 b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( 
c*e - b*f))*(p + 1))   Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si 
mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f 
 - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - 
b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f 
+ b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f 
*(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* 
(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b 
^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - 
(b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q 
, 0]
 

rule 1313

Int[1/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*( 
x_)^2]), x_Symbol] :> Simp[-2*e   Subst[Int[1/(e*(b*e - 4*a*f) - (b*d - a*e 
)*x^2), x], x, (e + 2*f*x)/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b, c, d, 
 e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && EqQ[c*e - b*f, 
0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1351\) vs. \(2(146)=292\).

Time = 4.97 (sec) , antiderivative size = 1352, normalized size of antiderivative = 8.35

method	result	size
default	\(\text {Expression too large to display}\)	\(1352\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 875 vs. \(2 (146) = 292\).

Time = 0.77 (sec) , antiderivative size = 2005, normalized size of antiderivative = 12.38 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/4*(sqrt(b^2*d*e^2 - a*b*e^3 - 4*(a*b*d*e - a^2*e^2)*f)*(a*b*e^3 + (b^2 
*e^2*f + 4*(b^2*d - 2*a*b*e)*f^2)*x^2 + 4*(a*b*d*e - 2*a^2*e^2)*f + (b^2*e 
^3 + 4*(b^2*d*e - 2*a*b*e^2)*f)*x)*log((8*b^2*d^2*e^4 - 8*a*b*d*e^5 + a^2* 
e^6 + 16*a^2*d^2*e^2*f^2 + (b^2*e^4*f^2 + 16*(b^2*d^2 - 8*a*b*d*e + 8*a^2* 
e^2)*f^4 + 8*(3*b^2*d*e^2 - 4*a*b*e^3)*f^3)*x^4 + 2*(b^2*e^5*f + 16*(b^2*d 
^2*e - 8*a*b*d*e^2 + 8*a^2*e^3)*f^3 + 8*(3*b^2*d*e^3 - 4*a*b*e^4)*f^2)*x^3 
 + (b^2*e^6 - 32*(3*a*b*d^2*e - 4*a^2*d*e^2)*f^3 + 16*(3*b^2*d^2*e^2 - 13* 
a*b*d*e^3 + 10*a^2*e^4)*f^2 + 2*(16*b^2*d*e^4 - 19*a*b*e^5)*f)*x^2 - 4*sqr 
t(b^2*d*e^2 - a*b*e^3 - 4*(a*b*d*e - a^2*e^2)*f)*(2*b*d*e^3 - a*e^4 - 4*a* 
d*e^2*f + 2*(b*e^2*f^2 + 4*(b*d - 2*a*e)*f^3)*x^3 + 3*(b*e^3*f + 4*(b*d*e 
- 2*a*e^2)*f^2)*x^2 + (b*e^4 - 8*a*d*e*f^2 + 2*(4*b*d*e^2 - 5*a*e^3)*f)*x) 
*sqrt(f*x^2 + e*x + d) - 8*(4*a*b*d^2*e^3 - 3*a^2*d*e^4)*f + 2*(4*b^2*d*e^ 
5 - 3*a*b*e^6 - 16*(3*a*b*d^2*e^2 - 4*a^2*d*e^3)*f^2 + 8*(2*b^2*d^2*e^3 - 
5*a*b*d*e^4 + 2*a^2*e^5)*f)*x)/(b^2*f^2*x^4 + 2*b^2*e*f*x^3 + 2*a*b*e^2*x 
+ a^2*e^2 + (b^2*e^2 + 2*a*b*e*f)*x^2)) + 4*(b^3*d*e^3 - a*b^2*e^4 - 4*(a* 
b^2*d*e^2 - a^2*b*e^3)*f - 2*(4*(a*b^2*d*e - a^2*b*e^2)*f^2 - (b^3*d*e^2 - 
 a*b^2*e^3)*f)*x)*sqrt(f*x^2 + e*x + d))/(a*b^4*d^2*e^5 - 2*a^2*b^3*d*e^6 
+ a^3*b^2*e^7 + 16*(a^3*b^2*d^2*e^3 - 2*a^4*b*d*e^4 + a^5*e^5)*f^2 + (16*( 
a^2*b^3*d^2*e^2 - 2*a^3*b^2*d*e^3 + a^4*b*e^4)*f^3 - 8*(a*b^4*d^2*e^3 - 2* 
a^2*b^3*d*e^4 + a^3*b^2*e^5)*f^2 + (b^5*d^2*e^4 - 2*a*b^4*d*e^5 + a^2*b...
 

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\int \frac {1}{\sqrt {d + e x + f x^{2}} \left (a e + b e x + b f x^{2}\right )^{2}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(d + e*x + f*x**2)*(a*e + b*e*x + b*f*x**2)**2), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\int { \frac {1}{{\left (b f x^{2} + b e x + a e\right )}^{2} \sqrt {f x^{2} + e x + d}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1305 vs. \(2 (146) = 292\).

Time = 0.32 (sec) , antiderivative size = 1305, normalized size of antiderivative = 8.06 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*((b*e^2 + 4*b*d*f - 8*a*e*f)*log(abs(-(sqrt(f)*x - sqrt(f*x^2 + e*x + 
d))^2*b*e^2*f - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*b*d*f^2 + 8*(sqrt( 
f)*x - sqrt(f*x^2 + e*x + d))^2*a*e*f^2 - (sqrt(f)*x - sqrt(f*x^2 + e*x + 
d))*b*e^3*sqrt(f) - 4*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*d*e*f^(3/2) + 
8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*a*e^2*f^(3/2) - 3*b*d*e^2*f + 2*a*e^ 
3*f + 4*b*d^2*f^2 + 4*sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f 
)*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*f^(3/2) + 4*sqrt(b^2*d*e^2 - a*b*e 
^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*e*f + 
sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*e^2*sqrt(f)))/sqrt(b 
^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f) - (b*e^2 + 4*b*d*f - 8*a*e 
*f)*log(abs(-(sqrt(f)*x - sqrt(f*x^2 + e*x + d))^2*b*e^2*f - 4*(sqrt(f)*x 
- sqrt(f*x^2 + e*x + d))^2*b*d*f^2 + 8*(sqrt(f)*x - sqrt(f*x^2 + e*x + d)) 
^2*a*e*f^2 - (sqrt(f)*x - sqrt(f*x^2 + e*x + d))*b*e^3*sqrt(f) - 4*(sqrt(f 
)*x - sqrt(f*x^2 + e*x + d))*b*d*e*f^(3/2) + 8*(sqrt(f)*x - sqrt(f*x^2 + e 
*x + d))*a*e^2*f^(3/2) - 3*b*d*e^2*f + 2*a*e^3*f + 4*b*d^2*f^2 - 4*sqrt(b^ 
2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f)*(sqrt(f)*x - sqrt(f*x^2 + e 
*x + d))^2*f^(3/2) - 4*sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2* 
f)*(sqrt(f)*x - sqrt(f*x^2 + e*x + d))*e*f - sqrt(b^2*d*e^2 - a*b*e^3 - 4* 
a*b*d*e*f + 4*a^2*e^2*f)*e^2*sqrt(f)))/sqrt(b^2*d*e^2 - a*b*e^3 - 4*a*b*d* 
e*f + 4*a^2*e^2*f))/(b^2*d*e^2 - a*b*e^3 - 4*a*b*d*e*f + 4*a^2*e^2*f) +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,f\,x^2+b\,e\,x+a\,e\right )}^2\,\sqrt {f\,x^2+e\,x+d}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 3406, normalized size of antiderivative = 21.02 \[ \int \frac {1}{\sqrt {d+e x+f x^2} \left (a e+b e x+b f x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*sqrt(e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d)*log( - sqrt(4*sqrt(f)*sqrt(e) 
*sqrt(4*a*f - b*e)*sqrt(a*e - b*d) - 8*a*e*f + 4*b*d*f + b*e**2) + 2*sqrt( 
f)*sqrt(b)*sqrt(d + e*x + f*x**2) + sqrt(b)*e + 2*sqrt(b)*f*x)*a**2*e**2*f 
 - 4*sqrt(e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d)*log( - sqrt(4*sqrt(f)*sqrt( 
e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d) - 8*a*e*f + 4*b*d*f + b*e**2) + 2*sqr 
t(f)*sqrt(b)*sqrt(d + e*x + f*x**2) + sqrt(b)*e + 2*sqrt(b)*f*x)*a*b*d*e*f 
 - sqrt(e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d)*log( - sqrt(4*sqrt(f)*sqrt(e) 
*sqrt(4*a*f - b*e)*sqrt(a*e - b*d) - 8*a*e*f + 4*b*d*f + b*e**2) + 2*sqrt( 
f)*sqrt(b)*sqrt(d + e*x + f*x**2) + sqrt(b)*e + 2*sqrt(b)*f*x)*a*b*e**3 + 
8*sqrt(e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d)*log( - sqrt(4*sqrt(f)*sqrt(e)* 
sqrt(4*a*f - b*e)*sqrt(a*e - b*d) - 8*a*e*f + 4*b*d*f + b*e**2) + 2*sqrt(f 
)*sqrt(b)*sqrt(d + e*x + f*x**2) + sqrt(b)*e + 2*sqrt(b)*f*x)*a*b*e**2*f*x 
 + 8*sqrt(e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d)*log( - sqrt(4*sqrt(f)*sqrt( 
e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d) - 8*a*e*f + 4*b*d*f + b*e**2) + 2*sqr 
t(f)*sqrt(b)*sqrt(d + e*x + f*x**2) + sqrt(b)*e + 2*sqrt(b)*f*x)*a*b*e*f** 
2*x**2 - 4*sqrt(e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d)*log( - sqrt(4*sqrt(f) 
*sqrt(e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d) - 8*a*e*f + 4*b*d*f + b*e**2) + 
 2*sqrt(f)*sqrt(b)*sqrt(d + e*x + f*x**2) + sqrt(b)*e + 2*sqrt(b)*f*x)*b** 
2*d*e*f*x - 4*sqrt(e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d)*log( - sqrt(4*sqrt 
(f)*sqrt(e)*sqrt(4*a*f - b*e)*sqrt(a*e - b*d) - 8*a*e*f + 4*b*d*f + b*e...