3.8.96 \(\int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} (a+b x+c x^2)^2} \, dx\) [796]

3.8.96.1 Optimal result

Integrand size = 32, antiderivative size = 571 \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=-\frac {\left (b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x\right ) \sqrt {1-d^2 x^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right )}-\frac {c \left (4 c^3+12 a c^2 d^2-a b \left (b+\sqrt {b^2-4 a c}\right ) d^4-c d^2 \left (5 b^2-b \sqrt {b^2-4 a c}-8 a^2 d^2\right )\right ) \text {arctanh}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )}+\frac {c \left (4 c^3+12 a c^2 d^2-2 a b^2 d^4-b \left (b+\sqrt {b^2-4 a c}\right ) d^2 \left (c-a d^2\right )-4 c d^2 \left (b^2-2 a^2 d^2\right )\right ) \text {arctanh}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (b^2 d^2-\left (c+a d^2\right )^2\right )} \]

output

-(b*(b^2*d^2-c*(3*a*d^2+c))-c*(2*a*c*d^2-b^2*d^2+2*c^2)*x)*(-d^2*x^2+1)^(1 
/2)/(-4*a*c+b^2)/(b^2*d^2-(a*d^2+c)^2)/(c*x^2+b*x+a)-1/2*c*arctanh(1/2*(2* 
c+d^2*x*(b-(-4*a*c+b^2)^(1/2)))*2^(1/2)/(-d^2*x^2+1)^(1/2)/(2*c^2+2*a*c*d^ 
2-b*d^2*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*(4*c^3+12*a*c^2*d^2-a*b*d^4*(b+(-4* 
a*c+b^2)^(1/2))-c*d^2*(5*b^2-8*a^2*d^2-b*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2) 
^(3/2)/(b^2*d^2-(a*d^2+c)^2)*2^(1/2)/(2*c^2+2*a*c*d^2-b*d^2*(b-(-4*a*c+b^2 
)^(1/2)))^(1/2)+1/2*c*arctanh(1/2*(2*c+d^2*x*(b+(-4*a*c+b^2)^(1/2)))*2^(1/ 
2)/(-d^2*x^2+1)^(1/2)/(2*c^2+2*a*c*d^2-b*d^2*(b+(-4*a*c+b^2)^(1/2)))^(1/2) 
)*(4*c^3+12*a*c^2*d^2-2*a*b^2*d^4-4*c*d^2*(-2*a^2*d^2+b^2)-b*d^2*(-a*d^2+c 
)*(b+(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(b^2*d^2-(a*d^2+c)^2)*2^(1/2) 
/(2*c^2+2*a*c*d^2-b*d^2*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
 

3.8.96.2 Mathematica [C] (verified)

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

Time = 2.67 (sec) , antiderivative size = 1548, normalized size of antiderivative = 2.71 \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \]

input

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

output

((-(b^3*d^2) + b*c*(c + 3*a*d^2) - b^2*c*d^2*x + 2*c^2*(c + a*d^2)*x)*Sqrt 
[1 - d^2*x^2])/((b^2 - 4*a*c)*(-c + d*(b - a*d))*(c + d*(b + a*d))*(a + x* 
(b + c*x))) + RootSum[a*d^4 - 2*b*d^2*#1 + 4*c*#1^2 + 2*a*d^2*#1^2 - 2*b*# 
1^3 + a*#1^4 & , (-4*b^2*Log[x] + 4*a*c*Log[x] - a^2*d^2*Log[x] + 4*b^2*Lo 
g[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 4*a*c*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1 
] + a^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 2*a*b*Log[x]*#1 + 2*a*b*L 
og[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - a^2*Log[x]*#1^2 + a^2*Log[-1 + Sqrt 
[1 - d^2*x^2] - x*#1]*#1^2)/(b*d^2 - 4*c*#1 - 2*a*d^2*#1 + 3*b*#1^2 - 2*a* 
#1^3) & ]/a^3 - RootSum[a*d^4 - 2*b*d^2*#1 + 4*c*#1^2 + 2*a*d^2*#1^2 - 2*b 
*#1^3 + a*#1^4 & , (4*b^4*c^2*Log[x] - 20*a*b^2*c^3*Log[x] + 16*a^2*c^4*Lo 
g[x] - 4*b^6*d^2*Log[x] + 28*a*b^4*c*d^2*Log[x] - 55*a^2*b^2*c^2*d^2*Log[x 
] + 30*a^3*c^3*d^2*Log[x] + 3*a^2*b^4*d^4*Log[x] - 16*a^3*b^2*c*d^4*Log[x] 
 + 14*a^4*c^2*d^4*Log[x] - 4*b^4*c^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 
20*a*b^2*c^3*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 16*a^2*c^4*Log[-1 + Sqrt 
[1 - d^2*x^2] - x*#1] + 4*b^6*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 28* 
a*b^4*c*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 55*a^2*b^2*c^2*d^2*Log[-1 
 + Sqrt[1 - d^2*x^2] - x*#1] - 30*a^3*c^3*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - 
 x*#1] - 3*a^2*b^4*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 16*a^3*b^2*c*d 
^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 14*a^4*c^2*d^4*Log[-1 + Sqrt[1 - d 
^2*x^2] - x*#1] + 2*a*b^3*c^2*Log[x]*#1 - 8*a^2*b*c^3*Log[x]*#1 - 2*a*b...
 

3.8.96.3 Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 564, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1188, 1306, 25, 1367, 488, 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.

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

\(\Big \downarrow \) 1188

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

\(\Big \downarrow \) 1306

\(\displaystyle -\frac {\int -\frac {-a b^2 d^4+6 a c^2 d^2-2 c \left (b^2-2 a^2 d^2\right ) d^2+b c \left (c-a d^2\right ) x d^2+2 c^3}{\left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}}dx}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {\sqrt {1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-a b^2 d^4+4 a^2 c d^4+6 a c^2 d^2-2 b^2 c d^2+b c \left (c-a d^2\right ) x d^2+2 c^3}{\left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}}dx}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {\sqrt {1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1367

\(\displaystyle \frac {\frac {c \left (-c d^2 \left (-8 a^2 d^2-b \sqrt {b^2-4 a c}+5 b^2\right )-a b d^4 \left (\sqrt {b^2-4 a c}+b\right )+12 a c^2 d^2+4 c^3\right ) \int \frac {1}{\left (b+2 c x-\sqrt {b^2-4 a c}\right ) \sqrt {1-d^2 x^2}}dx}{\sqrt {b^2-4 a c}}-\frac {c \left (-c d^2 \left (-8 a^2 d^2+b \sqrt {b^2-4 a c}+5 b^2\right )-a b d^4 \left (b-\sqrt {b^2-4 a c}\right )+12 a c^2 d^2+4 c^3\right ) \int \frac {1}{\left (b+2 c x+\sqrt {b^2-4 a c}\right ) \sqrt {1-d^2 x^2}}dx}{\sqrt {b^2-4 a c}}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {\sqrt {1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {c \left (-c d^2 \left (-8 a^2 d^2+b \sqrt {b^2-4 a c}+5 b^2\right )-a b d^4 \left (b-\sqrt {b^2-4 a c}\right )+12 a c^2 d^2+4 c^3\right ) \int \frac {1}{4 c^2-\left (b+\sqrt {b^2-4 a c}\right )^2 d^2-\frac {\left (\left (b+\sqrt {b^2-4 a c}\right ) x d^2+2 c\right )^2}{1-d^2 x^2}}d\frac {\left (b+\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {1-d^2 x^2}}}{\sqrt {b^2-4 a c}}-\frac {c \left (-c d^2 \left (-8 a^2 d^2-b \sqrt {b^2-4 a c}+5 b^2\right )-a b d^4 \left (\sqrt {b^2-4 a c}+b\right )+12 a c^2 d^2+4 c^3\right ) \int \frac {1}{4 c^2-\left (b-\sqrt {b^2-4 a c}\right )^2 d^2-\frac {\left (\left (b-\sqrt {b^2-4 a c}\right ) x d^2+2 c\right )^2}{1-d^2 x^2}}d\frac {\left (b-\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {1-d^2 x^2}}}{\sqrt {b^2-4 a c}}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {\sqrt {1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {c \left (-c d^2 \left (-8 a^2 d^2+b \sqrt {b^2-4 a c}+5 b^2\right )-a b d^4 \left (b-\sqrt {b^2-4 a c}\right )+12 a c^2 d^2+4 c^3\right ) \text {arctanh}\left (\frac {d^2 x \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-b d^2 \left (\sqrt {b^2-4 a c}+b\right )+2 a c d^2+2 c^2}}-\frac {c \left (-c d^2 \left (-8 a^2 d^2-b \sqrt {b^2-4 a c}+5 b^2\right )-a b d^4 \left (\sqrt {b^2-4 a c}+b\right )+12 a c^2 d^2+4 c^3\right ) \text {arctanh}\left (\frac {d^2 x \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {2} \sqrt {1-d^2 x^2} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {-b d^2 \left (b-\sqrt {b^2-4 a c}\right )+2 a c d^2+2 c^2}}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {\sqrt {1-d^2 x^2} \left (b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\)

input

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

output

-(((b*(b^2*d^2 - c*(c + 3*a*d^2)) - c*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)*Sqr 
t[1 - d^2*x^2])/((b^2 - 4*a*c)*(b^2*d^2 - (c + a*d^2)^2)*(a + b*x + c*x^2) 
)) + (-((c*(4*c^3 + 12*a*c^2*d^2 - a*b*(b + Sqrt[b^2 - 4*a*c])*d^4 - c*d^2 
*(5*b^2 - b*Sqrt[b^2 - 4*a*c] - 8*a^2*d^2))*ArcTanh[(2*c + (b - Sqrt[b^2 - 
 4*a*c])*d^2*x)/(Sqrt[2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c] 
)*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c* 
d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2])) + (c*(4*c^3 + 12*a*c^2*d^2 - a*b*(b 
 - Sqrt[b^2 - 4*a*c])*d^4 - c*d^2*(5*b^2 + b*Sqrt[b^2 - 4*a*c] - 8*a^2*d^2 
))*ArcTanh[(2*c + (b + Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqrt[2]*Sqrt[2*c^2 + 2*a 
*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2*x^2])])/(Sqrt[2]*Sqrt 
[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]))/(( 
b^2 - 4*a*c)*(b^2*d^2 - (c + a*d^2)^2))
 

3.8.96.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 488

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

rule 1188

Int[((d_) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d*f + e*g*x^2)^m*(a + b*x + c*x^2 
)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m, n] && EqQ[e*f 
 + d*g, 0] && (IntegerQ[m] || (GtQ[d, 0] && GtQ[f, 0]))
 

rule 1306

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

rule 1367

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

3.8.96.4 Maple [C] (warning: unable to verify)

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

Time = 1.02 (sec) , antiderivative size = 41837, normalized size of antiderivative = 73.27

method	result	size
default	\(\text {Expression too large to display}\)	\(41837\)

input

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

output

result too large to display
 

3.8.96.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35403 vs. \(2 (529) = 1058\).

Time = 33.26 (sec) , antiderivative size = 35403, normalized size of antiderivative = 62.00 \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

input

integrate(1/(c*x^2+b*x+a)^2/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="fri 
cas")
 

output

Too large to include
 

3.8.96.6 Sympy [F]

\[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\int \frac {1}{\sqrt {- d x + 1} \sqrt {d x + 1} \left (a + b x + c x^{2}\right )^{2}}\, dx \]

input

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

output

Integral(1/(sqrt(-d*x + 1)*sqrt(d*x + 1)*(a + b*x + c*x**2)**2), x)
 

3.8.96.7 Maxima [F]

\[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{2} \sqrt {d x + 1} \sqrt {-d x + 1}} \,d x } \]

input

integrate(1/(c*x^2+b*x+a)^2/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="max 
ima")
 

output

integrate(1/((c*x^2 + b*x + a)^2*sqrt(d*x + 1)*sqrt(-d*x + 1)), x)
 

3.8.96.8 Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

input

integrate(1/(c*x^2+b*x+a)^2/(-d*x+1)^(1/2)/(d*x+1)^(1/2),x, algorithm="gia 
c")
 

output

Timed out
 

3.8.96.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-d x} \sqrt {1+d x} \left (a+b x+c x^2\right )^2} \, dx=\text {Hanged} \]

input

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

output

\text{Hanged}