3.801 \(\int \frac{1}{(1-d x)^{3/2} (1+d x)^{3/2} (a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=939 \[ -\frac{\left (b \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-\left (2 c^4-d^2 \left (b^2+6 a^2 d^2\right ) c^2-\left (4 a^3 d^6+6 a b^2 d^4\right ) c+b^2 d^4 \left (2 b^2+a^2 d^2\right )\right ) x\right ) d^2}{\left (b^2-4 a c\right ) \left (a d^2-b d+c\right )^2 \left (a d^2+b d+c\right )^2 \sqrt{1-d^2 x^2}}+\frac{c \left (3 a b^3 \left (b+\sqrt{b^2-4 a c}\right ) d^6-2 a c^2 \left (7 b^2+5 \sqrt{b^2-4 a c} b-8 a^2 d^2\right ) d^4+b c \left (2 b^3+2 \sqrt{b^2-4 a c} b^2-17 a^2 d^2 b-11 a^2 \sqrt{b^2-4 a c} d^2\right ) d^4+24 a c^4 d^2-c^3 \left (9 b^2-\sqrt{b^2-4 a c} b-36 a^2 d^2\right ) d^2+4 c^5\right ) \tanh ^{-1}\left (\frac{\left (b-\sqrt{b^2-4 a c}\right ) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left (b-\sqrt{b^2-4 a c}\right ) d^2} \sqrt{1-d^2 x^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c^2+2 a d^2 c-b \left (b-\sqrt{b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2}-\frac{b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (c x^2+b x+a\right ) \sqrt{1-d^2 x^2}}+\frac{c \left (b \left (b+\sqrt{b^2-4 a c}\right ) d^4 \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-2 \left (3 a b^4 d^8+2 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+12 a c^4 d^4+2 c^5 d^2-c^3 \left (4 b^2 d^4-18 a^2 d^6\right )-4 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \tanh ^{-1}\left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left (b+\sqrt{b^2-4 a c}\right ) d^2} \sqrt{1-d^2 x^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c^2+2 a d^2 c-b \left (b+\sqrt{b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2 d^2} \]

[Out]

-((d^2*(b*(c^3 + 2*b^2*c*d^2 - 10*a*c^2*d^2 + 3*a*b^2*d^4 - 11*a^2*c*d^4) - (2*c^4 + b^2*d^4*(2*b^2 + a^2*d^2)
- c^2*d^2*(b^2 + 6*a^2*d^2) - c*(6*a*b^2*d^4 + 4*a^3*d^6))*x))/((b^2 - 4*a*c)*(c - b*d + a*d^2)^2*(c + b*d +
a*d^2)^2*Sqrt[1 - d^2*x^2])) - (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)/((b^2 - 4*a
*c)*(b^2*d^2 - (c + a*d^2)^2)*(a + b*x + c*x^2)*Sqrt[1 - d^2*x^2]) + (c*(4*c^5 + 24*a*c^4*d^2 + 3*a*b^3*(b + S
qrt[b^2 - 4*a*c])*d^6 - c^3*d^2*(9*b^2 - b*Sqrt[b^2 - 4*a*c] - 36*a^2*d^2) - 2*a*c^2*d^4*(7*b^2 + 5*b*Sqrt[b^2
- 4*a*c] - 8*a^2*d^2) + b*c*d^4*(2*b^3 + 2*b^2*Sqrt[b^2 - 4*a*c] - 17*a^2*b*d^2 - 11*a^2*Sqrt[b^2 - 4*a*c]*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]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2]
*(c^2 - b^2*d^2 + 2*a*c*d^2 + a^2*d^4)^2) + (c*(b*(b + Sqrt[b^2 - 4*a*c])*d^4*(c^3 + 2*b^2*c*d^2 - 10*a*c^2*d^
2 + 3*a*b^2*d^4 - 11*a^2*c*d^4) - 2*(2*c^5*d^2 + 12*a*c^4*d^4 + 3*a*b^4*d^8 + 2*b^2*c*d^6*(b^2 - 7*a^2*d^2) -
c^3*(4*b^2*d^4 - 18*a^2*d^6) - 4*c^2*(3*a*b^2*d^6 - 2*a^3*d^8)))*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]*(b^2 - 4*a*c)^
(3/2)*d^2*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*(c^2 - b^2*d^2 + 2*a*c*d^2 + a^2*d^4)^2)

________________________________________________________________________________________

Rubi [A]  time = 11.8441, antiderivative size = 938, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {899, 975, 1062, 1034, 725, 206} \[ -\frac{\left (b \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-\left (2 c^4-d^2 \left (b^2+6 a^2 d^2\right ) c^2-\left (4 a^3 d^6+6 a b^2 d^4\right ) c+b^2 d^4 \left (2 b^2+a^2 d^2\right )\right ) x\right ) d^2}{\left (b^2-4 a c\right ) \left (a d^2-b d+c\right )^2 \left (a d^2+b d+c\right )^2 \sqrt{1-d^2 x^2}}+\frac{c \left (3 a b^3 \left (b+\sqrt{b^2-4 a c}\right ) d^6-2 a c^2 \left (7 b^2+5 \sqrt{b^2-4 a c} b-8 a^2 d^2\right ) d^4+b c \left (2 b^3+2 \sqrt{b^2-4 a c} b^2-17 a^2 d^2 b-11 a^2 \sqrt{b^2-4 a c} d^2\right ) d^4+24 a c^4 d^2-c^3 \left (9 b^2-\sqrt{b^2-4 a c} b-36 a^2 d^2\right ) d^2+4 c^5\right ) \tanh ^{-1}\left (\frac{\left (b-\sqrt{b^2-4 a c}\right ) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left (b-\sqrt{b^2-4 a c}\right ) d^2} \sqrt{1-d^2 x^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c^2+2 a d^2 c-b \left (b-\sqrt{b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2}-\frac{b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (c x^2+b x+a\right ) \sqrt{1-d^2 x^2}}-\frac{c \left (6 a b^4 d^8+4 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+24 a c^4 d^4-b \left (b+\sqrt{b^2-4 a c}\right ) \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right ) d^4+4 c^5 d^2-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right ) \tanh ^{-1}\left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left (b+\sqrt{b^2-4 a c}\right ) d^2} \sqrt{1-d^2 x^2}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c^2+2 a d^2 c-b \left (b+\sqrt{b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2 d^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - d*x)^(3/2)*(1 + d*x)^(3/2)*(a + b*x + c*x^2)^2),x]

[Out]

-((d^2*(b*(c^3 + 2*b^2*c*d^2 - 10*a*c^2*d^2 + 3*a*b^2*d^4 - 11*a^2*c*d^4) - (2*c^4 + b^2*d^4*(2*b^2 + a^2*d^2)
- c^2*d^2*(b^2 + 6*a^2*d^2) - c*(6*a*b^2*d^4 + 4*a^3*d^6))*x))/((b^2 - 4*a*c)*(c - b*d + a*d^2)^2*(c + b*d +
a*d^2)^2*Sqrt[1 - d^2*x^2])) - (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)/((b^2 - 4*a
*c)*(b^2*d^2 - (c + a*d^2)^2)*(a + b*x + c*x^2)*Sqrt[1 - d^2*x^2]) + (c*(4*c^5 + 24*a*c^4*d^2 + 3*a*b^3*(b + S
qrt[b^2 - 4*a*c])*d^6 - c^3*d^2*(9*b^2 - b*Sqrt[b^2 - 4*a*c] - 36*a^2*d^2) - 2*a*c^2*d^4*(7*b^2 + 5*b*Sqrt[b^2
- 4*a*c] - 8*a^2*d^2) + b*c*d^4*(2*b^3 + 2*b^2*Sqrt[b^2 - 4*a*c] - 17*a^2*b*d^2 - 11*a^2*Sqrt[b^2 - 4*a*c]*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]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2]
*(c^2 - b^2*d^2 + 2*a*c*d^2 + a^2*d^4)^2) - (c*(4*c^5*d^2 + 24*a*c^4*d^4 + 6*a*b^4*d^8 + 4*b^2*c*d^6*(b^2 - 7*
a^2*d^2) - b*(b + Sqrt[b^2 - 4*a*c])*d^4*(c^3 + 2*b^2*c*d^2 - 10*a*c^2*d^2 + 3*a*b^2*d^4 - 11*a^2*c*d^4) - 4*c
^3*(2*b^2*d^4 - 9*a^2*d^6) - 8*c^2*(3*a*b^2*d^6 - 2*a^3*d^8))*ArcTanh[(2*c + (b + Sqrt[b^2 - 4*a*c])*d^2*x)/(S
qrt[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]*(b^2 - 4*a*c)^(3/
2)*d^2*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]*(c^2 - b^2*d^2 + 2*a*c*d^2 + a^2*d^4)^2)

Rule 899

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, 0] &&
EqQ[e*f + d*g, 0] && (IntegerQ[m] || (GtQ[d, 0] && GtQ[f, 0]))

Rule 975

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] - Dist[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 1062

Int[((a_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_
Symbol] :> Simp[((a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e) + (-(a*B))*(2*c^2*d - c*
(2*a*f)) + c*(A*(2*c^2*d - c*(2*a*f)) - B*(-2*a*c*e) + C*(-2*a*(c*d - a*f)))*x))/((-4*a*c)*(a*c*e^2 + (c*d - a
*f)^2)*(p + 1)), x] + Dist[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x +
f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (-(a*e))*(c*e))*(p + 1) + (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c
*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e) + (-(a*B))*(2*c^2*d - c*((Plus[2])*a*f)))
*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e) + (-(a*B))*(2*c^2*d - c*((Plus[2])*a*f)))*(p + q + 2) - (2*(A*c*(c*
d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(-(c*e*(2*p + q + 4))))*x - c*f*(2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e
- a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, A, B, C, q}, x] && NeQ[e^2 - 4*d*f, 0] &&
LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1034

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]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Dist[(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]

Rule 725

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx &=\int \frac{1}{\left (a+b x+c x^2\right )^2 \left (1-d^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{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}{\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 ) \sqrt{1-d^2 x^2}}-\frac{\int \frac{-2 c^3-6 a c^2 d^2+a b^2 d^4+2 c d^2 \left (b^2-2 a^2 d^2\right )+b d^2 \left (c^2-2 b^2 d^2+7 a c d^2\right ) x+2 c d^2 \left (2 c^2-b^2 d^2+2 a c d^2\right ) x^2}{\left (a+b x+c x^2\right ) \left (1-d^2 x^2\right )^{3/2}} \, dx}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )}\\ &=-\frac{d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2 \sqrt{1-d^2 x^2}}-\frac{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}{\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 ) \sqrt{1-d^2 x^2}}-\frac{\int \frac{2 \left (2 c^5 d^2+12 a c^4 d^4+3 a b^4 d^8+2 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-\frac{1}{2} c^3 \left (8 b^2 d^4-36 a^2 d^6\right )-4 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )+2 b c d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right ) x}{\left (a+b x+c x^2\right ) \sqrt{1-d^2 x^2}} \, dx}{2 \left (b^2-4 a c\right ) d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}\\ &=-\frac{d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2 \sqrt{1-d^2 x^2}}-\frac{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}{\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 ) \sqrt{1-d^2 x^2}}-\frac{\left (c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b-\sqrt{b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \int \frac{1}{\left (b-\sqrt{b^2-4 a c}+2 c x\right ) \sqrt{1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right )^{3/2} d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}+\frac{\left (c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b+\sqrt{b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \int \frac{1}{\left (b+\sqrt{b^2-4 a c}+2 c x\right ) \sqrt{1-d^2 x^2}} \, dx}{\left (b^2-4 a c\right )^{3/2} d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}\\ &=-\frac{d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2 \sqrt{1-d^2 x^2}}-\frac{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}{\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 ) \sqrt{1-d^2 x^2}}+\frac{\left (c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b-\sqrt{b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c^2-\left (b-\sqrt{b^2-4 a c}\right )^2 d^2-x^2} \, dx,x,\frac{2 c+\left (b-\sqrt{b^2-4 a c}\right ) d^2 x}{\sqrt{1-d^2 x^2}}\right )}{\left (b^2-4 a c\right )^{3/2} d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}-\frac{\left (c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b+\sqrt{b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c^2-\left (b+\sqrt{b^2-4 a c}\right )^2 d^2-x^2} \, dx,x,\frac{2 c+\left (b+\sqrt{b^2-4 a c}\right ) d^2 x}{\sqrt{1-d^2 x^2}}\right )}{\left (b^2-4 a c\right )^{3/2} d^2 \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2}\\ &=-\frac{d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right )^2 \sqrt{1-d^2 x^2}}-\frac{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}{\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 ) \sqrt{1-d^2 x^2}}+\frac{c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b-\sqrt{b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right ) \tanh ^{-1}\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} d^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 )^2}-\frac{c \left (4 c^5 d^2+24 a c^4 d^4+6 a b^4 d^8+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-b \left (b+\sqrt{b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right ) \tanh ^{-1}\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} d^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 )^2}\\ \end{align*}

Mathematica [A]  time = 9.33107, size = 890, normalized size = 0.95 \[ \frac{\frac{\sqrt{2} \left (-3 a b \left (b+\sqrt{b^2-4 a c}\right ) d^4+20 a c^2 d^2-c \left (7 b^2-3 \sqrt{b^2-4 a c} b-16 a^2 d^2\right ) d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{\left (b-\sqrt{b^2-4 a c}\right ) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left (b-\sqrt{b^2-4 a c}\right ) d^2} \sqrt{1-d^2 x^2}}\right ) c^3}{\sqrt{b^2-4 a c} \left (2 c^2+2 a d^2 c-b \left (b-\sqrt{b^2-4 a c}\right ) d^2\right )^{3/2}}-\frac{\sqrt{2} \left (-3 a b \left (b-\sqrt{b^2-4 a c}\right ) d^4+20 a c^2 d^2-c \left (7 b^2+3 \sqrt{b^2-4 a c} b-16 a^2 d^2\right ) d^2+4 c^3\right ) \tanh ^{-1}\left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) x d^2+2 c}{\sqrt{2} \sqrt{2 c^2+2 a d^2 c-b \left (b+\sqrt{b^2-4 a c}\right ) d^2} \sqrt{1-d^2 x^2}}\right ) c^3}{\sqrt{b^2-4 a c} \left (2 c^2+2 a d^2 c-b \left (b+\sqrt{b^2-4 a c}\right ) d^2\right )^{3/2}}-\frac{\left (3 b \left (c-a d^2\right ) d^2+\frac{-3 a b^2 d^4+16 a^2 c d^4+20 a c^2 d^2-7 b^2 c d^2+4 c^3}{\sqrt{b^2-4 a c}}\right ) \left (2 c-\left (b-\sqrt{b^2-4 a c}\right ) d^2 x\right ) c}{\left (4 c^2-\left (b-\sqrt{b^2-4 a c}\right )^2 d^2\right ) \sqrt{1-d^2 x^2}}-\frac{\left (3 b d^2 \left (c-a d^2\right )-\frac{-3 a b^2 d^4+16 a^2 c d^4+20 a c^2 d^2-7 b^2 c d^2+4 c^3}{\sqrt{b^2-4 a c}}\right ) \left (2 c-\left (b+\sqrt{b^2-4 a c}\right ) d^2 x\right ) c}{\left (4 c^2-\left (b+\sqrt{b^2-4 a c}\right )^2 d^2\right ) \sqrt{1-d^2 x^2}}+\frac{2 d^2 \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\sqrt{1-d^2 x^2}}}{\left (b^2-4 a c\right ) \left (\left (a d^2+c\right )^2-b^2 d^2\right )}-\frac{-d^2 b^3+c \left (3 a d^2+c\right ) b+c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\left (b^2-4 a c\right ) \left (\left (a d^2+c\right )^2-b^2 d^2\right ) \left (c x^2+b x+a\right ) \sqrt{1-d^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - d*x)^(3/2)*(1 + d*x)^(3/2)*(a + b*x + c*x^2)^2),x]

[Out]

-((-(b^3*d^2) + b*c*(c + 3*a*d^2) + c*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)/((b^2 - 4*a*c)*(-(b^2*d^2) + (c + a*d^2
)^2)*(a + b*x + c*x^2)*Sqrt[1 - d^2*x^2])) + ((2*d^2*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)/Sqrt[1 - d^2*x^2] - (c*(
3*b*d^2*(c - a*d^2) + (4*c^3 - 7*b^2*c*d^2 + 20*a*c^2*d^2 - 3*a*b^2*d^4 + 16*a^2*c*d^4)/Sqrt[b^2 - 4*a*c])*(2*
c - (b - Sqrt[b^2 - 4*a*c])*d^2*x))/((4*c^2 - (b - Sqrt[b^2 - 4*a*c])^2*d^2)*Sqrt[1 - d^2*x^2]) - (c*(3*b*d^2*
(c - a*d^2) - (4*c^3 - 7*b^2*c*d^2 + 20*a*c^2*d^2 - 3*a*b^2*d^4 + 16*a^2*c*d^4)/Sqrt[b^2 - 4*a*c])*(2*c - (b +
Sqrt[b^2 - 4*a*c])*d^2*x))/((4*c^2 - (b + Sqrt[b^2 - 4*a*c])^2*d^2)*Sqrt[1 - d^2*x^2]) + (Sqrt[2]*c^3*(4*c^3
+ 20*a*c^2*d^2 - 3*a*b*(b + Sqrt[b^2 - 4*a*c])*d^4 - c*d^2*(7*b^2 - 3*b*Sqrt[b^2 - 4*a*c] - 16*a^2*d^2))*ArcTa
nh[(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[b^2 - 4*a*c]*(2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2)^(3/2)) - (Sqrt[2]*c^3*
(4*c^3 + 20*a*c^2*d^2 - 3*a*b*(b - Sqrt[b^2 - 4*a*c])*d^4 - c*d^2*(7*b^2 + 3*b*Sqrt[b^2 - 4*a*c] - 16*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[b^2 - 4*a*c]*(2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2)^(3/2)))/((b^2 -
4*a*c)*(-(b^2*d^2) + (c + a*d^2)^2))

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Maple [C]  time = 4.968, size = 108969, normalized size = 116.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-d*x+1)^(3/2)/(d*x+1)^(3/2)/(c*x^2+b*x+a)^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{2}{\left (d x + 1\right )}^{\frac{3}{2}}{\left (-d x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-d*x+1)^(3/2)/(d*x+1)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^2*(d*x + 1)^(3/2)*(-d*x + 1)^(3/2)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-d*x+1)^(3/2)/(d*x+1)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-d*x+1)**(3/2)/(d*x+1)**(3/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-d*x+1)^(3/2)/(d*x+1)^(3/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

Timed out