\(\int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} (a+b x+c x^2)^2} \, dx\) [801]

Optimal result

Integrand size = 32, antiderivative size = 939 \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c-b d+a d^2\right )^2 \left (c+b d+a d^2\right )^2 \sqrt {1-d^2 x^2}}-\frac {b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}}+\frac {c \left (4 c^5+24 a c^4 d^2+3 a b^3 \left (b+\sqrt {b^2-4 a c}\right ) d^6-c^3 d^2 \left (9 b^2-b \sqrt {b^2-4 a c}-36 a^2 d^2\right )-2 a c^2 d^4 \left (7 b^2+5 b \sqrt {b^2-4 a c}-8 a^2 d^2\right )+b c d^4 \left (2 b^3+2 b^2 \sqrt {b^2-4 a c}-17 a^2 b d^2-11 a^2 \sqrt {b^2-4 a c} d^2\right )\right ) \text {arctanh}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (c^2-b^2 d^2+2 a c d^2+a^2 d^4\right )^2}+\frac {c \left (b \left (b+\sqrt {b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-2 \left (2 c^5 d^2+12 a c^4 d^4+3 a b^4 d^8+2 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-c^3 \left (4 b^2 d^4-18 a^2 d^6\right )-4 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \text {arctanh}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} d^2 \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (c^2-b^2 d^2+2 a c d^2+a^2 d^4\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 11.13 (sec) , antiderivative size = 938, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {913, 989, 1076, 1048, 739, 212} \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {\left (b \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-\left (2 c^4-d^2 \left (b^2+6 a^2 d^2\right ) c^2-\left (4 a^3 d^6+6 a b^2 d^4\right ) c+b^2 d^4 \left (2 b^2+a^2 d^2\right )\right ) x\right ) d^2}{\left (b^2-4 a c\right ) \left (a d^2-b d+c\right )^2 \left (a d^2+b d+c\right )^2 \sqrt {1-d^2 x^2}}+\frac {c \left (3 a b^3 \left (b+\sqrt {b^2-4 a c}\right ) d^6-2 a c^2 \left (7 b^2+5 \sqrt {b^2-4 a c} b-8 a^2 d^2\right ) d^4+b c \left (2 b^3+2 \sqrt {b^2-4 a c} b^2-17 a^2 d^2 b-11 a^2 \sqrt {b^2-4 a c} d^2\right ) d^4+24 a c^4 d^2-c^3 \left (9 b^2-\sqrt {b^2-4 a c} b-36 a^2 d^2\right ) d^2+4 c^5\right ) \text {arctanh}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}}-\frac {c \left (6 a b^4 d^8+4 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+24 a c^4 d^4-b \left (b+\sqrt {b^2-4 a c}\right ) \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right ) d^4+4 c^5 d^2-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right ) \text {arctanh}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (a^2 d^4-b^2 d^2+2 a c d^2+c^2\right )^2 d^2} \]

[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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

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 913

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 989

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 1048

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 1076

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)/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 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), 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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 [C] (verified)

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

Time = 7.47 (sec) , antiderivative size = 3830, normalized size of antiderivative = 4.08 \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(-((Sqrt[1 - d^2*x^2]*(b^5*d^4*(-1 + 2*d^2*x^2) + 2*c*(c + a*d^2)^2*x*(-2*a^2*d^4 - 2*a*c*d^4*x^2 + c^2*(-1 +
d^2*x^2)) + b^2*d^2*x*(a^3*d^6 + a*c^2*d^2*(13 - 6*d^2*x^2) + c^3*(2 - d^2*x^2) + a^2*c*d^4*(6 + d^2*x^2)) + b
^4*d^4*x*(-(a*d^2) + c*(-3 + 2*d^2*x^2)) + b*c*(c + a*d^2)*(-4*a^2*d^4*(-2 + d^2*x^2) + c^2*(-1 + d^2*x^2) + a
*c*d^2*(-5 + 9*d^2*x^2)) + b^3*(a^2*d^6*(-2 + d^2*x^2) + c^2*(2*d^2 - 3*d^4*x^2) + a*c*(3*d^4 - 9*d^6*x^2))))/
((b^2 - 4*a*c)*(-1 + d*x)*(1 + d*x)*(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^4*c^4*Log[x] + 20*a*b^2*c^5*Log[x] - 16*a^2*c^6*Log[x] + 8*b^6*c^2*d^2*Log[x] -
56*a*b^4*c^3*d^2*Log[x] + 107*a^2*b^2*c^4*d^2*Log[x] - 46*a^3*c^5*d^2*Log[x] - 4*b^8*d^4*Log[x] + 36*a*b^6*c*
d^4*Log[x] - 110*a^2*b^4*c^2*d^4*Log[x] + 132*a^3*b^2*c^3*d^4*Log[x] - 44*a^4*c^4*d^4*Log[x] + 3*a^2*b^6*d^6*L
og[x] - 22*a^3*b^4*c*d^6*Log[x] + 43*a^4*b^2*c^2*d^6*Log[x] - 14*a^5*c^3*d^6*Log[x] + 4*b^4*c^4*Log[-1 + Sqrt[
1 - d^2*x^2] - x*#1] - 20*a*b^2*c^5*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 16*a^2*c^6*Log[-1 + Sqrt[1 - d^2*x^2]
- x*#1] - 8*b^6*c^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 56*a*b^4*c^3*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*
#1] - 107*a^2*b^2*c^4*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 46*a^3*c^5*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#
1] + 4*b^8*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 36*a*b^6*c*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 110*a^
2*b^4*c^2*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 132*a^3*b^2*c^3*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 44
*a^4*c^4*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 3*a^2*b^6*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 22*a^3*b^
4*c*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 43*a^4*b^2*c^2*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 14*a^5*c^
3*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 2*a*b^3*c^4*Log[x]*#1 + 8*a^2*b*c^5*Log[x]*#1 + 4*a*b^5*c^2*d^2*Log
[x]*#1 - 24*a^2*b^3*c^3*d^2*Log[x]*#1 + 34*a^3*b*c^4*d^2*Log[x]*#1 - 2*a*b^7*d^4*Log[x]*#1 + 16*a^2*b^5*c*d^4*
Log[x]*#1 - 42*a^3*b^3*c^2*d^4*Log[x]*#1 + 36*a^4*b*c^3*d^4*Log[x]*#1 + 2*a^3*b^5*d^6*Log[x]*#1 - 10*a^4*b^3*c
*d^6*Log[x]*#1 + 10*a^5*b*c^2*d^6*Log[x]*#1 + 2*a*b^3*c^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - 8*a^2*b*c^5*
Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - 4*a*b^5*c^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 24*a^2*b^3*c^3
*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - 34*a^3*b*c^4*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 2*a*b^7*
d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - 16*a^2*b^5*c*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 42*a^3*b^
3*c^2*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - 36*a^4*b*c^3*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - 2*a
^3*b^5*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 10*a^4*b^3*c*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - 10
*a^5*b*c^2*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - a^2*b^2*c^4*Log[x]*#1^2 + 2*a^3*c^5*Log[x]*#1^2 + 2*a^2
*b^4*c^2*d^2*Log[x]*#1^2 - 8*a^3*b^2*c^3*d^2*Log[x]*#1^2 + 4*a^4*c^4*d^2*Log[x]*#1^2 - a^2*b^6*d^4*Log[x]*#1^2
+ 6*a^3*b^4*c*d^4*Log[x]*#1^2 - 9*a^4*b^2*c^2*d^4*Log[x]*#1^2 + 2*a^5*c^3*d^4*Log[x]*#1^2 + a^2*b^2*c^4*Log[-
1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 - 2*a^3*c^5*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 - 2*a^2*b^4*c^2*d^2*Log
[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 + 8*a^3*b^2*c^3*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 - 4*a^4*c^4*d
^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 + a^2*b^6*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 - 6*a^3*b^4*c
*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 + 9*a^4*b^2*c^2*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 - 2*a
^5*c^3*d^4*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*(-b^2 + 4*a*c)) + 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*c^4*
Log[x] + 4*a*c^5*Log[x] + 8*b^4*c^2*d^2*Log[x] - 24*a*b^2*c^3*d^2*Log[x] + 11*a^2*c^4*d^2*Log[x] - 4*b^6*d^4*L
og[x] + 20*a*b^4*c*d^4*Log[x] - 30*a^2*b^2*c^2*d^4*Log[x] + 8*a^3*c^3*d^4*Log[x] + 3*a^2*b^4*d^6*Log[x] - 8*a^
3*b^2*c*d^6*Log[x] - a^4*c^2*d^6*Log[x] + 3*a^4*b^2*d^8*Log[x] - 2*a^5*c*d^8*Log[x] + 4*b^2*c^4*Log[-1 + Sqrt[
1 - d^2*x^2] - x*#1] - 4*a*c^5*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 8*b^4*c^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] -
x*#1] + 24*a*b^2*c^3*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 11*a^2*c^4*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#
1] + 4*b^6*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 20*a*b^4*c*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 30*a^2
*b^2*c^2*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 8*a^3*c^3*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 3*a^2*b^4
*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 8*a^3*b^2*c*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + a^4*c^2*d^6*Log
[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 3*a^4*b^2*d^8*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 2*a^5*c*d^8*Log[-1 + Sqrt
[1 - d^2*x^2] - x*#1] - 2*a*b*c^4*Log[x]*#1 + 4*a*b^3*c^2*d^2*Log[x]*#1 - 8*a^2*b*c^3*d^2*Log[x]*#1 - 2*a*b^5*
d^4*Log[x]*#1 + 8*a^2*b^3*c*d^4*Log[x]*#1 - 14*a^3*b*c^2*d^4*Log[x]*#1 + 2*a^3*b^3*d^6*Log[x]*#1 - 8*a^4*b*c*d
^6*Log[x]*#1 + 2*a*b*c^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 - 4*a*b^3*c^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] -
x*#1]*#1 + 8*a^2*b*c^3*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 2*a*b^5*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*
#1]*#1 - 8*a^2*b^3*c*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 14*a^3*b*c^2*d^4*Log[-1 + Sqrt[1 - d^2*x^2] -
x*#1]*#1 - 2*a^3*b^3*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1 + 8*a^4*b*c*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x
*#1]*#1 - a^2*c^4*Log[x]*#1^2 + 2*a^2*b^2*c^2*d^2*Log[x]*#1^2 - 4*a^3*c^3*d^2*Log[x]*#1^2 - a^2*b^4*d^4*Log[x]
*#1^2 + 4*a^3*b^2*c*d^4*Log[x]*#1^2 - 5*a^4*c^2*d^4*Log[x]*#1^2 + 3*a^4*b^2*d^6*Log[x]*#1^2 - 2*a^5*c*d^6*Log[
x]*#1^2 + a^2*c^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 - 2*a^2*b^2*c^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#
1]*#1^2 + 4*a^3*c^3*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 + a^2*b^4*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#
1]*#1^2 - 4*a^3*b^2*c*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 + 5*a^4*c^2*d^4*Log[-1 + Sqrt[1 - d^2*x^2] -
x*#1]*#1^2 - 3*a^4*b^2*d^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1]*#1^2 + 2*a^5*c*d^6*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)/((c + d*(-b + a*d))^2*(c + d*(b + a*
d))^2)

Maple [C] (warning: unable to verify)

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

Time = 3.58 (sec) , antiderivative size = 108969, normalized size of antiderivative = 116.05

method	result	size
default	\(\text {Expression too large to display}\)	\(108969\)

[In]

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

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{2} {\left (d x + 1\right )}^{\frac {3}{2}} {\left (-d x + 1\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\int \frac {1}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^2} \,d x \]

[In]

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

[Out]

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