3.2488 \(\int \frac{(A+B x) (d+e x)^5}{(a+b x+c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=942 \[ \frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^5}{c^{7/2}}+\frac{2 \left (5 B e^3 \left (c d^2-3 a e^2\right ) b^5+4 B c^2 d^3 e^2 b^4-8 c e \left (16 A c^2 e d^3+B \left (11 c^2 d^4+7 a c e^2 d^2-20 a^2 e^4\right )\right ) b^3+32 c^3 d^2 \left (2 B c d^3+8 A c e d^2+17 a B e^2 d+16 a A e^3\right ) b^2-16 c^2 \left (8 A c d \left (c^2 d^4+6 a c e^2 d^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c e^2 d^2+33 a^2 e^4\right )\right ) b+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )+\left (-15 B e^5 b^6+10 B c d e^4 b^5+2 B c e^3 \left (3 c d^2+85 a e^2\right ) b^4+16 c^2 d e^2 \left (6 B c d^2+8 A c e d-7 a B e^2\right ) b^3-16 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c e^2 d^2+39 a^2 e^4\right )\right ) b^2+32 c^3 \left (4 A e \left (5 c^2 d^4+6 a c e^2 d^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c e^2 d^3+29 a^2 e^4 d\right )\right ) b-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c e^2 d^2-3 a^2 e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{c x^2+b x+a}}+\frac{2 (d+e x)^2 \left (B e \left (3 c d^2-5 a e^2\right ) b^3-4 c d \left (2 B c d^2+4 A c e d+a B e^2\right ) b^2+4 c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right ) b-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+\left (-5 B e^3 b^4+2 B c d e^2 b^3+2 c e \left (7 B c d^2+8 A c e d+19 a B e^2\right ) b^2-8 c^2 \left (2 B c d^3+6 A c e d^2+7 a B e^2 d+2 a A e^3\right ) b+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac{2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}} \]

[Out]

(2*(d + e*x)^4*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/
(5*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(d + e*x)^2*(b^3*B*e*(3*c*d^2 - 5*a*e^2) - 4*b^2*c*d*(2*B*c*d
^2 + 4*A*c*d*e + a*B*e^2) - 16*a*c^2*e*(5*a*B*d*e + 2*A*(c*d^2 + a*e^2)) + 4*b*c*(9*a*B*e*(c*d^2 + a*e^2) + 4*
A*c*d*(c*d^2 + 3*a*e^2)) + (2*b^3*B*c*d*e^2 - 5*b^4*B*e^3 + 2*b^2*c*e*(7*B*c*d^2 + 8*A*c*d*e + 19*a*B*e^2) - 8
*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 7*a*B*d*e^2 + 2*a*A*e^3) + 8*c^2*(5*a*B*e*(c*d^2 - a*e^2) + 4*A*c*d*(c*d^2 +
a*e^2)))*x))/(15*c^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (2*(4*b^4*B*c^2*d^3*e^2 + 5*b^5*B*e^3*(c*d^2
- 3*a*e^2) + 32*b^2*c^3*d^2*(2*B*c*d^3 + 8*A*c*d^2*e + 17*a*B*d*e^2 + 16*a*A*e^3) + 64*a*c^3*e*(4*A*(c*d^2 + a
*e^2)^2 + 5*a*B*d*e*(c*d^2 + 4*a*e^2)) - 8*b^3*c*e*(16*A*c^2*d^3*e + B*(11*c^2*d^4 + 7*a*c*d^2*e^2 - 20*a^2*e^
4)) - 16*b*c^2*(8*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 + 5*a^2*e^4) + a*B*e*(18*c^2*d^4 + 71*a*c*d^2*e^2 + 33*a^2*e^
4)) + (10*b^5*B*c*d*e^4 - 15*b^6*B*e^5 + 2*b^4*B*c*e^3*(3*c*d^2 + 85*a*e^2) + 16*b^3*c^2*d*e^2*(6*B*c*d^2 + 8*
A*c*d*e - 7*a*B*e^2) - 32*c^3*(8*A*c*d*(c*d^2 + a*e^2)^2 + 5*a*B*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 - 3*a^2*e^4)) -
16*b^2*c^2*e*(16*A*c*d*e*(2*c*d^2 + a*e^2) + B*(15*c^2*d^4 + 29*a*c*d^2*e^2 + 39*a^2*e^4)) + 32*b*c^3*(4*A*e*(
5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4) + B*(4*c^2*d^5 + 28*a*c*d^3*e^2 + 29*a^2*d*e^4)))*x))/(15*c^3*(b^2 - 4*a*
c)^3*Sqrt[a + b*x + c*x^2]) + (B*e^5*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(7/2)

________________________________________________________________________________________

Rubi [A]  time = 0.960573, antiderivative size = 942, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {818, 777, 621, 206} \[ \frac{B \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{c x^2+b x+a}}\right ) e^5}{c^{7/2}}+\frac{2 \left (5 B e^3 \left (c d^2-3 a e^2\right ) b^5+4 B c^2 d^3 e^2 b^4-8 c e \left (16 A c^2 e d^3+B \left (11 c^2 d^4+7 a c e^2 d^2-20 a^2 e^4\right )\right ) b^3+32 c^3 d^2 \left (2 B c d^3+8 A c e d^2+17 a B e^2 d+16 a A e^3\right ) b^2-16 c^2 \left (8 A c d \left (c^2 d^4+6 a c e^2 d^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c e^2 d^2+33 a^2 e^4\right )\right ) b+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )+\left (-15 B e^5 b^6+10 B c d e^4 b^5+2 B c e^3 \left (3 c d^2+85 a e^2\right ) b^4+16 c^2 d e^2 \left (6 B c d^2+8 A c e d-7 a B e^2\right ) b^3-16 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c e^2 d^2+39 a^2 e^4\right )\right ) b^2+32 c^3 \left (4 A e \left (5 c^2 d^4+6 a c e^2 d^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c e^2 d^3+29 a^2 e^4 d\right )\right ) b-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c e^2 d^2-3 a^2 e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{c x^2+b x+a}}+\frac{2 (d+e x)^2 \left (B e \left (3 c d^2-5 a e^2\right ) b^3-4 c d \left (2 B c d^2+4 A c e d+a B e^2\right ) b^2+4 c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right ) b-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+\left (-5 B e^3 b^4+2 B c d e^2 b^3+2 c e \left (7 B c d^2+8 A c e d+19 a B e^2\right ) b^2-8 c^2 \left (2 B c d^3+6 A c e d^2+7 a B e^2 d+2 a A e^3\right ) b+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (c x^2+b x+a\right )^{3/2}}+\frac{2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x]

[Out]

(2*(d + e*x)^4*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/
(5*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) + (2*(d + e*x)^2*(b^3*B*e*(3*c*d^2 - 5*a*e^2) - 4*b^2*c*d*(2*B*c*d
^2 + 4*A*c*d*e + a*B*e^2) - 16*a*c^2*e*(5*a*B*d*e + 2*A*(c*d^2 + a*e^2)) + 4*b*c*(9*a*B*e*(c*d^2 + a*e^2) + 4*
A*c*d*(c*d^2 + 3*a*e^2)) + (2*b^3*B*c*d*e^2 - 5*b^4*B*e^3 + 2*b^2*c*e*(7*B*c*d^2 + 8*A*c*d*e + 19*a*B*e^2) - 8
*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 7*a*B*d*e^2 + 2*a*A*e^3) + 8*c^2*(5*a*B*e*(c*d^2 - a*e^2) + 4*A*c*d*(c*d^2 +
a*e^2)))*x))/(15*c^2*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2)) + (2*(4*b^4*B*c^2*d^3*e^2 + 5*b^5*B*e^3*(c*d^2
- 3*a*e^2) + 32*b^2*c^3*d^2*(2*B*c*d^3 + 8*A*c*d^2*e + 17*a*B*d*e^2 + 16*a*A*e^3) + 64*a*c^3*e*(4*A*(c*d^2 + a
*e^2)^2 + 5*a*B*d*e*(c*d^2 + 4*a*e^2)) - 8*b^3*c*e*(16*A*c^2*d^3*e + B*(11*c^2*d^4 + 7*a*c*d^2*e^2 - 20*a^2*e^
4)) - 16*b*c^2*(8*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 + 5*a^2*e^4) + a*B*e*(18*c^2*d^4 + 71*a*c*d^2*e^2 + 33*a^2*e^
4)) + (10*b^5*B*c*d*e^4 - 15*b^6*B*e^5 + 2*b^4*B*c*e^3*(3*c*d^2 + 85*a*e^2) + 16*b^3*c^2*d*e^2*(6*B*c*d^2 + 8*
A*c*d*e - 7*a*B*e^2) - 32*c^3*(8*A*c*d*(c*d^2 + a*e^2)^2 + 5*a*B*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 - 3*a^2*e^4)) -
16*b^2*c^2*e*(16*A*c*d*e*(2*c*d^2 + a*e^2) + B*(15*c^2*d^4 + 29*a*c*d^2*e^2 + 39*a^2*e^4)) + 32*b*c^3*(4*A*e*(
5*c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4) + B*(4*c^2*d^5 + 28*a*c*d^3*e^2 + 29*a^2*d*e^4)))*x))/(15*c^3*(b^2 - 4*a*
c)^3*Sqrt[a + b*x + c*x^2]) + (B*e^5*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(7/2)

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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{(A+B x) (d+e x)^5}{\left (a+b x+c x^2\right )^{7/2}} \, dx &=\frac{2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 \int \frac{(d+e x)^3 \left (\frac{1}{2} \left (-16 A c^2 d^2-2 b B e \left (\frac{3 b d}{2}-4 a e\right )+8 b c d (B d+2 A e)-4 a c e (5 B d+4 A e)\right )+\frac{5}{2} B \left (b^2-4 a c\right ) e^2 x\right )}{\left (a+b x+c x^2\right )^{5/2}} \, dx}{5 c \left (b^2-4 a c\right )}\\ &=\frac{2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 (d+e x)^2 \left (b^3 B e \left (3 c d^2-5 a e^2\right )-4 b^2 c d \left (2 B c d^2+4 A c d e+a B e^2\right )-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+4 b c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right )+\left (2 b^3 B c d e^2-5 b^4 B e^3+2 b^2 c e \left (7 B c d^2+8 A c d e+19 a B e^2\right )-8 b c^2 \left (2 B c d^3+6 A c d^2 e+7 a B d e^2+2 a A e^3\right )+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{4 \int \frac{(d+e x) \left (\frac{1}{4} \left (-5 b^4 B d e^3+8 b^2 c d e \left (11 B c d^2+16 A c d e+6 a B e^2\right )-4 b^3 B \left (c d^2 e^2-5 a e^4\right )+16 c^2 \left (8 A \left (c d^2+a e^2\right )^2+5 a B d e \left (2 c d^2+5 a e^2\right )\right )-16 b c \left (16 A c d e \left (c d^2+a e^2\right )+B \left (4 c^2 d^4+23 a c d^2 e^2+9 a^2 e^4\right )\right )\right )+\frac{15}{4} B \left (b^2-4 a c\right )^2 e^4 x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{15 c^2 \left (b^2-4 a c\right )^2}\\ &=\frac{2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 (d+e x)^2 \left (b^3 B e \left (3 c d^2-5 a e^2\right )-4 b^2 c d \left (2 B c d^2+4 A c d e+a B e^2\right )-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+4 b c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right )+\left (2 b^3 B c d e^2-5 b^4 B e^3+2 b^2 c e \left (7 B c d^2+8 A c d e+19 a B e^2\right )-8 b c^2 \left (2 B c d^3+6 A c d^2 e+7 a B d e^2+2 a A e^3\right )+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 \left (4 b^4 B c^2 d^3 e^2+5 b^5 B e^3 \left (c d^2-3 a e^2\right )+32 b^2 c^3 d^2 \left (2 B c d^3+8 A c d^2 e+17 a B d e^2+16 a A e^3\right )+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )-8 b^3 c e \left (16 A c^2 d^3 e+B \left (11 c^2 d^4+7 a c d^2 e^2-20 a^2 e^4\right )\right )-16 b c^2 \left (8 A c d \left (c^2 d^4+6 a c d^2 e^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c d^2 e^2+33 a^2 e^4\right )\right )+\left (10 b^5 B c d e^4-15 b^6 B e^5+2 b^4 B c e^3 \left (3 c d^2+85 a e^2\right )+16 b^3 c^2 d e^2 \left (6 B c d^2+8 A c d e-7 a B e^2\right )-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c d^2 e^2-3 a^2 e^4\right )\right )-16 b^2 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c d^2 e^2+39 a^2 e^4\right )\right )+32 b c^3 \left (4 A e \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c d^3 e^2+29 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{\left (B e^5\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{c^3}\\ &=\frac{2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 (d+e x)^2 \left (b^3 B e \left (3 c d^2-5 a e^2\right )-4 b^2 c d \left (2 B c d^2+4 A c d e+a B e^2\right )-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+4 b c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right )+\left (2 b^3 B c d e^2-5 b^4 B e^3+2 b^2 c e \left (7 B c d^2+8 A c d e+19 a B e^2\right )-8 b c^2 \left (2 B c d^3+6 A c d^2 e+7 a B d e^2+2 a A e^3\right )+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 \left (4 b^4 B c^2 d^3 e^2+5 b^5 B e^3 \left (c d^2-3 a e^2\right )+32 b^2 c^3 d^2 \left (2 B c d^3+8 A c d^2 e+17 a B d e^2+16 a A e^3\right )+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )-8 b^3 c e \left (16 A c^2 d^3 e+B \left (11 c^2 d^4+7 a c d^2 e^2-20 a^2 e^4\right )\right )-16 b c^2 \left (8 A c d \left (c^2 d^4+6 a c d^2 e^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c d^2 e^2+33 a^2 e^4\right )\right )+\left (10 b^5 B c d e^4-15 b^6 B e^5+2 b^4 B c e^3 \left (3 c d^2+85 a e^2\right )+16 b^3 c^2 d e^2 \left (6 B c d^2+8 A c d e-7 a B e^2\right )-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c d^2 e^2-3 a^2 e^4\right )\right )-16 b^2 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c d^2 e^2+39 a^2 e^4\right )\right )+32 b c^3 \left (4 A e \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c d^3 e^2+29 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{\left (2 B e^5\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{c^3}\\ &=\frac{2 (d+e x)^4 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{5 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac{2 (d+e x)^2 \left (b^3 B e \left (3 c d^2-5 a e^2\right )-4 b^2 c d \left (2 B c d^2+4 A c d e+a B e^2\right )-16 a c^2 e \left (5 a B d e+2 A \left (c d^2+a e^2\right )\right )+4 b c \left (9 a B e \left (c d^2+a e^2\right )+4 A c d \left (c d^2+3 a e^2\right )\right )+\left (2 b^3 B c d e^2-5 b^4 B e^3+2 b^2 c e \left (7 B c d^2+8 A c d e+19 a B e^2\right )-8 b c^2 \left (2 B c d^3+6 A c d^2 e+7 a B d e^2+2 a A e^3\right )+8 c^2 \left (5 a B e \left (c d^2-a e^2\right )+4 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{15 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac{2 \left (4 b^4 B c^2 d^3 e^2+5 b^5 B e^3 \left (c d^2-3 a e^2\right )+32 b^2 c^3 d^2 \left (2 B c d^3+8 A c d^2 e+17 a B d e^2+16 a A e^3\right )+64 a c^3 e \left (4 A \left (c d^2+a e^2\right )^2+5 a B d e \left (c d^2+4 a e^2\right )\right )-8 b^3 c e \left (16 A c^2 d^3 e+B \left (11 c^2 d^4+7 a c d^2 e^2-20 a^2 e^4\right )\right )-16 b c^2 \left (8 A c d \left (c^2 d^4+6 a c d^2 e^2+5 a^2 e^4\right )+a B e \left (18 c^2 d^4+71 a c d^2 e^2+33 a^2 e^4\right )\right )+\left (10 b^5 B c d e^4-15 b^6 B e^5+2 b^4 B c e^3 \left (3 c d^2+85 a e^2\right )+16 b^3 c^2 d e^2 \left (6 B c d^2+8 A c d e-7 a B e^2\right )-32 c^3 \left (8 A c d \left (c d^2+a e^2\right )^2+5 a B e \left (2 c^2 d^4+5 a c d^2 e^2-3 a^2 e^4\right )\right )-16 b^2 c^2 e \left (16 A c d e \left (2 c d^2+a e^2\right )+B \left (15 c^2 d^4+29 a c d^2 e^2+39 a^2 e^4\right )\right )+32 b c^3 \left (4 A e \left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right )+B \left (4 c^2 d^5+28 a c d^3 e^2+29 a^2 d e^4\right )\right )\right ) x\right )}{15 c^3 \left (b^2-4 a c\right )^3 \sqrt{a+b x+c x^2}}+\frac{B e^5 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 9.7022, size = 1608, normalized size = 1.71 \[ \frac{\frac{2 \sqrt{c} \left (A \left (\left (3 d^5+25 e x d^4+150 e^2 x^2 d^3-150 e^3 x^3 d^2-25 e^4 x^4 d-3 e^5 x^5\right ) b^5+10 (a e-c d x) \left (d^4+20 e x d^3-90 e^2 x^2 d^2+20 e^3 x^3 d+e^4 x^4\right ) b^4+40 (d-e x) \left (2 a^2 \left (d^2-14 e x d+e^2 x^2\right ) e^2+2 c^2 d^2 x^2 \left (d^2-14 e x d+e^2 x^2\right )-a c (d-e x)^2 \left (d^2+18 e x d+e^2 x^2\right )\right ) b^3+80 \left (-2 a^3 \left (3 d^2-10 e x d+3 e^2 x^2\right ) e^3-3 a^2 c \left (d^4-10 e x d^3+10 e^2 x^2 d^2-10 e^3 x^3 d+e^4 x^4\right ) e+2 c^3 d^3 x^3 \left (3 d^2-10 e x d+3 e^2 x^2\right )+3 a c^2 d x \left (d^4-10 e x d^3+10 e^2 x^2 d^2-10 e^3 x^3 d+e^4 x^4\right )\right ) b^2+80 (d-e x) \left (8 a^4 e^4+4 a^3 c \left (3 d^2-2 e x d+3 e^2 x^2\right ) e^2+8 c^4 d^4 x^4+3 a^2 c^2 (d-e x)^4+4 a c^3 d^2 x^2 \left (3 d^2-2 e x d+3 e^2 x^2\right )\right ) b+32 \left (-8 a^5 e^5-20 a^4 c \left (d^2+e^2 x^2\right ) e^3-5 a^3 c^2 \left (3 d^4+10 e^2 x^2 d^2+3 e^4 x^4\right ) e+8 c^5 d^5 x^5+20 a c^4 d^3 x^3 \left (d^2+e^2 x^2\right )+5 a^2 c^3 d x \left (3 d^4+10 e^2 x^2 d^2+3 e^4 x^4\right )\right )\right ) c^3+B \left (-16 c^2 e^4 (80 c d-33 b e+30 c e x) a^5-80 c e^2 \left (2 \left (4 d^3+20 e^2 x^2 d+7 e^3 x^3\right ) c^3+b e \left (-16 d^2+40 e x d-3 e^2 x^2\right ) c^2-21 b^2 e^3 x c+2 b^3 e^3\right ) a^4-\left (32 \left (3 d^5+50 e^2 x^2 d^3+75 e^4 x^4 d+23 e^5 x^5\right ) c^5+80 b e \left (-6 d^4+20 e x d^3-40 e^2 x^2 d^2+60 e^3 x^3 d+5 e^4 x^4\right ) c^4+80 b^2 e^2 \left (6 d^3-40 e x d^2+30 e^2 x^2 d-27 e^3 x^3\right ) c^3-1400 b^3 e^5 x^2 c^2+490 b^4 e^5 x c-15 b^5 e^5\right ) a^3+\left (45 e^5 x b^6-465 c e^5 x^2 b^5-150 c^2 e^5 x^3 b^4+40 c^3 e \left (d^4-30 e x d^3+60 e^2 x^2 d^2-10 e^3 x^3 d+30 e^4 x^4\right ) b^3-48 c^4 \left (d^5-25 e x d^4+50 e^2 x^2 d^3-100 e^3 x^3 d^2+25 e^4 x^4 d-19 e^5 x^5\right ) b^2-240 c^5 d x \left (d^4-5 e x d^3+10 e^2 x^2 d^2-10 e^3 x^3 d+5 e^4 x^4\right ) b+160 c^6 d^2 e x^3 \left (5 d^2+6 e^2 x^2\right )\right ) a^2+\left (45 e^5 x^2 b^7-100 c e^5 x^3 b^6-375 c^2 e^5 x^4 b^5+2 c^3 \left (d^5+50 e x d^4-450 e^2 x^2 d^3+200 e^3 x^3 d^2+25 e^4 x^4 d-129 e^5 x^5\right ) b^4+40 c^4 d x \left (-3 d^4+25 e x d^3-50 e^2 x^2 d^2+30 e^3 x^3 d+5 e^4 x^4\right ) b^3+240 c^5 d^2 x^2 \left (-2 d^3+5 e x d^2-10 e^2 x^2 d+2 e^3 x^3\right ) b^2-160 c^6 d^3 x^3 \left (2 d^2-5 e x d+6 e^2 x^2\right ) b+320 c^7 d^4 e x^5\right ) a+b x \left (15 e^5 x^2 b^7+35 c e^5 x^3 b^6+23 c^2 e^5 x^4 b^5+5 c^3 d \left (d^4+15 e x d^3-30 e^2 x^2 d^2-10 e^3 x^3 d-3 e^4 x^4\right ) b^4-10 c^4 d^2 x \left (4 d^3-45 e x d^2+20 e^2 x^2 d+2 e^3 x^3\right ) b^3-40 c^5 d^3 x^2 \left (6 d^2-15 e x d+2 e^2 x^2\right ) b^2+80 c^6 d^4 x^3 (3 e x-4 d) b-128 c^7 d^5 x^4\right )\right )\right )}{(a+x (b+c x))^{5/2}}-15 B \left (b^2-4 a c\right )^3 e^5 \log \left (b+2 c x+2 \sqrt{c} \sqrt{a+x (b+c x)}\right )}{15 c^{7/2} \left (4 a c-b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x + c*x^2)^(7/2),x]

[Out]

((2*Sqrt[c]*(A*c^3*(10*b^4*(a*e - c*d*x)*(d^4 + 20*d^3*e*x - 90*d^2*e^2*x^2 + 20*d*e^3*x^3 + e^4*x^4) + b^5*(3
*d^5 + 25*d^4*e*x + 150*d^3*e^2*x^2 - 150*d^2*e^3*x^3 - 25*d*e^4*x^4 - 3*e^5*x^5) + 40*b^3*(d - e*x)*(2*a^2*e^
2*(d^2 - 14*d*e*x + e^2*x^2) + 2*c^2*d^2*x^2*(d^2 - 14*d*e*x + e^2*x^2) - a*c*(d - e*x)^2*(d^2 + 18*d*e*x + e^
2*x^2)) + 80*b*(d - e*x)*(8*a^4*e^4 + 8*c^4*d^4*x^4 + 3*a^2*c^2*(d - e*x)^4 + 4*a^3*c*e^2*(3*d^2 - 2*d*e*x + 3
*e^2*x^2) + 4*a*c^3*d^2*x^2*(3*d^2 - 2*d*e*x + 3*e^2*x^2)) + 80*b^2*(-2*a^3*e^3*(3*d^2 - 10*d*e*x + 3*e^2*x^2)
+ 2*c^3*d^3*x^3*(3*d^2 - 10*d*e*x + 3*e^2*x^2) - 3*a^2*c*e*(d^4 - 10*d^3*e*x + 10*d^2*e^2*x^2 - 10*d*e^3*x^3
+ e^4*x^4) + 3*a*c^2*d*x*(d^4 - 10*d^3*e*x + 10*d^2*e^2*x^2 - 10*d*e^3*x^3 + e^4*x^4)) + 32*(-8*a^5*e^5 + 8*c^
5*d^5*x^5 - 20*a^4*c*e^3*(d^2 + e^2*x^2) + 20*a*c^4*d^3*x^3*(d^2 + e^2*x^2) - 5*a^3*c^2*e*(3*d^4 + 10*d^2*e^2*
x^2 + 3*e^4*x^4) + 5*a^2*c^3*d*x*(3*d^4 + 10*d^2*e^2*x^2 + 3*e^4*x^4))) + B*(-16*a^5*c^2*e^4*(80*c*d - 33*b*e
+ 30*c*e*x) - 80*a^4*c*e^2*(2*b^3*e^3 - 21*b^2*c*e^3*x + b*c^2*e*(-16*d^2 + 40*d*e*x - 3*e^2*x^2) + 2*c^3*(4*d
^3 + 20*d*e^2*x^2 + 7*e^3*x^3)) + b*x*(15*b^7*e^5*x^2 + 35*b^6*c*e^5*x^3 - 128*c^7*d^5*x^4 + 23*b^5*c^2*e^5*x^
4 + 80*b*c^6*d^4*x^3*(-4*d + 3*e*x) - 40*b^2*c^5*d^3*x^2*(6*d^2 - 15*d*e*x + 2*e^2*x^2) - 10*b^3*c^4*d^2*x*(4*
d^3 - 45*d^2*e*x + 20*d*e^2*x^2 + 2*e^3*x^3) + 5*b^4*c^3*d*(d^4 + 15*d^3*e*x - 30*d^2*e^2*x^2 - 10*d*e^3*x^3 -
3*e^4*x^4)) + a*(45*b^7*e^5*x^2 - 100*b^6*c*e^5*x^3 - 375*b^5*c^2*e^5*x^4 + 320*c^7*d^4*e*x^5 - 160*b*c^6*d^3
*x^3*(2*d^2 - 5*d*e*x + 6*e^2*x^2) + 240*b^2*c^5*d^2*x^2*(-2*d^3 + 5*d^2*e*x - 10*d*e^2*x^2 + 2*e^3*x^3) + 40*
b^3*c^4*d*x*(-3*d^4 + 25*d^3*e*x - 50*d^2*e^2*x^2 + 30*d*e^3*x^3 + 5*e^4*x^4) + 2*b^4*c^3*(d^5 + 50*d^4*e*x -
450*d^3*e^2*x^2 + 200*d^2*e^3*x^3 + 25*d*e^4*x^4 - 129*e^5*x^5)) + a^2*(45*b^6*e^5*x - 465*b^5*c*e^5*x^2 - 150
*b^4*c^2*e^5*x^3 + 160*c^6*d^2*e*x^3*(5*d^2 + 6*e^2*x^2) - 240*b*c^5*d*x*(d^4 - 5*d^3*e*x + 10*d^2*e^2*x^2 - 1
0*d*e^3*x^3 + 5*e^4*x^4) + 40*b^3*c^3*e*(d^4 - 30*d^3*e*x + 60*d^2*e^2*x^2 - 10*d*e^3*x^3 + 30*e^4*x^4) - 48*b
^2*c^4*(d^5 - 25*d^4*e*x + 50*d^3*e^2*x^2 - 100*d^2*e^3*x^3 + 25*d*e^4*x^4 - 19*e^5*x^5)) - a^3*(-15*b^5*e^5 +
490*b^4*c*e^5*x - 1400*b^3*c^2*e^5*x^2 + 80*b^2*c^3*e^2*(6*d^3 - 40*d^2*e*x + 30*d*e^2*x^2 - 27*e^3*x^3) + 80
*b*c^4*e*(-6*d^4 + 20*d^3*e*x - 40*d^2*e^2*x^2 + 60*d*e^3*x^3 + 5*e^4*x^4) + 32*c^5*(3*d^5 + 50*d^3*e^2*x^2 +
75*d*e^4*x^4 + 23*e^5*x^5)))))/(a + x*(b + c*x))^(5/2) - 15*B*(b^2 - 4*a*c)^3*e^5*Log[b + 2*c*x + 2*Sqrt[c]*Sq
rt[a + x*(b + c*x)]])/(15*c^(7/2)*(-b^2 + 4*a*c)^3)

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Maple [B]  time = 0.021, size = 7765, normalized size = 8.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x)

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),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((B*x+A)*(e*x+d)**5/(c*x**2+b*x+a)**(7/2),x)

[Out]

Timed out

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Giac [B]  time = 1.45314, size = 3409, normalized size = 3.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")

[Out]

-B*e^5*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2) + 2/15*((((((128*B*b*c^7*d^5 - 256
*A*c^8*d^5 - 240*B*b^2*c^6*d^4*e - 320*B*a*c^7*d^4*e + 640*A*b*c^7*d^4*e + 80*B*b^3*c^5*d^3*e^2 + 960*B*a*b*c^
6*d^3*e^2 - 480*A*b^2*c^6*d^3*e^2 - 640*A*a*c^7*d^3*e^2 + 20*B*b^4*c^4*d^2*e^3 - 480*B*a*b^2*c^5*d^2*e^3 + 80*
A*b^3*c^5*d^2*e^3 - 960*B*a^2*c^6*d^2*e^3 + 960*A*a*b*c^6*d^2*e^3 + 15*B*b^5*c^3*d*e^4 - 200*B*a*b^3*c^4*d*e^4
+ 10*A*b^4*c^4*d*e^4 + 1200*B*a^2*b*c^5*d*e^4 - 240*A*a*b^2*c^5*d*e^4 - 480*A*a^2*c^6*d*e^4 - 23*B*b^6*c^2*e^
5 + 258*B*a*b^4*c^3*e^5 + 3*A*b^5*c^3*e^5 - 912*B*a^2*b^2*c^4*e^5 - 40*A*a*b^3*c^4*e^5 + 736*B*a^3*c^5*e^5 + 2
40*A*a^2*b*c^5*e^5)*x/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6) + 5*(64*B*b^2*c^6*d^5 - 128*A*b*c
^7*d^5 - 120*B*b^3*c^5*d^4*e - 160*B*a*b*c^6*d^4*e + 320*A*b^2*c^6*d^4*e + 40*B*b^4*c^4*d^3*e^2 + 480*B*a*b^2*
c^5*d^3*e^2 - 240*A*b^3*c^5*d^3*e^2 - 320*A*a*b*c^6*d^3*e^2 + 10*B*b^5*c^3*d^2*e^3 - 240*B*a*b^3*c^4*d^2*e^3 +
40*A*b^4*c^4*d^2*e^3 - 480*B*a^2*b*c^5*d^2*e^3 + 480*A*a*b^2*c^5*d^2*e^3 - 10*B*a*b^4*c^3*d*e^4 + 5*A*b^5*c^3
*d*e^4 + 240*B*a^2*b^2*c^4*d*e^4 - 120*A*a*b^3*c^4*d*e^4 + 480*B*a^3*c^5*d*e^4 - 240*A*a^2*b*c^5*d*e^4 - 7*B*b
^7*c*e^5 + 75*B*a*b^5*c^2*e^5 - 240*B*a^2*b^3*c^3*e^5 - 2*A*a*b^4*c^3*e^5 + 80*B*a^3*b*c^4*e^5 + 48*A*a^2*b^2*
c^4*e^5 + 96*A*a^3*c^5*e^5)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x + 5*(48*B*b^3*c^5*d^5 +
64*B*a*b*c^6*d^5 - 96*A*b^2*c^6*d^5 - 128*A*a*c^7*d^5 - 90*B*b^4*c^4*d^4*e - 240*B*a*b^2*c^5*d^4*e + 240*A*b^3
*c^5*d^4*e - 160*B*a^2*c^6*d^4*e + 320*A*a*b*c^6*d^4*e + 30*B*b^5*c^3*d^3*e^2 + 400*B*a*b^3*c^4*d^3*e^2 - 180*
A*b^4*c^4*d^3*e^2 + 480*B*a^2*b*c^5*d^3*e^2 - 480*A*a*b^2*c^5*d^3*e^2 - 320*A*a^2*c^6*d^3*e^2 - 80*B*a*b^4*c^3
*d^2*e^3 + 30*A*b^5*c^3*d^2*e^3 - 960*B*a^2*b^2*c^4*d^2*e^3 + 400*A*a*b^3*c^4*d^2*e^3 + 480*A*a^2*b*c^5*d^2*e^
3 + 80*B*a^2*b^3*c^3*d*e^4 - 40*A*a*b^4*c^3*d*e^4 + 960*B*a^3*b*c^4*d*e^4 - 480*A*a^2*b^2*c^4*d*e^4 - 3*B*b^8*
e^5 + 20*B*a*b^6*c*e^5 + 30*B*a^2*b^4*c^2*e^5 - 432*B*a^3*b^2*c^3*e^5 + 16*A*a^2*b^3*c^3*e^5 + 224*B*a^4*c^4*e
^5 + 192*A*a^3*b*c^4*e^5)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x + 5*(8*B*b^4*c^4*d^5 + 96*
B*a*b^2*c^5*d^5 - 16*A*b^3*c^5*d^5 - 192*A*a*b*c^6*d^5 - 15*B*b^5*c^3*d^4*e - 200*B*a*b^3*c^4*d^4*e + 40*A*b^4
*c^4*d^4*e - 240*B*a^2*b*c^5*d^4*e + 480*A*a*b^2*c^5*d^4*e + 180*B*a*b^4*c^3*d^3*e^2 - 30*A*b^5*c^3*d^3*e^2 +
480*B*a^2*b^2*c^4*d^3*e^2 - 400*A*a*b^3*c^4*d^3*e^2 + 320*B*a^3*c^5*d^3*e^2 - 480*A*a^2*b*c^5*d^3*e^2 - 480*B*
a^2*b^3*c^3*d^2*e^3 + 180*A*a*b^4*c^3*d^2*e^3 - 640*B*a^3*b*c^4*d^2*e^3 + 480*A*a^2*b^2*c^4*d^2*e^3 + 320*A*a^
3*c^5*d^2*e^3 + 480*B*a^3*b^2*c^3*d*e^4 - 240*A*a^2*b^3*c^3*d*e^4 + 640*B*a^4*c^4*d*e^4 - 320*A*a^3*b*c^4*d*e^
4 - 9*B*a*b^7*e^5 + 93*B*a^2*b^5*c*e^5 - 280*B*a^3*b^3*c^2*e^5 - 48*B*a^4*b*c^3*e^5 + 96*A*a^3*b^2*c^3*e^5 + 1
28*A*a^4*c^4*e^5)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x - 5*(B*b^5*c^3*d^5 - 24*B*a*b^3*c^
4*d^5 - 2*A*b^4*c^4*d^5 - 48*B*a^2*b*c^5*d^5 + 48*A*a*b^2*c^5*d^5 + 96*A*a^2*c^6*d^5 + 20*B*a*b^4*c^3*d^4*e +
5*A*b^5*c^3*d^4*e + 240*B*a^2*b^2*c^4*d^4*e - 120*A*a*b^3*c^4*d^4*e - 240*A*a^2*b*c^5*d^4*e - 240*B*a^2*b^3*c^
3*d^3*e^2 + 40*A*a*b^4*c^3*d^3*e^2 - 320*B*a^3*b*c^4*d^3*e^2 + 480*A*a^2*b^2*c^4*d^3*e^2 + 640*B*a^3*b^2*c^3*d
^2*e^3 - 240*A*a^2*b^3*c^3*d^2*e^3 - 320*A*a^3*b*c^4*d^2*e^3 - 640*B*a^4*b*c^3*d*e^4 + 320*A*a^3*b^2*c^3*d*e^4
+ 9*B*a^2*b^6*e^5 - 98*B*a^3*b^4*c*e^5 + 336*B*a^4*b^2*c^2*e^5 - 96*B*a^5*c^3*e^5 - 128*A*a^4*b*c^3*e^5)/(b^6
*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*x - (2*B*a*b^4*c^3*d^5 + 3*A*b^5*c^3*d^5 - 48*B*a^2*b^2*c^
4*d^5 - 40*A*a*b^3*c^4*d^5 - 96*B*a^3*c^5*d^5 + 240*A*a^2*b*c^5*d^5 + 40*B*a^2*b^3*c^3*d^4*e + 10*A*a*b^4*c^3*
d^4*e + 480*B*a^3*b*c^4*d^4*e - 240*A*a^2*b^2*c^4*d^4*e - 480*A*a^3*c^5*d^4*e - 480*B*a^3*b^2*c^3*d^3*e^2 + 80
*A*a^2*b^3*c^3*d^3*e^2 - 640*B*a^4*c^4*d^3*e^2 + 960*A*a^3*b*c^4*d^3*e^2 + 1280*B*a^4*b*c^3*d^2*e^3 - 480*A*a^
3*b^2*c^3*d^2*e^3 - 640*A*a^4*c^4*d^2*e^3 - 1280*B*a^5*c^3*d*e^4 + 640*A*a^4*b*c^3*d*e^4 + 15*B*a^3*b^5*e^5 -
160*B*a^4*b^3*c*e^5 + 528*B*a^5*b*c^2*e^5 - 256*A*a^5*c^3*e^5)/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a
^3*c^6))/(c*x^2 + b*x + a)^(5/2)