3.1180 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^6} \, dx\)
Optimal. Leaf size=269 \[ \frac{3 b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{128 d^3 (d+e x)^2 (c d-b e)^3}-\frac{3 b^4 (A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{7/2} (c d-b e)^{7/2}}-\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{16 d^2 (d+e x)^4 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{5 d (d+e x)^5 (c d-b e)} \]
[Out]
(3*b^2*(b*B*d - 2*A*c*d + A*b*e)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^3*(c*d - b*e)^3*(d + e*x)^2
) - ((b*B*d - 2*A*c*d + A*b*e)*(b*d + (2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(16*d^2*(c*d - b*e)^2*(d + e*x)^4)
+ ((B*d - A*e)*(b*x + c*x^2)^(5/2))/(5*d*(c*d - b*e)*(d + e*x)^5) - (3*b^4*(b*B*d - 2*A*c*d + A*b*e)*ArcTanh[
(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(256*d^(7/2)*(c*d - b*e)^(7/2))
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Rubi [A] time = 0.231235, antiderivative size = 269, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used =
{806, 720, 724, 206} \[ \frac{3 b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{128 d^3 (d+e x)^2 (c d-b e)^3}-\frac{3 b^4 (A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{7/2} (c d-b e)^{7/2}}-\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{16 d^2 (d+e x)^4 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{5 d (d+e x)^5 (c d-b e)} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^6,x]
[Out]
(3*b^2*(b*B*d - 2*A*c*d + A*b*e)*(b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^3*(c*d - b*e)^3*(d + e*x)^2
) - ((b*B*d - 2*A*c*d + A*b*e)*(b*d + (2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(16*d^2*(c*d - b*e)^2*(d + e*x)^4)
+ ((B*d - A*e)*(b*x + c*x^2)^(5/2))/(5*d*(c*d - b*e)*(d + e*x)^5) - (3*b^4*(b*B*d - 2*A*c*d + A*b*e)*ArcTanh[
(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(256*d^(7/2)*(c*d - b*e)^(7/2))
Rule 806
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]
Rule 720
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 0]
Rule 724
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 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) \left (b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx &=\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{5 d (c d-b e) (d+e x)^5}-\frac{(b B d-2 A c d+A b e) \int \frac{\left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{2 d (c d-b e)}\\ &=-\frac{(b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{16 d^2 (c d-b e)^2 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{5 d (c d-b e) (d+e x)^5}+\frac{\left (3 b^2 (b B d-2 A c d+A b e)\right ) \int \frac{\sqrt{b x+c x^2}}{(d+e x)^3} \, dx}{32 d^2 (c d-b e)^2}\\ &=\frac{3 b^2 (b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{128 d^3 (c d-b e)^3 (d+e x)^2}-\frac{(b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{16 d^2 (c d-b e)^2 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{5 d (c d-b e) (d+e x)^5}-\frac{\left (3 b^4 (b B d-2 A c d+A b e)\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{256 d^3 (c d-b e)^3}\\ &=\frac{3 b^2 (b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{128 d^3 (c d-b e)^3 (d+e x)^2}-\frac{(b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{16 d^2 (c d-b e)^2 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{5 d (c d-b e) (d+e x)^5}+\frac{\left (3 b^4 (b B d-2 A c d+A b e)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{128 d^3 (c d-b e)^3}\\ &=\frac{3 b^2 (b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{128 d^3 (c d-b e)^3 (d+e x)^2}-\frac{(b B d-2 A c d+A b e) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{16 d^2 (c d-b e)^2 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{5 d (c d-b e) (d+e x)^5}-\frac{3 b^4 (b B d-2 A c d+A b e) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{256 d^{7/2} (c d-b e)^{7/2}}\\ \end{align*}
Mathematica [A] time = 1.15917, size = 268, normalized size = 1. \[ \frac{(x (b+c x))^{3/2} \left (\frac{5 (A b e-2 A c d+b B d) \left (\frac{b^2 \sqrt{x} \sqrt{b+c x} (5 b d+3 b e x+2 c d x)}{8 d^2 (d+e x)^2 (b e-c d)}+\frac{3 b^4 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{8 d^{5/2} (b e-c d)^{3/2}}-\frac{2 x^{3/2} (b+c x)^{5/2}}{(d+e x)^4}+\frac{b \sqrt{x} (b+c x)^{5/2}}{(d+e x)^3 (c d-b e)}\right )}{8 (b+c x)^{3/2} (b e-c d)}+\frac{2 x^{5/2} (b+c x) (A e-B d)}{(d+e x)^5}\right )}{10 d x^{3/2} (b e-c d)} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^6,x]
[Out]
((x*(b + c*x))^(3/2)*((2*(-(B*d) + A*e)*x^(5/2)*(b + c*x))/(d + e*x)^5 + (5*(b*B*d - 2*A*c*d + A*b*e)*((-2*x^(
3/2)*(b + c*x)^(5/2))/(d + e*x)^4 + (b*Sqrt[x]*(b + c*x)^(5/2))/((c*d - b*e)*(d + e*x)^3) + (b^2*Sqrt[x]*Sqrt[
b + c*x]*(5*b*d + 2*c*d*x + 3*b*e*x))/(8*d^2*(-(c*d) + b*e)*(d + e*x)^2) + (3*b^4*ArcTan[(Sqrt[-(c*d) + b*e]*S
qrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(8*d^(5/2)*(-(c*d) + b*e)^(3/2))))/(8*(-(c*d) + b*e)*(b + c*x)^(3/2))))/(10*
d*(-(c*d) + b*e)*x^(3/2))
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Maple [B] time = 0.026, size = 22107, normalized size = 82.2 \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)*(c*x^2+b*x)^(3/2)/(e*x+d)^6,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)*(c*x^2+b*x)^(3/2)/(e*x+d)^6,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.8338, size = 4832, normalized size = 17.96 \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)*(c*x^2+b*x)^(3/2)/(e*x+d)^6,x, algorithm="fricas")
[Out]
[-1/1280*(15*(A*b^5*d^5*e + (B*b^5 - 2*A*b^4*c)*d^6 + (A*b^5*e^6 + (B*b^5 - 2*A*b^4*c)*d*e^5)*x^5 + 5*(A*b^5*d
*e^5 + (B*b^5 - 2*A*b^4*c)*d^2*e^4)*x^4 + 10*(A*b^5*d^2*e^4 + (B*b^5 - 2*A*b^4*c)*d^3*e^3)*x^3 + 10*(A*b^5*d^3
*e^3 + (B*b^5 - 2*A*b^4*c)*d^4*e^2)*x^2 + 5*(A*b^5*d^4*e^2 + (B*b^5 - 2*A*b^4*c)*d^5*e)*x)*sqrt(c*d^2 - b*d*e)
*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(15*A*b^5*d^5*e^2 - 15*(
B*b^4*c - 2*A*b^3*c^2)*d^7 + 15*(B*b^5 - 3*A*b^4*c)*d^6*e - (128*B*c^5*d^7 + 15*A*b^5*d*e^6 - 16*(29*B*b*c^4 -
2*A*c^5)*d^6*e + 8*(73*B*b^2*c^3 - 12*A*b*c^4)*d^5*e^2 - 2*(129*B*b^3*c^2 - 38*A*b^2*c^3)*d^4*e^3 - (5*B*b^4*
c - 8*A*b^3*c^2)*d^3*e^4 + 5*(3*B*b^5 - 7*A*b^4*c)*d^2*e^5)*x^4 - 2*(35*A*b^5*d^2*e^5 + 8*(11*B*b*c^4 + 10*A*c
^5)*d^7 - 8*(43*B*b^2*c^3 + 31*A*b*c^4)*d^6*e + (489*B*b^3*c^2 + 214*A*b^2*c^3)*d^5*e^2 - (268*B*b^4*c - A*b^3
*c^2)*d^4*e^3 + (35*B*b^5 - 82*A*b^4*c)*d^3*e^4)*x^3 - 2*(64*A*b^5*d^3*e^4 + 4*(B*b^2*c^3 + 30*A*b*c^4)*d^7 -
(27*B*b^3*c^2 + 454*A*b^2*c^3)*d^6*e + 3*(29*B*b^4*c + 189*A*b^3*c^2)*d^5*e^2 - (64*B*b^5 + 297*A*b^4*c)*d^4*e
^3)*x^2 + 10*(7*A*b^5*d^4*e^3 + (B*b^3*c^2 - 2*A*b^2*c^3)*d^7 - (8*B*b^4*c - 17*A*b^3*c^2)*d^6*e + (7*B*b^5 -
22*A*b^4*c)*d^5*e^2)*x)*sqrt(c*x^2 + b*x))/(c^4*d^13 - 4*b*c^3*d^12*e + 6*b^2*c^2*d^11*e^2 - 4*b^3*c*d^10*e^3
+ b^4*d^9*e^4 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 + 6*b^2*c^2*d^6*e^7 - 4*b^3*c*d^5*e^8 + b^4*d^4*e^9)*x^5 + 5*(c
^4*d^9*e^4 - 4*b*c^3*d^8*e^5 + 6*b^2*c^2*d^7*e^6 - 4*b^3*c*d^6*e^7 + b^4*d^5*e^8)*x^4 + 10*(c^4*d^10*e^3 - 4*b
*c^3*d^9*e^4 + 6*b^2*c^2*d^8*e^5 - 4*b^3*c*d^7*e^6 + b^4*d^6*e^7)*x^3 + 10*(c^4*d^11*e^2 - 4*b*c^3*d^10*e^3 +
6*b^2*c^2*d^9*e^4 - 4*b^3*c*d^8*e^5 + b^4*d^7*e^6)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*d^11*e^2 + 6*b^2*c^2*d^10*e^3
- 4*b^3*c*d^9*e^4 + b^4*d^8*e^5)*x), -1/640*(15*(A*b^5*d^5*e + (B*b^5 - 2*A*b^4*c)*d^6 + (A*b^5*e^6 + (B*b^5
- 2*A*b^4*c)*d*e^5)*x^5 + 5*(A*b^5*d*e^5 + (B*b^5 - 2*A*b^4*c)*d^2*e^4)*x^4 + 10*(A*b^5*d^2*e^4 + (B*b^5 - 2*A
*b^4*c)*d^3*e^3)*x^3 + 10*(A*b^5*d^3*e^3 + (B*b^5 - 2*A*b^4*c)*d^4*e^2)*x^2 + 5*(A*b^5*d^4*e^2 + (B*b^5 - 2*A*
b^4*c)*d^5*e)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + (15*A*
b^5*d^5*e^2 - 15*(B*b^4*c - 2*A*b^3*c^2)*d^7 + 15*(B*b^5 - 3*A*b^4*c)*d^6*e - (128*B*c^5*d^7 + 15*A*b^5*d*e^6
- 16*(29*B*b*c^4 - 2*A*c^5)*d^6*e + 8*(73*B*b^2*c^3 - 12*A*b*c^4)*d^5*e^2 - 2*(129*B*b^3*c^2 - 38*A*b^2*c^3)*d
^4*e^3 - (5*B*b^4*c - 8*A*b^3*c^2)*d^3*e^4 + 5*(3*B*b^5 - 7*A*b^4*c)*d^2*e^5)*x^4 - 2*(35*A*b^5*d^2*e^5 + 8*(1
1*B*b*c^4 + 10*A*c^5)*d^7 - 8*(43*B*b^2*c^3 + 31*A*b*c^4)*d^6*e + (489*B*b^3*c^2 + 214*A*b^2*c^3)*d^5*e^2 - (2
68*B*b^4*c - A*b^3*c^2)*d^4*e^3 + (35*B*b^5 - 82*A*b^4*c)*d^3*e^4)*x^3 - 2*(64*A*b^5*d^3*e^4 + 4*(B*b^2*c^3 +
30*A*b*c^4)*d^7 - (27*B*b^3*c^2 + 454*A*b^2*c^3)*d^6*e + 3*(29*B*b^4*c + 189*A*b^3*c^2)*d^5*e^2 - (64*B*b^5 +
297*A*b^4*c)*d^4*e^3)*x^2 + 10*(7*A*b^5*d^4*e^3 + (B*b^3*c^2 - 2*A*b^2*c^3)*d^7 - (8*B*b^4*c - 17*A*b^3*c^2)*d
^6*e + (7*B*b^5 - 22*A*b^4*c)*d^5*e^2)*x)*sqrt(c*x^2 + b*x))/(c^4*d^13 - 4*b*c^3*d^12*e + 6*b^2*c^2*d^11*e^2 -
4*b^3*c*d^10*e^3 + b^4*d^9*e^4 + (c^4*d^8*e^5 - 4*b*c^3*d^7*e^6 + 6*b^2*c^2*d^6*e^7 - 4*b^3*c*d^5*e^8 + b^4*d
^4*e^9)*x^5 + 5*(c^4*d^9*e^4 - 4*b*c^3*d^8*e^5 + 6*b^2*c^2*d^7*e^6 - 4*b^3*c*d^6*e^7 + b^4*d^5*e^8)*x^4 + 10*(
c^4*d^10*e^3 - 4*b*c^3*d^9*e^4 + 6*b^2*c^2*d^8*e^5 - 4*b^3*c*d^7*e^6 + b^4*d^6*e^7)*x^3 + 10*(c^4*d^11*e^2 - 4
*b*c^3*d^10*e^3 + 6*b^2*c^2*d^9*e^4 - 4*b^3*c*d^8*e^5 + b^4*d^7*e^6)*x^2 + 5*(c^4*d^12*e - 4*b*c^3*d^11*e^2 +
6*b^2*c^2*d^10*e^3 - 4*b^3*c*d^9*e^4 + b^4*d^8*e^5)*x)]
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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)*(c*x**2+b*x)**(3/2)/(e*x+d)**6,x)
[Out]
Timed out
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Giac [B] time = 1.71517, size = 5689, normalized size = 21.15 \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)*(c*x^2+b*x)^(3/2)/(e*x+d)^6,x, algorithm="giac")
[Out]
-3/128*(B*b^5*d - 2*A*b^4*c*d + A*b^5*e)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 +
b*d*e))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*sqrt(-c*d^2 + b*d*e)) + 1/640*(10240*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^6*B*c^(13/2)*d^9*e + 4096*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*c^7*d^10 + 10240*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^7*B*c^6*d^8*e^2 + 9728*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^6*d^9*e + 1024*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^5*A*c^7*d^9*e + 10240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b*c^(13/2)*d^10 + 5120*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^(11/2)*d^7*e^3 - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b*c^(11/2)*d^8
*e^2 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*c^(13/2)*d^8*e^2 - 11520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b
^2*c^(11/2)*d^9*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b*c^(13/2)*d^9*e + 10240*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^3*B*b^2*c^6*d^10 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*c^5*d^6*e^4 - 21760*(sqrt(c)*x - sqrt(c*x^
2 + b*x))^7*B*b*c^5*d^7*e^3 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*c^6*d^7*e^3 - 38144*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^5*B*b^2*c^5*d^8*e^2 + 3072*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b*c^6*d^8*e^2 - 21760*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^3*B*b^3*c^5*d^9*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^6*d^9*e + 5120*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^2*B*b^3*c^(11/2)*d^10 - 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b*c^(9/2)*d^6*e^4 +
1280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*c^(11/2)*d^6*e^4 - 19200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^2*c^(9
/2)*d^7*e^3 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b*c^(11/2)*d^7*e^3 - 19200*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^4*B*b^3*c^(9/2)*d^8*e^2 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^(11/2)*d^8*e^2 - 12800*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^2*B*b^4*c^(9/2)*d^9*e + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^3*c^(11/2)*d^9*e + 12
80*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^4*c^5*d^10 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b*c^4*d^5*e^5 + 6
400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^2*c^4*d^6*e^4 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b*c^5*d^6*e
^4 + 19200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*c^4*d^7*e^3 - 7936*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*
c^5*d^7*e^3 + 6400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*c^4*d^8*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
3*A*b^3*c^5*d^8*e^2 - 3360*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^5*c^4*d^9*e + 320*(sqrt(c)*x - sqrt(c*x^2 + b*x
))*A*b^4*c^5*d^9*e + 128*B*b^5*c^(9/2)*d^10 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^2*c^(7/2)*d^5*e^5 - 3
840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b*c^(9/2)*d^5*e^5 + 25600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^3*c^(7
/2)*d^6*e^4 - 6400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^2*c^(9/2)*d^6*e^4 + 25600*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^4*B*b^4*c^(7/2)*d^7*e^3 - 6400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*c^(9/2)*d^7*e^3 + 8240*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^2*B*b^5*c^(7/2)*d^8*e^2 - 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^4*c^(9/2)*d^8*e^2 -
336*B*b^6*c^(7/2)*d^9*e + 32*A*b^5*c^(9/2)*d^9*e + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^2*c^3*d^4*e^6 +
8960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^3*c^3*d^5*e^5 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^2*c^4*d^
5*e^5 + 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^4*c^3*d^6*e^4 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b
^3*c^4*d^6*e^4 + 9120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^5*c^3*d^7*e^3 - 1600*(sqrt(c)*x - sqrt(c*x^2 + b*x
))^3*A*b^4*c^4*d^7*e^3 + 2480*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^6*c^3*d^8*e^2 - 640*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*A*b^5*c^4*d^8*e^2 + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^3*c^(5/2)*d^4*e^6 + 3840*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^8*A*b^2*c^(7/2)*d^4*e^6 + 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^3*c^(7/2)*d^5*e^5 + 80*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^5*c^(5/2)*d^6*e^4 + 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^4*c^(7/2)
*d^6*e^4 + 840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6*c^(5/2)*d^7*e^3 + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2
*A*b^5*c^(7/2)*d^7*e^3 + 248*B*b^7*c^(5/2)*d^8*e^2 - 64*A*b^6*c^(7/2)*d^8*e^2 - 1280*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^9*B*b^3*c^2*d^3*e^7 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^4*c^2*d^4*e^6 + 8960*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^7*A*b^3*c^3*d^4*e^6 - 3548*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^5*c^2*d^5*e^5 + 4280*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^5*A*b^4*c^3*d^5*e^5 - 1600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^6*c^2*d^6*e^4 + 2080*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^5*c^3*d^6*e^4 - 100*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^7*c^2*d^7*e^3 + 1
20*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^6*c^3*d^7*e^3 - 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^4*c^(3/2)*d^
3*e^7 - 1280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^3*c^(5/2)*d^3*e^7 - 3070*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*
B*b^5*c^(3/2)*d^4*e^6 + 7420*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^4*c^(5/2)*d^4*e^6 - 2670*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^4*B*b^6*c^(3/2)*d^5*e^5 + 2860*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^5*c^(5/2)*d^5*e^5 - 650*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^7*c^(3/2)*d^6*e^4 + 860*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^6*c^(5/2)*d^6
*e^4 - 10*B*b^8*c^(3/2)*d^7*e^3 + 12*A*b^7*c^(5/2)*d^7*e^3 - 2090*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^5*c*d^
3*e^7 - 4780*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^4*c^2*d^3*e^7 - 2114*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^
6*c*d^4*e^6 + 1448*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^5*c^2*d^4*e^6 - 1070*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
3*B*b^7*c*d^5*e^5 + 540*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^6*c^2*d^5*e^5 - 150*(sqrt(c)*x - sqrt(c*x^2 + b*
x))*B*b^8*c*d^6*e^4 + 200*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^7*c^2*d^6*e^4 + 135*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^8*B*b^5*sqrt(c)*d^2*e^8 - 270*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^4*c^(3/2)*d^2*e^8 - 790*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^6*B*b^6*sqrt(c)*d^3*e^7 - 5330*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^5*c^(3/2)*d^3*e^7 - 640
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^7*sqrt(c)*d^4*e^6 - 1390*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^6*c^(3/2
)*d^4*e^6 - 210*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^8*sqrt(c)*d^5*e^5 - 230*(sqrt(c)*x - sqrt(c*x^2 + b*x))^
2*A*b^7*c^(3/2)*d^5*e^5 - 15*B*b^9*sqrt(c)*d^6*e^4 + 20*A*b^8*c^(3/2)*d^6*e^4 + 15*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^9*B*b^5*d*e^9 - 30*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^4*c*d*e^9 + 70*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7
*B*b^6*d^2*e^8 + 330*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^5*c*d^2*e^8 - 128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5
*B*b^7*d^3*e^7 - 2626*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^6*c*d^3*e^7 - 70*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3
*B*b^8*d^4*e^6 - 930*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^7*c*d^4*e^6 - 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*
b^9*d^5*e^5 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^8*c*d^5*e^5 + 135*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^
5*sqrt(c)*d*e^9 + 490*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^6*sqrt(c)*d^2*e^8 - 640*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^4*A*b^7*sqrt(c)*d^3*e^7 - 210*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^8*sqrt(c)*d^4*e^6 - 15*A*b^9*sqrt(c)
*d^5*e^5 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^5*e^10 + 70*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^6*d*e^9
+ 128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^7*d^2*e^8 - 70*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^8*d^3*e^7 - 1
5*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^9*d^4*e^6)/((c^3*d^6*e^5 - 3*b*c^2*d^5*e^6 + 3*b^2*c*d^4*e^7 - b^3*d^3*e
^8)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^5)