3.1182 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=565 \[ -\frac{b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (2 b^2 c d e (21 A e+10 B d)+b^3 \left (-e^2\right ) (9 A e+5 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{1024 d^5 (d+e x)^2 (c d-b e)^5}+\frac{\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{840 d^3 (d+e x)^5 (c d-b e)^3}+\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (2 b^2 c d e (21 A e+10 B d)+b^3 \left (-e^2\right ) (9 A e+5 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{384 d^4 (d+e x)^4 (c d-b e)^4}+\frac{b^4 \left (2 b^2 c d e (21 A e+10 B d)+b^3 \left (-e^2\right ) (9 A e+5 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2048 d^{11/2} (c d-b e)^{11/2}}-\frac{\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{84 d^2 (d+e x)^6 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)} \]

[Out]

-(b^2*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(b
*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(1024*d^5*(c*d - b*e)^5*(d + e*x)^2) + ((48*A*c^3*d^3 - 24*b*c^2*d^2*
(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(b*d + (2*c*d - b*e)*x)*(b*x + c*x^2)
^(3/2))/(384*d^4*(c*d - b*e)^4*(d + e*x)^4) + ((B*d - A*e)*(b*x + c*x^2)^(5/2))/(7*d*(c*d - b*e)*(d + e*x)^7)
- ((9*A*e*(2*c*d - b*e) - B*d*(4*c*d + 5*b*e))*(b*x + c*x^2)^(5/2))/(84*d^2*(c*d - b*e)^2*(d + e*x)^6) + ((B*d
*(8*c^2*d^2 + 90*b*c*d*e - 35*b^2*e^2) - 3*A*e*(68*c^2*d^2 - 68*b*c*d*e + 21*b^2*e^2))*(b*x + c*x^2)^(5/2))/(8
40*d^3*(c*d - b*e)^3*(d + e*x)^5) + (b^4*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e)
+ 2*b^2*c*d*e*(10*B*d + 21*A*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])
])/(2048*d^(11/2)*(c*d - b*e)^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 1.0149, antiderivative size = 565, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {834, 806, 720, 724, 206} \[ -\frac{b^2 \sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (2 b^2 c d e (21 A e+10 B d)+b^3 \left (-e^2\right ) (9 A e+5 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{1024 d^5 (d+e x)^2 (c d-b e)^5}+\frac{\left (b x+c x^2\right )^{5/2} \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{840 d^3 (d+e x)^5 (c d-b e)^3}+\frac{\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d) \left (2 b^2 c d e (21 A e+10 B d)+b^3 \left (-e^2\right ) (9 A e+5 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right )}{384 d^4 (d+e x)^4 (c d-b e)^4}+\frac{b^4 \left (2 b^2 c d e (21 A e+10 B d)+b^3 \left (-e^2\right ) (9 A e+5 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2048 d^{11/2} (c d-b e)^{11/2}}-\frac{\left (b x+c x^2\right )^{5/2} (9 A e (2 c d-b e)-B d (5 b e+4 c d))}{84 d^2 (d+e x)^6 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{5/2} (B d-A e)}{7 d (d+e x)^7 (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

-(b^2*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(b
*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(1024*d^5*(c*d - b*e)^5*(d + e*x)^2) + ((48*A*c^3*d^3 - 24*b*c^2*d^2*
(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*e))*(b*d + (2*c*d - b*e)*x)*(b*x + c*x^2)
^(3/2))/(384*d^4*(c*d - b*e)^4*(d + e*x)^4) + ((B*d - A*e)*(b*x + c*x^2)^(5/2))/(7*d*(c*d - b*e)*(d + e*x)^7)
- ((9*A*e*(2*c*d - b*e) - B*d*(4*c*d + 5*b*e))*(b*x + c*x^2)^(5/2))/(84*d^2*(c*d - b*e)^2*(d + e*x)^6) + ((B*d
*(8*c^2*d^2 + 90*b*c*d*e - 35*b^2*e^2) - 3*A*e*(68*c^2*d^2 - 68*b*c*d*e + 21*b^2*e^2))*(b*x + c*x^2)^(5/2))/(8
40*d^3*(c*d - b*e)^3*(d + e*x)^5) + (b^4*(48*A*c^3*d^3 - 24*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e)
+ 2*b^2*c*d*e*(10*B*d + 21*A*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])
])/(2048*d^(11/2)*(c*d - b*e)^(11/2))

Rule 834

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

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)^8} \, dx &=\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac{\int \frac{\left (\frac{1}{2} (-14 A c d+b (5 B d+9 A e))-2 c (B d-A e) x\right ) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx}{7 d (c d-b e)}\\ &=\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac{(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac{\int \frac{\left (\frac{1}{4} \left (168 A c^2 d^2+7 b^2 e (5 B d+9 A e)-2 b c d (40 B d+93 A e)\right )-\frac{1}{2} c (9 A e (2 c d-b e)-B d (4 c d+5 b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx}{42 d^2 (c d-b e)^2}\\ &=\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac{(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac{\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}+\frac{\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{(d+e x)^5} \, dx}{48 d^3 (c d-b e)^3}\\ &=\frac{\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{384 d^4 (c d-b e)^4 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac{(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac{\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}-\frac{\left (b^2 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right )\right ) \int \frac{\sqrt{b x+c x^2}}{(d+e x)^3} \, dx}{256 d^4 (c d-b e)^4}\\ &=-\frac{b^2 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{1024 d^5 (c d-b e)^5 (d+e x)^2}+\frac{\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{384 d^4 (c d-b e)^4 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac{(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac{\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}+\frac{\left (b^4 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2048 d^5 (c d-b e)^5}\\ &=-\frac{b^2 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{1024 d^5 (c d-b e)^5 (d+e x)^2}+\frac{\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{384 d^4 (c d-b e)^4 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac{(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac{\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}-\frac{\left (b^4 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right )\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 )}{1024 d^5 (c d-b e)^5}\\ &=-\frac{b^2 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{1024 d^5 (c d-b e)^5 (d+e x)^2}+\frac{\left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{384 d^4 (c d-b e)^4 (d+e x)^4}+\frac{(B d-A e) \left (b x+c x^2\right )^{5/2}}{7 d (c d-b e) (d+e x)^7}-\frac{(9 A e (2 c d-b e)-B d (4 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{84 d^2 (c d-b e)^2 (d+e x)^6}+\frac{\left (B d \left (8 c^2 d^2+90 b c d e-35 b^2 e^2\right )-3 A e \left (68 c^2 d^2-68 b c d e+21 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{5/2}}{840 d^3 (c d-b e)^3 (d+e x)^5}+\frac{b^4 \left (48 A c^3 d^3-24 b c^2 d^2 (B d+3 A e)-b^3 e^2 (5 B d+9 A e)+2 b^2 c d e (10 B d+21 A e)\right ) \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 )}{2048 d^{11/2} (c d-b e)^{11/2}}\\ \end{align*}

Mathematica [A]  time = 4.94706, size = 464, normalized size = 0.82 \[ \frac{(x (b+c x))^{3/2} \left (-\frac{\frac{2 x^{5/2} (b+c x) \left (B d \left (-35 b^2 e^2+90 b c d e+8 c^2 d^2\right )-3 A e \left (21 b^2 e^2-68 b c d e+68 c^2 d^2\right )\right )}{(d+e x)^5}+\frac{35 \left (2 b^2 c d e (21 A e+10 B d)+b^3 \left (-e^2\right ) (9 A e+5 B d)-24 b c^2 d^2 (3 A e+B d)+48 A c^3 d^3\right ) \left (\frac{b^2 \sqrt{x} \sqrt{b+c x} (5 b d+3 b e x+2 c d x)}{8 d^2 (d+e x)^2 (b e-c d)}+\frac{3 b^4 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{8 d^{5/2} (b e-c d)^{3/2}}-\frac{2 x^{3/2} (b+c x)^{5/2}}{(d+e x)^4}+\frac{b \sqrt{x} (b+c x)^{5/2}}{(d+e x)^3 (c d-b e)}\right )}{8 (b+c x)^{3/2} (b e-c d)}}{240 d^2 (c d-b e)^2}-\frac{x^{5/2} (b+c x) (9 A e (b e-2 c d)+B d (5 b e+4 c d))}{12 d (d+e x)^6 (c d-b e)}+\frac{x^{5/2} (b+c x) (A e-B d)}{(d+e x)^7}\right )}{7 d x^{3/2} (b e-c d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

((x*(b + c*x))^(3/2)*(((-(B*d) + A*e)*x^(5/2)*(b + c*x))/(d + e*x)^7 - ((9*A*e*(-2*c*d + b*e) + B*d*(4*c*d + 5
*b*e))*x^(5/2)*(b + c*x))/(12*d*(c*d - b*e)*(d + e*x)^6) - ((2*(B*d*(8*c^2*d^2 + 90*b*c*d*e - 35*b^2*e^2) - 3*
A*e*(68*c^2*d^2 - 68*b*c*d*e + 21*b^2*e^2))*x^(5/2)*(b + c*x))/(d + e*x)^5 + (35*(48*A*c^3*d^3 - 24*b*c^2*d^2*
(B*d + 3*A*e) - b^3*e^2*(5*B*d + 9*A*e) + 2*b^2*c*d*e*(10*B*d + 21*A*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]*Sqrt[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)))/(240*d^2*(c*d - b*e)^2)))/(7*d*(-(
c*d) + b*e)*x^(3/2))

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Maple [B]  time = 0.052, size = 37630, normalized size = 66.6 \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)^8,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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.5422, size = 11826, normalized size = 20.93 \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)^8,x, algorithm="fricas")

[Out]

[-1/215040*(105*(9*A*b^7*d^7*e^3 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^10 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^9*e + (5
*B*b^7 - 42*A*b^6*c)*d^8*e^2 + (9*A*b^7*e^10 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^3*e^7 - 4*(5*B*b^6*c - 18*A*b^5*
c^2)*d^2*e^8 + (5*B*b^7 - 42*A*b^6*c)*d*e^9)*x^7 + 7*(9*A*b^7*d*e^9 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^4*e^6 - 4
*(5*B*b^6*c - 18*A*b^5*c^2)*d^3*e^7 + (5*B*b^7 - 42*A*b^6*c)*d^2*e^8)*x^6 + 21*(9*A*b^7*d^2*e^8 + 24*(B*b^5*c^
2 - 2*A*b^4*c^3)*d^5*e^5 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^4*e^6 + (5*B*b^7 - 42*A*b^6*c)*d^3*e^7)*x^5 + 35*(9*
A*b^7*d^3*e^7 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^6*e^4 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^5*e^5 + (5*B*b^7 - 42*A*
b^6*c)*d^4*e^6)*x^4 + 35*(9*A*b^7*d^4*e^6 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^7*e^3 - 4*(5*B*b^6*c - 18*A*b^5*c^2
)*d^6*e^4 + (5*B*b^7 - 42*A*b^6*c)*d^5*e^5)*x^3 + 21*(9*A*b^7*d^5*e^5 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^8*e^2 -
4*(5*B*b^6*c - 18*A*b^5*c^2)*d^7*e^3 + (5*B*b^7 - 42*A*b^6*c)*d^6*e^4)*x^2 + 7*(9*A*b^7*d^6*e^4 + 24*(B*b^5*c
^2 - 2*A*b^4*c^3)*d^9*e - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^8*e^2 + (5*B*b^7 - 42*A*b^6*c)*d^7*e^3)*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*(945*A*b^7*d^7*
e^4 - 2520*(B*b^4*c^3 - 2*A*b^3*c^4)*d^11 + 420*(11*B*b^5*c^2 - 30*A*b^4*c^3)*d^10*e - 105*(25*B*b^6*c - 114*A
*b^5*c^2)*d^9*e^2 + 105*(5*B*b^7 - 51*A*b^6*c)*d^8*e^3 - (1024*B*c^7*d^9*e^2 + 945*A*b^7*d*e^10 - 384*(13*B*b*
c^6 - 2*A*c^7)*d^8*e^3 + 384*(23*B*b^2*c^5 - 8*A*b*c^6)*d^7*e^4 - 96*(59*B*b^3*c^4 - 34*A*b^2*c^5)*d^6*e^5 - 1
20*(5*B*b^4*c^3 - 8*A*b^3*c^4)*d^5*e^6 + 6*(525*B*b^5*c^2 - 1174*A*b^4*c^3)*d^4*e^7 - 7*(325*B*b^6*c - 1272*A*
b^5*c^2)*d^3*e^8 + 525*(B*b^7 - 9*A*b^6*c)*d^2*e^9)*x^6 - 2*(3584*B*c^7*d^10*e + 3150*A*b^7*d^2*e^9 - 64*(277*
B*b*c^6 - 42*A*c^7)*d^9*e^2 + 96*(335*B*b^2*c^5 - 114*A*b*c^6)*d^8*e^3 - 48*(459*B*b^3*c^4 - 254*A*b^2*c^5)*d^
7*e^4 - 12*(57*B*b^4*c^3 - 212*A*b^3*c^4)*d^6*e^5 + 3*(3515*B*b^5*c^2 - 7878*A*b^4*c^3)*d^5*e^6 - 5*(1519*B*b^
6*c - 5955*A*b^5*c^2)*d^4*e^7 + 7*(250*B*b^7 - 2253*A*b^6*c)*d^3*e^8)*x^5 - (21504*B*c^7*d^11 + 17829*A*b^7*d^
3*e^8 - 896*(121*B*b*c^6 - 18*A*c^7)*d^10*e + 192*(1059*B*b^2*c^5 - 350*A*b*c^6)*d^9*e^2 - 48*(3161*B*b^3*c^4
- 1658*A*b^2*c^5)*d^8*e^3 + 24*(439*B*b^4*c^3 + 316*A*b^3*c^4)*d^7*e^4 + 48*(1207*B*b^5*c^2 - 2785*A*b^4*c^3)*
d^6*e^5 - 33*(1305*B*b^6*c - 5126*A*b^5*c^2)*d^5*e^6 + (9905*B*b^7 - 89403*A*b^6*c)*d^4*e^7)*x^4 - 4*(6912*A*b
^7*d^4*e^7 + 672*(11*B*b*c^6 + 10*A*c^7)*d^11 - 14448*(3*B*b^2*c^5 + 2*A*b*c^6)*d^10*e + 4*(25775*B*b^3*c^4 +
9282*A*b^2*c^5)*d^9*e^2 - 12*(9883*B*b^4*c^3 + 187*A*b^3*c^4)*d^8*e^3 + 3*(24291*B*b^5*c^2 - 16846*A*b^4*c^3)*
d^7*e^4 - (25265*B*b^6*c - 65643*A*b^5*c^2)*d^6*e^5 + 15*(256*B*b^7 - 2315*A*b^6*c)*d^5*e^6)*x^3 - 7*(3597*A*b
^7*d^5*e^6 + 192*(B*b^2*c^5 + 30*A*b*c^6)*d^11 - 176*(11*B*b^3*c^4 + 186*A*b^2*c^5)*d^10*e + 8*(1237*B*b^4*c^3
+ 8718*A*b^3*c^4)*d^9*e^2 - 6*(2357*B*b^5*c^2 + 13498*A*b^4*c^3)*d^8*e^3 + 5*(1481*B*b^6*c + 11292*A*b^5*c^2)
*d^7*e^4 - (1415*B*b^7 + 21837*A*b^6*c)*d^6*e^5)*x^2 + 70*(90*A*b^7*d^6*e^5 + 24*(B*b^3*c^4 - 2*A*b^2*c^5)*d^1
1 - 4*(71*B*b^4*c^3 - 150*A*b^3*c^4)*d^10*e + 3*(155*B*b^5*c^2 - 438*A*b^4*c^3)*d^9*e^2 - 3*(85*B*b^6*c - 397*
A*b^5*c^2)*d^8*e^3 + (50*B*b^7 - 519*A*b^6*c)*d^7*e^4)*x)*sqrt(c*x^2 + b*x))/(c^6*d^19 - 6*b*c^5*d^18*e + 15*b
^2*c^4*d^17*e^2 - 20*b^3*c^3*d^16*e^3 + 15*b^4*c^2*d^15*e^4 - 6*b^5*c*d^14*e^5 + b^6*d^13*e^6 + (c^6*d^12*e^7
- 6*b*c^5*d^11*e^8 + 15*b^2*c^4*d^10*e^9 - 20*b^3*c^3*d^9*e^10 + 15*b^4*c^2*d^8*e^11 - 6*b^5*c*d^7*e^12 + b^6*
d^6*e^13)*x^7 + 7*(c^6*d^13*e^6 - 6*b*c^5*d^12*e^7 + 15*b^2*c^4*d^11*e^8 - 20*b^3*c^3*d^10*e^9 + 15*b^4*c^2*d^
9*e^10 - 6*b^5*c*d^8*e^11 + b^6*d^7*e^12)*x^6 + 21*(c^6*d^14*e^5 - 6*b*c^5*d^13*e^6 + 15*b^2*c^4*d^12*e^7 - 20
*b^3*c^3*d^11*e^8 + 15*b^4*c^2*d^10*e^9 - 6*b^5*c*d^9*e^10 + b^6*d^8*e^11)*x^5 + 35*(c^6*d^15*e^4 - 6*b*c^5*d^
14*e^5 + 15*b^2*c^4*d^13*e^6 - 20*b^3*c^3*d^12*e^7 + 15*b^4*c^2*d^11*e^8 - 6*b^5*c*d^10*e^9 + b^6*d^9*e^10)*x^
4 + 35*(c^6*d^16*e^3 - 6*b*c^5*d^15*e^4 + 15*b^2*c^4*d^14*e^5 - 20*b^3*c^3*d^13*e^6 + 15*b^4*c^2*d^12*e^7 - 6*
b^5*c*d^11*e^8 + b^6*d^10*e^9)*x^3 + 21*(c^6*d^17*e^2 - 6*b*c^5*d^16*e^3 + 15*b^2*c^4*d^15*e^4 - 20*b^3*c^3*d^
14*e^5 + 15*b^4*c^2*d^13*e^6 - 6*b^5*c*d^12*e^7 + b^6*d^11*e^8)*x^2 + 7*(c^6*d^18*e - 6*b*c^5*d^17*e^2 + 15*b^
2*c^4*d^16*e^3 - 20*b^3*c^3*d^15*e^4 + 15*b^4*c^2*d^14*e^5 - 6*b^5*c*d^13*e^6 + b^6*d^12*e^7)*x), -1/107520*(1
05*(9*A*b^7*d^7*e^3 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^10 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^9*e + (5*B*b^7 - 42*A
*b^6*c)*d^8*e^2 + (9*A*b^7*e^10 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^3*e^7 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^2*e^8
+ (5*B*b^7 - 42*A*b^6*c)*d*e^9)*x^7 + 7*(9*A*b^7*d*e^9 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^4*e^6 - 4*(5*B*b^6*c -
18*A*b^5*c^2)*d^3*e^7 + (5*B*b^7 - 42*A*b^6*c)*d^2*e^8)*x^6 + 21*(9*A*b^7*d^2*e^8 + 24*(B*b^5*c^2 - 2*A*b^4*c
^3)*d^5*e^5 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^4*e^6 + (5*B*b^7 - 42*A*b^6*c)*d^3*e^7)*x^5 + 35*(9*A*b^7*d^3*e^7
+ 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^6*e^4 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^5*e^5 + (5*B*b^7 - 42*A*b^6*c)*d^4*e^
6)*x^4 + 35*(9*A*b^7*d^4*e^6 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^7*e^3 - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^6*e^4 + (
5*B*b^7 - 42*A*b^6*c)*d^5*e^5)*x^3 + 21*(9*A*b^7*d^5*e^5 + 24*(B*b^5*c^2 - 2*A*b^4*c^3)*d^8*e^2 - 4*(5*B*b^6*c
- 18*A*b^5*c^2)*d^7*e^3 + (5*B*b^7 - 42*A*b^6*c)*d^6*e^4)*x^2 + 7*(9*A*b^7*d^6*e^4 + 24*(B*b^5*c^2 - 2*A*b^4*
c^3)*d^9*e - 4*(5*B*b^6*c - 18*A*b^5*c^2)*d^8*e^2 + (5*B*b^7 - 42*A*b^6*c)*d^7*e^3)*x)*sqrt(-c*d^2 + b*d*e)*ar
ctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + (945*A*b^7*d^7*e^4 - 2520*(B*b^4*c^3 - 2*A*b^3
*c^4)*d^11 + 420*(11*B*b^5*c^2 - 30*A*b^4*c^3)*d^10*e - 105*(25*B*b^6*c - 114*A*b^5*c^2)*d^9*e^2 + 105*(5*B*b^
7 - 51*A*b^6*c)*d^8*e^3 - (1024*B*c^7*d^9*e^2 + 945*A*b^7*d*e^10 - 384*(13*B*b*c^6 - 2*A*c^7)*d^8*e^3 + 384*(2
3*B*b^2*c^5 - 8*A*b*c^6)*d^7*e^4 - 96*(59*B*b^3*c^4 - 34*A*b^2*c^5)*d^6*e^5 - 120*(5*B*b^4*c^3 - 8*A*b^3*c^4)*
d^5*e^6 + 6*(525*B*b^5*c^2 - 1174*A*b^4*c^3)*d^4*e^7 - 7*(325*B*b^6*c - 1272*A*b^5*c^2)*d^3*e^8 + 525*(B*b^7 -
9*A*b^6*c)*d^2*e^9)*x^6 - 2*(3584*B*c^7*d^10*e + 3150*A*b^7*d^2*e^9 - 64*(277*B*b*c^6 - 42*A*c^7)*d^9*e^2 + 9
6*(335*B*b^2*c^5 - 114*A*b*c^6)*d^8*e^3 - 48*(459*B*b^3*c^4 - 254*A*b^2*c^5)*d^7*e^4 - 12*(57*B*b^4*c^3 - 212*
A*b^3*c^4)*d^6*e^5 + 3*(3515*B*b^5*c^2 - 7878*A*b^4*c^3)*d^5*e^6 - 5*(1519*B*b^6*c - 5955*A*b^5*c^2)*d^4*e^7 +
7*(250*B*b^7 - 2253*A*b^6*c)*d^3*e^8)*x^5 - (21504*B*c^7*d^11 + 17829*A*b^7*d^3*e^8 - 896*(121*B*b*c^6 - 18*A
*c^7)*d^10*e + 192*(1059*B*b^2*c^5 - 350*A*b*c^6)*d^9*e^2 - 48*(3161*B*b^3*c^4 - 1658*A*b^2*c^5)*d^8*e^3 + 24*
(439*B*b^4*c^3 + 316*A*b^3*c^4)*d^7*e^4 + 48*(1207*B*b^5*c^2 - 2785*A*b^4*c^3)*d^6*e^5 - 33*(1305*B*b^6*c - 51
26*A*b^5*c^2)*d^5*e^6 + (9905*B*b^7 - 89403*A*b^6*c)*d^4*e^7)*x^4 - 4*(6912*A*b^7*d^4*e^7 + 672*(11*B*b*c^6 +
10*A*c^7)*d^11 - 14448*(3*B*b^2*c^5 + 2*A*b*c^6)*d^10*e + 4*(25775*B*b^3*c^4 + 9282*A*b^2*c^5)*d^9*e^2 - 12*(9
883*B*b^4*c^3 + 187*A*b^3*c^4)*d^8*e^3 + 3*(24291*B*b^5*c^2 - 16846*A*b^4*c^3)*d^7*e^4 - (25265*B*b^6*c - 6564
3*A*b^5*c^2)*d^6*e^5 + 15*(256*B*b^7 - 2315*A*b^6*c)*d^5*e^6)*x^3 - 7*(3597*A*b^7*d^5*e^6 + 192*(B*b^2*c^5 + 3
0*A*b*c^6)*d^11 - 176*(11*B*b^3*c^4 + 186*A*b^2*c^5)*d^10*e + 8*(1237*B*b^4*c^3 + 8718*A*b^3*c^4)*d^9*e^2 - 6*
(2357*B*b^5*c^2 + 13498*A*b^4*c^3)*d^8*e^3 + 5*(1481*B*b^6*c + 11292*A*b^5*c^2)*d^7*e^4 - (1415*B*b^7 + 21837*
A*b^6*c)*d^6*e^5)*x^2 + 70*(90*A*b^7*d^6*e^5 + 24*(B*b^3*c^4 - 2*A*b^2*c^5)*d^11 - 4*(71*B*b^4*c^3 - 150*A*b^3
*c^4)*d^10*e + 3*(155*B*b^5*c^2 - 438*A*b^4*c^3)*d^9*e^2 - 3*(85*B*b^6*c - 397*A*b^5*c^2)*d^8*e^3 + (50*B*b^7
- 519*A*b^6*c)*d^7*e^4)*x)*sqrt(c*x^2 + b*x))/(c^6*d^19 - 6*b*c^5*d^18*e + 15*b^2*c^4*d^17*e^2 - 20*b^3*c^3*d^
16*e^3 + 15*b^4*c^2*d^15*e^4 - 6*b^5*c*d^14*e^5 + b^6*d^13*e^6 + (c^6*d^12*e^7 - 6*b*c^5*d^11*e^8 + 15*b^2*c^4
*d^10*e^9 - 20*b^3*c^3*d^9*e^10 + 15*b^4*c^2*d^8*e^11 - 6*b^5*c*d^7*e^12 + b^6*d^6*e^13)*x^7 + 7*(c^6*d^13*e^6
- 6*b*c^5*d^12*e^7 + 15*b^2*c^4*d^11*e^8 - 20*b^3*c^3*d^10*e^9 + 15*b^4*c^2*d^9*e^10 - 6*b^5*c*d^8*e^11 + b^6
*d^7*e^12)*x^6 + 21*(c^6*d^14*e^5 - 6*b*c^5*d^13*e^6 + 15*b^2*c^4*d^12*e^7 - 20*b^3*c^3*d^11*e^8 + 15*b^4*c^2*
d^10*e^9 - 6*b^5*c*d^9*e^10 + b^6*d^8*e^11)*x^5 + 35*(c^6*d^15*e^4 - 6*b*c^5*d^14*e^5 + 15*b^2*c^4*d^13*e^6 -
20*b^3*c^3*d^12*e^7 + 15*b^4*c^2*d^11*e^8 - 6*b^5*c*d^10*e^9 + b^6*d^9*e^10)*x^4 + 35*(c^6*d^16*e^3 - 6*b*c^5*
d^15*e^4 + 15*b^2*c^4*d^14*e^5 - 20*b^3*c^3*d^13*e^6 + 15*b^4*c^2*d^12*e^7 - 6*b^5*c*d^11*e^8 + b^6*d^10*e^9)*
x^3 + 21*(c^6*d^17*e^2 - 6*b*c^5*d^16*e^3 + 15*b^2*c^4*d^15*e^4 - 20*b^3*c^3*d^14*e^5 + 15*b^4*c^2*d^13*e^6 -
6*b^5*c*d^12*e^7 + b^6*d^11*e^8)*x^2 + 7*(c^6*d^18*e - 6*b*c^5*d^17*e^2 + 15*b^2*c^4*d^16*e^3 - 20*b^3*c^3*d^1
5*e^4 + 15*b^4*c^2*d^14*e^5 - 6*b^5*c*d^13*e^6 + b^6*d^12*e^7)*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)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.00373, size = 10684, normalized size = 18.91 \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)^8,x, algorithm="giac")

[Out]

-1/1024*(24*B*b^5*c^2*d^3 - 48*A*b^4*c^3*d^3 - 20*B*b^6*c*d^2*e + 72*A*b^5*c^2*d^2*e + 5*B*b^7*d*e^2 - 42*A*b^
6*c*d*e^2 + 9*A*b^7*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^5*d
^10 - 5*b*c^4*d^9*e + 10*b^2*c^3*d^8*e^2 - 10*b^3*c^2*d^7*e^3 + 5*b^4*c*d^6*e^4 - b^5*d^5*e^5)*sqrt(-c*d^2 + b
*d*e)) + 1/107520*(458752*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^(19/2)*d^13*e + 131072*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^7*B*c^10*d^14 + 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*c^9*d^12*e^2 + 868352*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^7*B*b*c^9*d^13*e + 98304*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*c^10*d^13*e + 458752*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^6*B*b*c^(19/2)*d^14 + 573440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*c^(17/2)*d^11*e^3 - 57344*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b*c^(17/2)*d^12*e^2 + 344064*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*c^(19/2)*d^
12*e^2 - 57344*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^2*c^(17/2)*d^13*e + 344064*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^6*A*b*c^(19/2)*d^13*e + 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^9*d^14 + 286720*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^11*B*c^8*d^10*e^4 - 1519616*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b*c^8*d^11*e^3 + 516096*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^9*A*c^9*d^11*e^3 - 2990080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^2*c^8*d^12*e^2 + 73728
0*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b*c^9*d^12*e^2 - 1519616*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*c^8*d^1
3*e + 516096*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^9*d^13*e + 573440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b
^3*c^(17/2)*d^14 - 1792000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b*c^(15/2)*d^10*e^4 + 430080*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^10*A*c^(17/2)*d^10*e^4 - 3627008*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^2*c^(15/2)*d^11*e^3 + 25
8048*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b*c^(17/2)*d^11*e^3 - 3627008*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^3
*c^(15/2)*d^12*e^2 + 258048*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^2*c^(17/2)*d^12*e^2 - 1792000*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^4*B*b^4*c^(15/2)*d^13*e + 430080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*c^(17/2)*d^13*e + 2
86720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*c^8*d^14 - 1433600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b*c^7*d^
9*e^5 - 960512*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^2*c^7*d^10*e^4 - 688128*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9
*A*b*c^8*d^10*e^4 + 55296*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^3*c^7*d^11*e^3 - 1683456*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^7*A*b^2*c^8*d^11*e^3 - 960512*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^4*c^7*d^12*e^2 - 688128*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^5*A*b^3*c^8*d^12*e^2 - 1025024*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^5*c^7*d^13*e + 2
15040*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*c^8*d^13*e + 86016*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5*c^(15
/2)*d^14 + 358400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^2*c^(13/2)*d^9*e^5 - 2150400*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^10*A*b*c^(15/2)*d^9*e^5 + 4515840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^3*c^(13/2)*d^10*e^4 - 2795520*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^2*c^(15/2)*d^10*e^4 + 4515840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^4*c^
(13/2)*d^11*e^3 - 2795520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^3*c^(15/2)*d^11*e^3 + 1078784*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^4*B*b^5*c^(13/2)*d^12*e^2 - 967680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^4*c^(15/2)*d^12*e^2 -
326144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6*c^(13/2)*d^13*e + 64512*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^
5*c^(15/2)*d^13*e + 14336*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^6*c^7*d^14 + 2867200*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^11*B*b^2*c^6*d^8*e^6 + 3512320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^3*c^6*d^9*e^5 - 4300800*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^9*A*b^2*c^7*d^9*e^5 + 5806080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^4*c^6*d^10*e^4 - 21196
80*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^3*c^7*d^10*e^4 + 4076800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^5*c^6*
d^11*e^3 - 1774080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^4*c^7*d^11*e^3 + 1028608*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^3*B*b^6*c^6*d^12*e^2 - 580608*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^5*c^7*d^12*e^2 - 55552*(sqrt(c)*x - sq
rt(c*x^2 + b*x))*B*b^7*c^6*d^13*e + 10752*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^6*c^7*d^13*e + 1024*B*b^7*c^(13/
2)*d^14 + 5017600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^3*c^(11/2)*d^8*e^6 + 4300800*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^10*A*b^2*c^(13/2)*d^8*e^6 + 1720320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^4*c^(11/2)*d^9*e^5 - 1935360
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^3*c^(13/2)*d^9*e^5 + 1975680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^5*c^
(11/2)*d^10*e^4 + 564480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^4*c^(13/2)*d^10*e^4 + 1511552*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^4*B*b^6*c^(11/2)*d^11*e^3 - 413952*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^5*c^(13/2)*d^11*e^3 +
380800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^7*c^(11/2)*d^12*e^2 - 188160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*
b^6*c^(13/2)*d^12*e^2 - 3968*B*b^8*c^(11/2)*d^13*e + 768*A*b^7*c^(13/2)*d^13*e - 2867200*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^11*B*b^3*c^5*d^7*e^7 + 1863680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^4*c^5*d^8*e^6 + 13762560*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^9*A*b^3*c^6*d^8*e^6 - 2088576*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^5*c^5*d^9*e^5 +
2327808*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^4*c^6*d^9*e^5 - 815360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^6*
c^5*d^10*e^4 + 1881600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^5*c^6*d^10*e^4 + 184576*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^3*B*b^7*c^5*d^11*e^3 + 75264*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^6*c^6*d^11*e^3 + 68096*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*B*b^8*c^5*d^12*e^2 - 32256*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^7*c^6*d^12*e^2 - 7884800*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^4*c^(9/2)*d^7*e^7 - 4300800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b^3*c^(11
/2)*d^7*e^7 - 2283456*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^5*c^(9/2)*d^8*e^6 + 16695168*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^8*A*b^4*c^(11/2)*d^8*e^6 - 2968896*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^6*c^(9/2)*d^9*e^5 + 306700
8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^5*c^(11/2)*d^9*e^5 - 1017856*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^7*c
^(9/2)*d^10*e^4 + 1430016*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^6*c^(11/2)*d^10*e^4 - 33152*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^2*B*b^8*c^(9/2)*d^11*e^3 + 64512*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^7*c^(11/2)*d^11*e^3 + 486
4*B*b^9*c^(9/2)*d^12*e^2 - 2304*A*b^8*c^(11/2)*d^12*e^2 + 1433600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b^4*c^4
*d^6*e^8 - 7790944*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^5*c^4*d^7*e^7 - 18193728*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^9*A*b^4*c^5*d^7*e^7 - 2131136*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^6*c^4*d^8*e^6 + 7469952*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^7*A*b^5*c^5*d^8*e^6 - 1485344*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^7*c^4*d^9*e^5 + 716352*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^6*c^5*d^9*e^5 - 459200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^8*c^4*d^10*e
^4 + 618240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^7*c^5*d^10*e^4 - 11200*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^9
*c^4*d^11*e^3 + 13440*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^8*c^5*d^11*e^3 + 5338480*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^10*B*b^5*c^(7/2)*d^6*e^8 + 1078560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b^4*c^(9/2)*d^6*e^8 - 2001328*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^6*c^(7/2)*d^7*e^7 - 28144032*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^5*c^(9/2
)*d^7*e^7 + 333872*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^7*c^(7/2)*d^8*e^6 - 2198112*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^6*A*b^6*c^(9/2)*d^8*e^6 - 160720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^8*c^(7/2)*d^9*e^5 - 971040*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^4*A*b^7*c^(9/2)*d^9*e^5 - 123200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^9*c^(7/2)*d^
10*e^4 + 168000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^8*c^(9/2)*d^10*e^4 - 800*B*b^10*c^(7/2)*d^11*e^3 + 960*A
*b^9*c^(9/2)*d^11*e^3 - 106960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b^5*c^3*d^5*e^9 - 359520*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^11*A*b^4*c^4*d^5*e^9 + 6953408*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^6*c^3*d^6*e^8 + 10063872*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^5*c^4*d^6*e^8 + 1909360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^7*c^3*d^7*e
^7 - 20156640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^6*c^4*d^7*e^7 + 1255408*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*
B*b^8*c^3*d^8*e^6 - 3698688*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^7*c^4*d^8*e^6 + 192640*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^3*B*b^9*c^3*d^9*e^5 - 913920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^8*c^4*d^9*e^5 - 19600*(sqrt(c)*x
- sqrt(c*x^2 + b*x))*B*b^10*c^3*d^10*e^4 + 26880*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^9*c^4*d^10*e^4 + 32760*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^12*B*b^5*c^(5/2)*d^4*e^10 - 65520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*A*b^4*c^(7
/2)*d^4*e^10 - 1337000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^6*c^(5/2)*d^5*e^9 + 808080*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^10*A*b^5*c^(7/2)*d^5*e^9 + 3935624*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^7*c^(5/2)*d^6*e^8 + 196193
76*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^6*c^(7/2)*d^6*e^8 + 1520624*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^8*c
^(5/2)*d^7*e^7 - 5499984*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^7*c^(7/2)*d^7*e^7 + 673400*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^4*B*b^9*c^(5/2)*d^8*e^6 - 1233120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^8*c^(7/2)*d^8*e^6 + 104440
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^10*c^(5/2)*d^9*e^5 - 351456*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^9*c^(
7/2)*d^9*e^5 - 1400*B*b^11*c^(5/2)*d^10*e^4 + 1920*A*b^10*c^(7/2)*d^10*e^4 + 2520*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^13*B*b^5*c^2*d^3*e^11 - 5040*(sqrt(c)*x - sqrt(c*x^2 + b*x))^13*A*b^4*c^3*d^3*e^11 - 133000*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^11*B*b^6*c^2*d^4*e^10 + 505680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^5*c^3*d^4*e^10 - 22137
64*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^7*c^2*d^5*e^9 - 1120056*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^6*c^3*d
^5*e^9 + 595552*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^8*c^2*d^6*e^8 + 16718928*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^7*A*b^7*c^3*d^6*e^8 + 229936*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^9*c^2*d^7*e^7 + 1320144*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^5*A*b^8*c^3*d^7*e^7 + 133000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^10*c^2*d^8*e^6 + 122304*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^3*A*b^9*c^3*d^8*e^6 + 21980*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^11*c^2*d^9*e^5 - 66
696*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^10*c^3*d^9*e^5 - 27300*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*B*b^6*c^(3/2
)*d^3*e^11 + 98280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*A*b^5*c^(5/2)*d^3*e^11 - 42350*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^10*B*b^7*c^(3/2)*d^4*e^10 - 383460*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b^6*c^(5/2)*d^4*e^10 - 1534330*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^8*c^(3/2)*d^5*e^9 - 5388180*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^7*c^(5
/2)*d^5*e^9 - 422380*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^9*c^(3/2)*d^6*e^8 + 7011480*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^6*A*b^8*c^(5/2)*d^6*e^8 - 141400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^10*c^(3/2)*d^7*e^7 + 1334256*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^9*c^(5/2)*d^7*e^7 - 4550*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^11*c^(3/2)
*d^8*e^6 + 183372*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^10*c^(5/2)*d^8*e^6 + 1750*B*b^12*c^(3/2)*d^9*e^5 - 512
4*A*b^11*c^(5/2)*d^9*e^5 - 2100*(sqrt(c)*x - sqrt(c*x^2 + b*x))^13*B*b^6*c*d^2*e^12 + 7560*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^13*A*b^5*c^2*d^2*e^12 + 23450*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b^7*c*d^3*e^11 - 264180*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^11*A*b^6*c^2*d^3*e^11 + 133070*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^8*c*d^4*e^10 -
960960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^7*c^2*d^4*e^10 - 549660*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^9*c
*d^5*e^9 - 5471640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^8*c^2*d^5*e^9 - 243320*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^5*B*b^10*c*d^6*e^8 + 1126440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^9*c^2*d^6*e^8 - 67550*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^3*B*b^11*c*d^7*e^7 + 317100*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^10*c^2*d^7*e^7 - 5250*(sqrt(c)*
x - sqrt(c*x^2 + b*x))*B*b^12*c*d^8*e^6 + 45360*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^11*c^2*d^8*e^6 + 6825*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^12*B*b^7*sqrt(c)*d^2*e^12 - 57330*(sqrt(c)*x - sqrt(c*x^2 + b*x))^12*A*b^6*c^(3/2)
*d^2*e^12 + 38500*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^8*sqrt(c)*d^3*e^11 - 122430*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^10*A*b^7*c^(3/2)*d^3*e^11 + 89145*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^9*sqrt(c)*d^4*e^10 - 30240*(sqr
t(c)*x - sqrt(c*x^2 + b*x))^8*A*b^8*c^(3/2)*d^4*e^10 - 107520*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^10*sqrt(c)
*d^5*e^9 - 2566620*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^9*c^(3/2)*d^5*e^9 - 49525*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^4*B*b^11*sqrt(c)*d^6*e^8 - 195090*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^10*c^(3/2)*d^6*e^8 - 10500*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*B*b^12*sqrt(c)*d^7*e^7 + 4410*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^11*c^(3/2)*d^7
*e^7 - 525*B*b^13*sqrt(c)*d^8*e^6 + 3780*A*b^12*c^(3/2)*d^8*e^6 + 525*(sqrt(c)*x - sqrt(c*x^2 + b*x))^13*B*b^7
*d*e^13 - 4410*(sqrt(c)*x - sqrt(c*x^2 + b*x))^13*A*b^6*c*d*e^13 + 3500*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b
^8*d^2*e^12 + 38010*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^7*c*d^2*e^12 + 9905*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^9*B*b^9*d^3*e^11 + 227640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^8*c*d^3*e^11 + 15360*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^7*B*b^10*d^4*e^10 + 429876*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^9*c*d^4*e^10 - 9905*(sqrt(c)*x - sqrt
(c*x^2 + b*x))^5*B*b^11*d^5*e^9 - 641130*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^10*c*d^5*e^9 - 3500*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^3*B*b^12*d^6*e^8 - 117390*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^11*c*d^6*e^8 - 525*(sqrt(
c)*x - sqrt(c*x^2 + b*x))*B*b^13*d^7*e^7 - 8820*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^12*c*d^7*e^7 + 12285*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^12*A*b^7*sqrt(c)*d*e^13 + 69300*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b^8*sqrt(c)*d^
2*e^12 + 160461*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^9*sqrt(c)*d^3*e^11 + 193536*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^6*A*b^10*sqrt(c)*d^4*e^10 - 89145*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^11*sqrt(c)*d^5*e^9 - 18900*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^2*A*b^12*sqrt(c)*d^6*e^8 - 945*A*b^13*sqrt(c)*d^7*e^7 + 945*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^13*A*b^7*e^14 + 6300*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^8*d*e^13 + 17829*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^9*A*b^9*d^2*e^12 + 27648*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^10*d^3*e^11 + 25179*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^5*A*b^11*d^4*e^10 - 6300*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^12*d^5*e^9 - 945*(sqrt(c)*x - sqrt(c
*x^2 + b*x))*A*b^13*d^6*e^8)/((c^5*d^10*e^5 - 5*b*c^4*d^9*e^6 + 10*b^2*c^3*d^8*e^7 - 10*b^3*c^2*d^7*e^8 + 5*b^
4*c*d^6*e^9 - b^5*d^5*e^10)*((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)^7)