3.12.9 \(\int \frac {A+B x}{(d+e x)^{5/2} (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=644 \[ -\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)+48 A c^2 d^2\right )}{4 b^5 d^{9/2}}+\frac {c^{7/2} \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}+\frac {b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac {e \left (5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)+3 b^2 c^2 d^2 e (9 A e+29 B d)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac {e \left (-5 b^5 e^4 (4 B d-7 A e)+8 b^4 c d e^3 (7 B d-10 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{4 b^4 d^4 \sqrt {d+e x} (c d-b e)^4} \]

________________________________________________________________________________________

Rubi [A]  time = 1.79, antiderivative size = 644, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {822, 828, 826, 1166, 208} \begin {gather*} \frac {c x \left (b^2 c d e (23 B d-2 A e)+b^3 \left (-e^2\right ) (4 B d-7 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac {e \left (-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (4 A e+5 B d)+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-12 b c^4 d^4 (5 A e+B d)+24 A c^5 d^5\right )}{4 b^4 d^4 \sqrt {d+e x} (c d-b e)^4}+\frac {e \left (3 b^2 c^2 d^2 e (9 A e+29 B d)-9 b^3 c d e^2 (4 B d-5 A e)+5 b^4 e^3 (4 B d-7 A e)-36 b c^3 d^3 (4 A e+B d)+72 A c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac {c^{7/2} \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)+48 A c^2 d^2\right )}{4 b^5 d^{9/2}}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(e*(72*A*c^4*d^4 + 5*b^4*e^3*(4*B*d - 7*A*e) - 9*b^3*c*d*e^2*(4*B*d - 5*A*e) - 36*b*c^3*d^3*(B*d + 4*A*e) + 3*
b^2*c^2*d^2*e*(29*B*d + 9*A*e)))/(12*b^4*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) + (e*(24*A*c^5*d^5 + 8*b^4*c*d*e^3
*(7*B*d - 10*A*e) - 5*b^5*e^4*(4*B*d - 7*A*e) - 6*b^3*c^2*d^2*e^2*(4*B*d - 3*A*e) + 7*b^2*c^3*d^3*e*(5*B*d + 4
*A*e) - 12*b*c^4*d^4*(B*d + 5*A*e)))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (A*b*(c*d - b*e) + c*(2*A*c*d -
b*(B*d + A*e))*x)/(2*b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*(b*x + c*x^2)^2) + (b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*
e*(4*B*d - 7*A*e) - 3*b*c*d*(2*B*d + A*e)) + c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 7*A*e) + b^2*c*d*e*(23*B*d - 2
*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e))*x)/(4*b^4*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)*(b*x + c*x^2)) - ((48*A*c^2*d^
2 - 5*b^2*e*(4*B*d - 7*A*e) - 12*b*c*d*(2*B*d - 5*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(
7/2)*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*ArcTanh[(Sqrt[c
]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

Rule 208

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

Rule 822

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

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

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

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {A+B x}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx &=-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )-\frac {9}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^2 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )+\frac {5}{4} c e \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^3 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )+\frac {1}{4} c e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^4 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )+\frac {1}{4} c e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^4 (c d-b e)^4}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} e (c d-b e)^4 \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )-\frac {1}{4} c d e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )+\frac {1}{4} c e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 d^4 (c d-b e)^4}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\left (c \left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 d^4}-\frac {\left (c^4 \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 (c d-b e)^4}\\ &=\frac {e \left (72 A c^4 d^4+5 b^4 e^3 (4 B d-7 A e)-9 b^3 c d e^2 (4 B d-5 A e)-36 b c^3 d^3 (B d+4 A e)+3 b^2 c^2 d^2 e (29 B d+9 A e)\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e \left (24 A c^5 d^5+8 b^4 c d e^3 (7 B d-10 A e)-5 b^5 e^4 (4 B d-7 A e)-6 b^3 c^2 d^2 e^2 (4 B d-3 A e)+7 b^2 c^3 d^3 e (5 B d+4 A e)-12 b c^4 d^4 (B d+5 A e)\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-7 A e)-3 b c d (2 B d+A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-7 A e)+b^2 c d e (23 B d-2 A e)-12 b c^2 d^2 (B d+3 A e)\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\left (48 A c^2 d^2-5 b^2 e (4 B d-7 A e)-12 b c d (2 B d-5 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}+\frac {c^{7/2} \left (48 A c^3 d^2-99 b^3 B e^2-12 b c^2 d (2 B d+13 A e)+11 b^2 c e (8 B d+13 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.57, size = 388, normalized size = 0.60 \begin {gather*} \frac {6 A b^4 d^2 (b e-c d)^3+3 b^3 d x (b e-c d)^3 (-7 A b e-8 A c d+4 b B d)+x^2 \left (3 b^2 c d (c d-b e)^2 \left (b^2 e (4 B d-7 A e)-3 b c d (A e+2 B d)+12 A c^2 d^2\right )+(b+c x) \left (3 b c d (b e-c d) \left (b^3 e^2 (4 B d-7 A e)+b^2 c d e (2 A e-23 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )-(b+c x) \left (c^2 d^3 \left (11 b^2 c e (13 A e+8 B d)-12 b c^2 d (13 A e+2 B d)+48 A c^3 d^2-99 b^3 B e^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c (d+e x)}{c d-b e}\right )-(c d-b e)^3 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {e x}{d}+1\right ) \left (5 b^2 e (7 A e-4 B d)+12 b c d (5 A e-2 B d)+48 A c^2 d^2\right )\right )\right )\right )}{12 b^5 d^3 x^2 (b+c x)^2 (d+e x)^{3/2} (c d-b e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(6*A*b^4*d^2*(-(c*d) + b*e)^3 + 3*b^3*d*(-(c*d) + b*e)^3*(4*b*B*d - 8*A*c*d - 7*A*b*e)*x + x^2*(3*b^2*c*d*(c*d
- b*e)^2*(12*A*c^2*d^2 + b^2*e*(4*B*d - 7*A*e) - 3*b*c*d*(2*B*d + A*e)) + (b + c*x)*(3*b*c*d*(-(c*d) + b*e)*(
-24*A*c^3*d^3 + b^3*e^2*(4*B*d - 7*A*e) + b^2*c*d*e*(-23*B*d + 2*A*e) + 12*b*c^2*d^2*(B*d + 3*A*e)) - (b + c*x
)*(c^2*d^3*(48*A*c^3*d^2 - 99*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 13*A*e) + 11*b^2*c*e*(8*B*d + 13*A*e))*Hypergeom
etric2F1[-3/2, 1, -1/2, (c*(d + e*x))/(c*d - b*e)] - (c*d - b*e)^3*(48*A*c^2*d^2 + 12*b*c*d*(-2*B*d + 5*A*e) +
5*b^2*e*(-4*B*d + 7*A*e))*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (e*x)/d]))))/(12*b^5*d^3*(c*d - b*e)^3*x^2*(b
+ c*x)^2*(d + e*x)^(3/2))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 2.70, size = 1590, normalized size = 2.47 \begin {gather*} -\frac {72 A c^7 (d+e x)^2 d^8-36 b B c^6 (d+e x)^2 d^8-8 b^4 B c^3 e^3 d^7-216 A c^7 (d+e x)^3 d^7+108 b B c^6 (d+e x)^3 d^7-288 A b c^6 e (d+e x)^2 d^7+159 b^2 B c^5 e (d+e x)^2 d^7+8 A b^4 c^3 e^4 d^6+24 b^5 B c^2 e^4 d^6+216 A c^7 (d+e x)^4 d^6-108 b B c^6 (d+e x)^4 d^6+756 A b c^6 e (d+e x)^3 d^6-423 b^2 B c^5 e (d+e x)^3 d^6+381 A b^2 c^5 e^2 (d+e x)^2 d^6-243 b^3 B c^4 e^2 (d+e x)^2 d^6-88 b^4 B c^3 e^3 (d+e x) d^6-24 A b^5 c^2 e^5 d^5-24 b^6 B c e^5 d^5-72 A c^7 (d+e x)^5 d^5+36 b B c^6 (d+e x)^5 d^5-648 A b c^6 e (d+e x)^4 d^5+369 b^2 B c^5 e (d+e x)^4 d^5-822 A b^2 c^5 e^2 (d+e x)^3 d^5+546 b^3 B c^4 e^2 (d+e x)^3 d^5-135 A b^3 c^4 e^3 (d+e x)^2 d^5+672 b^4 B c^3 e^3 (d+e x)^2 d^5+112 A b^4 c^3 e^4 (d+e x) d^5+208 b^5 B c^2 e^4 (d+e x) d^5+8 b^7 B e^6 d^4+24 A b^6 c e^6 d^4+180 A b c^6 e (d+e x)^5 d^4-105 b^2 B c^5 e (d+e x)^5 d^4+525 A b^2 c^5 e^2 (d+e x)^4 d^4-375 b^3 B c^4 e^2 (d+e x)^4 d^4+165 A b^3 c^4 e^3 (d+e x)^3 d^4-1168 b^4 B c^3 e^3 (d+e x)^3 d^4-768 A b^4 c^3 e^4 (d+e x)^2 d^4-996 b^5 B c^2 e^4 (d+e x)^2 d^4-280 A b^5 c^2 e^5 (d+e x) d^4-152 b^6 B c e^5 (d+e x) d^4-8 A b^7 e^7 d^3-84 A b^2 c^5 e^2 (d+e x)^5 d^3+72 b^3 B c^4 e^2 (d+e x)^5 d^3+30 A b^3 c^4 e^3 (d+e x)^4 d^3+760 b^4 B c^3 e^3 (d+e x)^4 d^3+1372 A b^4 c^3 e^4 (d+e x)^3 d^3+1260 b^5 B c^2 e^4 (d+e x)^3 d^3+1425 A b^5 c^2 e^5 (d+e x)^2 d^3+544 b^6 B c e^5 (d+e x)^2 d^3+32 b^7 B e^6 (d+e x) d^3+224 A b^6 c e^6 (d+e x) d^3-54 A b^3 c^4 e^3 (d+e x)^5 d^2-168 b^4 B c^3 e^3 (d+e x)^5 d^2-988 A b^4 c^3 e^4 (d+e x)^4 d^2-556 b^5 B c^2 e^4 (d+e x)^4 d^2-1845 A b^5 c^2 e^5 (d+e x)^3 d^2-488 b^6 B c e^5 (d+e x)^3 d^2-100 b^7 B e^6 (d+e x)^2 d^2-862 A b^6 c e^6 (d+e x)^2 d^2-56 A b^7 e^7 (d+e x) d^2+240 A b^4 c^3 e^4 (d+e x)^5 d+60 b^5 B c^2 e^4 (d+e x)^5 d+865 A b^5 c^2 e^5 (d+e x)^4 d+120 b^6 B c e^5 (d+e x)^4 d+60 b^7 B e^6 (d+e x)^3 d+800 A b^6 c e^6 (d+e x)^3 d+175 A b^7 e^7 (d+e x)^2 d-105 A b^5 c^2 e^5 (d+e x)^5-210 A b^6 c e^6 (d+e x)^4-105 A b^7 e^7 (d+e x)^3}{12 b^4 d^4 e (b e-c d)^4 x^2 (d+e x)^{3/2} (-c d+b e+c (d+e x))^2}+\frac {\left (48 A d^2 c^{13/2}-24 b B d^2 c^{11/2}-156 A b d e c^{11/2}+143 A b^2 e^2 c^{9/2}+88 b^2 B d e c^{9/2}-99 b^3 B e^2 c^{7/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {b e-c d} \sqrt {d+e x}}{c d-b e}\right )}{4 b^5 (b e-c d)^{9/2}}+\frac {\left (-35 A e^2 b^2+20 B d e b^2+24 B c d^2 b-60 A c d e b-48 A c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

-1/12*(-8*b^4*B*c^3*d^7*e^3 + 24*b^5*B*c^2*d^6*e^4 + 8*A*b^4*c^3*d^6*e^4 - 24*b^6*B*c*d^5*e^5 - 24*A*b^5*c^2*d
^5*e^5 + 8*b^7*B*d^4*e^6 + 24*A*b^6*c*d^4*e^6 - 8*A*b^7*d^3*e^7 - 88*b^4*B*c^3*d^6*e^3*(d + e*x) + 208*b^5*B*c
^2*d^5*e^4*(d + e*x) + 112*A*b^4*c^3*d^5*e^4*(d + e*x) - 152*b^6*B*c*d^4*e^5*(d + e*x) - 280*A*b^5*c^2*d^4*e^5
*(d + e*x) + 32*b^7*B*d^3*e^6*(d + e*x) + 224*A*b^6*c*d^3*e^6*(d + e*x) - 56*A*b^7*d^2*e^7*(d + e*x) - 36*b*B*
c^6*d^8*(d + e*x)^2 + 72*A*c^7*d^8*(d + e*x)^2 + 159*b^2*B*c^5*d^7*e*(d + e*x)^2 - 288*A*b*c^6*d^7*e*(d + e*x)
^2 - 243*b^3*B*c^4*d^6*e^2*(d + e*x)^2 + 381*A*b^2*c^5*d^6*e^2*(d + e*x)^2 + 672*b^4*B*c^3*d^5*e^3*(d + e*x)^2
- 135*A*b^3*c^4*d^5*e^3*(d + e*x)^2 - 996*b^5*B*c^2*d^4*e^4*(d + e*x)^2 - 768*A*b^4*c^3*d^4*e^4*(d + e*x)^2 +
544*b^6*B*c*d^3*e^5*(d + e*x)^2 + 1425*A*b^5*c^2*d^3*e^5*(d + e*x)^2 - 100*b^7*B*d^2*e^6*(d + e*x)^2 - 862*A*
b^6*c*d^2*e^6*(d + e*x)^2 + 175*A*b^7*d*e^7*(d + e*x)^2 + 108*b*B*c^6*d^7*(d + e*x)^3 - 216*A*c^7*d^7*(d + e*x
)^3 - 423*b^2*B*c^5*d^6*e*(d + e*x)^3 + 756*A*b*c^6*d^6*e*(d + e*x)^3 + 546*b^3*B*c^4*d^5*e^2*(d + e*x)^3 - 82
2*A*b^2*c^5*d^5*e^2*(d + e*x)^3 - 1168*b^4*B*c^3*d^4*e^3*(d + e*x)^3 + 165*A*b^3*c^4*d^4*e^3*(d + e*x)^3 + 126
0*b^5*B*c^2*d^3*e^4*(d + e*x)^3 + 1372*A*b^4*c^3*d^3*e^4*(d + e*x)^3 - 488*b^6*B*c*d^2*e^5*(d + e*x)^3 - 1845*
A*b^5*c^2*d^2*e^5*(d + e*x)^3 + 60*b^7*B*d*e^6*(d + e*x)^3 + 800*A*b^6*c*d*e^6*(d + e*x)^3 - 105*A*b^7*e^7*(d
+ e*x)^3 - 108*b*B*c^6*d^6*(d + e*x)^4 + 216*A*c^7*d^6*(d + e*x)^4 + 369*b^2*B*c^5*d^5*e*(d + e*x)^4 - 648*A*b
*c^6*d^5*e*(d + e*x)^4 - 375*b^3*B*c^4*d^4*e^2*(d + e*x)^4 + 525*A*b^2*c^5*d^4*e^2*(d + e*x)^4 + 760*b^4*B*c^3
*d^3*e^3*(d + e*x)^4 + 30*A*b^3*c^4*d^3*e^3*(d + e*x)^4 - 556*b^5*B*c^2*d^2*e^4*(d + e*x)^4 - 988*A*b^4*c^3*d^
2*e^4*(d + e*x)^4 + 120*b^6*B*c*d*e^5*(d + e*x)^4 + 865*A*b^5*c^2*d*e^5*(d + e*x)^4 - 210*A*b^6*c*e^6*(d + e*x
)^4 + 36*b*B*c^6*d^5*(d + e*x)^5 - 72*A*c^7*d^5*(d + e*x)^5 - 105*b^2*B*c^5*d^4*e*(d + e*x)^5 + 180*A*b*c^6*d^
4*e*(d + e*x)^5 + 72*b^3*B*c^4*d^3*e^2*(d + e*x)^5 - 84*A*b^2*c^5*d^3*e^2*(d + e*x)^5 - 168*b^4*B*c^3*d^2*e^3*
(d + e*x)^5 - 54*A*b^3*c^4*d^2*e^3*(d + e*x)^5 + 60*b^5*B*c^2*d*e^4*(d + e*x)^5 + 240*A*b^4*c^3*d*e^4*(d + e*x
)^5 - 105*A*b^5*c^2*e^5*(d + e*x)^5)/(b^4*d^4*e*(-(c*d) + b*e)^4*x^2*(d + e*x)^(3/2)*(-(c*d) + b*e + c*(d + e*
x))^2) + ((-24*b*B*c^(11/2)*d^2 + 48*A*c^(13/2)*d^2 + 88*b^2*B*c^(9/2)*d*e - 156*A*b*c^(11/2)*d*e - 99*b^3*B*c
^(7/2)*e^2 + 143*A*b^2*c^(9/2)*e^2)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4*b^5*(-(
c*d) + b*e)^(9/2)) + ((24*b*B*c*d^2 - 48*A*c^2*d^2 + 20*b^2*B*d*e - 60*A*b*c*d*e - 35*A*b^2*e^2)*ArcTanh[Sqrt[
d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2))

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [B]  time = 1.09, size = 1600, normalized size = 2.48

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/4*(24*B*b*c^6*d^2 - 48*A*c^7*d^2 - 88*B*b^2*c^5*d*e + 156*A*b*c^6*d*e + 99*B*b^3*c^4*e^2 - 143*A*b^2*c^5*e^2
)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^4*d^4 - 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d
*e^3 + b^9*e^4)*sqrt(-c^2*d + b*c*e)) + 2/3*(15*(x*e + d)*B*c*d^2*e^4 + B*c*d^3*e^4 - 6*(x*e + d)*B*b*d*e^5 -
18*(x*e + d)*A*c*d*e^5 - B*b*d^2*e^5 - A*c*d^2*e^5 + 9*(x*e + d)*A*b*e^6 + A*b*d*e^6)/((c^4*d^8 - 4*b*c^3*d^7*
e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*(x*e + d)^(3/2)) - 1/4*(12*(x*e + d)^(7/2)*B*b*c^6*d^5*
e - 24*(x*e + d)^(7/2)*A*c^7*d^5*e - 36*(x*e + d)^(5/2)*B*b*c^6*d^6*e + 72*(x*e + d)^(5/2)*A*c^7*d^6*e + 36*(x
*e + d)^(3/2)*B*b*c^6*d^7*e - 72*(x*e + d)^(3/2)*A*c^7*d^7*e - 12*sqrt(x*e + d)*B*b*c^6*d^8*e + 24*sqrt(x*e +
d)*A*c^7*d^8*e - 35*(x*e + d)^(7/2)*B*b^2*c^5*d^4*e^2 + 60*(x*e + d)^(7/2)*A*b*c^6*d^4*e^2 + 123*(x*e + d)^(5/
2)*B*b^2*c^5*d^5*e^2 - 216*(x*e + d)^(5/2)*A*b*c^6*d^5*e^2 - 141*(x*e + d)^(3/2)*B*b^2*c^5*d^6*e^2 + 252*(x*e
+ d)^(3/2)*A*b*c^6*d^6*e^2 + 53*sqrt(x*e + d)*B*b^2*c^5*d^7*e^2 - 96*sqrt(x*e + d)*A*b*c^6*d^7*e^2 + 24*(x*e +
d)^(7/2)*B*b^3*c^4*d^3*e^3 - 28*(x*e + d)^(7/2)*A*b^2*c^5*d^3*e^3 - 125*(x*e + d)^(5/2)*B*b^3*c^4*d^4*e^3 + 1
75*(x*e + d)^(5/2)*A*b^2*c^5*d^4*e^3 + 182*(x*e + d)^(3/2)*B*b^3*c^4*d^5*e^3 - 274*(x*e + d)^(3/2)*A*b^2*c^5*d
^5*e^3 - 81*sqrt(x*e + d)*B*b^3*c^4*d^6*e^3 + 127*sqrt(x*e + d)*A*b^2*c^5*d^6*e^3 - 16*(x*e + d)^(7/2)*B*b^4*c
^3*d^2*e^4 - 18*(x*e + d)^(7/2)*A*b^3*c^4*d^2*e^4 + 96*(x*e + d)^(5/2)*B*b^4*c^3*d^3*e^4 + 10*(x*e + d)^(5/2)*
A*b^3*c^4*d^3*e^4 - 160*(x*e + d)^(3/2)*B*b^4*c^3*d^4*e^4 + 55*(x*e + d)^(3/2)*A*b^3*c^4*d^4*e^4 + 80*sqrt(x*e
+ d)*B*b^4*c^3*d^5*e^4 - 45*sqrt(x*e + d)*A*b^3*c^4*d^5*e^4 + 4*(x*e + d)^(7/2)*B*b^5*c^2*d*e^5 + 32*(x*e + d
)^(7/2)*A*b^4*c^3*d*e^5 - 44*(x*e + d)^(5/2)*B*b^5*c^2*d^2*e^5 - 140*(x*e + d)^(5/2)*A*b^4*c^3*d^2*e^5 + 100*(
x*e + d)^(3/2)*B*b^5*c^2*d^3*e^5 + 180*(x*e + d)^(3/2)*A*b^4*c^3*d^3*e^5 - 60*sqrt(x*e + d)*B*b^5*c^2*d^4*e^5
- 80*sqrt(x*e + d)*A*b^4*c^3*d^4*e^5 - 11*(x*e + d)^(7/2)*A*b^5*c^2*e^6 + 8*(x*e + d)^(5/2)*B*b^6*c*d*e^6 + 99
*(x*e + d)^(5/2)*A*b^5*c^2*d*e^6 - 32*(x*e + d)^(3/2)*B*b^6*c*d^2*e^6 - 199*(x*e + d)^(3/2)*A*b^5*c^2*d^2*e^6
+ 24*sqrt(x*e + d)*B*b^6*c*d^3*e^6 + 123*sqrt(x*e + d)*A*b^5*c^2*d^3*e^6 - 22*(x*e + d)^(5/2)*A*b^6*c*e^7 + 4*
(x*e + d)^(3/2)*B*b^7*d*e^7 + 80*(x*e + d)^(3/2)*A*b^6*c*d*e^7 - 4*sqrt(x*e + d)*B*b^7*d^2*e^7 - 66*sqrt(x*e +
d)*A*b^6*c*d^2*e^7 - 11*(x*e + d)^(3/2)*A*b^7*e^8 + 13*sqrt(x*e + d)*A*b^7*d*e^8)/((b^4*c^4*d^8 - 4*b^5*c^3*d
^7*e + 6*b^6*c^2*d^6*e^2 - 4*b^7*c*d^5*e^3 + b^8*d^4*e^4)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)
*b*e - b*d*e)^2) - 1/4*(24*B*b*c*d^2 - 48*A*c^2*d^2 + 20*B*b^2*d*e - 60*A*b*c*d*e - 35*A*b^2*e^2)*arctan(sqrt(
x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^4)

________________________________________________________________________________________

maple [A]  time = 0.09, size = 1130, normalized size = 1.75

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x)

[Out]

-29/4*e^2*c^5/(b*e-c*d)^4/b^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d+2*e*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*B*(e*x+d)^
(1/2)*d^2+39*e*c^6/(b*e-c*d)^4/b^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d-22*e*c^
5/(b*e-c*d)^4/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d-3*e*c^7/(b*e-c*d)^4/b^4/
(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^2-2*e*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d+3*e*c^7/(b*e-c*d)^4/
b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d+37/4*e^2*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d-35/4*e^2/b^3/
d^(9/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A+5*e/b^3/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B-12/b^5/d^(5/2)*arcta
nh((e*x+d)^(1/2)/d^(1/2))*A*c^2+6/b^4/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c+11/4/b^3/d^4/x^2*A*(e*x+d)^(3
/2)-13/4/b^3/d^3/x^2*(e*x+d)^(1/2)*A-2/3*e^4/(b*e-c*d)^3/d^2/(e*x+d)^(3/2)*B+2/3*e^5/(b*e-c*d)^3/d^3/(e*x+d)^(
3/2)*A-1/e/b^3/d^3/x^2*B*(e*x+d)^(3/2)+1/e/b^3/d^2/x^2*(e*x+d)^(1/2)*B+10*e^4/(b*e-c*d)^4/d^2/(e*x+d)^(1/2)*B*
c+6*e^6/(b*e-c*d)^4/d^4/(e*x+d)^(1/2)*A*b-12*e^5/(b*e-c*d)^4/d^3/(e*x+d)^(1/2)*A*c-4*e^5/(b*e-c*d)^4/d^3/(e*x+
d)^(1/2)*B*b-15*e/b^4/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-3/e/b^4/d^2/x^2*(e*x+d)^(1/2)*A*c-12*c^7/(b*e
-c*d)^4/b^5/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^2+6*c^6/(b*e-c*d)^4/b^4/((b*e-
c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^2-23/4*e^2*c^6/(b*e-c*d)^4/b^3/(c*e*x+b*e)^2*(e*
x+d)^(3/2)*A+19/4*e^2*c^5/(b*e-c*d)^4/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B-25/4*e^3*c^5/(b*e-c*d)^4/b^2/(c*e*x+b*
e)^2*A*(e*x+d)^(1/2)+21/4*e^3*c^4/(b*e-c*d)^4/b/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)-143/4*e^2*c^5/(b*e-c*d)^4/b^3/((
b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A+99/4*e^2*c^4/(b*e-c*d)^4/b^2/((b*e-c*d)*c)^(1/
2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B+3/e/b^4/d^3/x^2*A*(e*x+d)^(3/2)*c

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

________________________________________________________________________________________

mupad [B]  time = 6.93, size = 24572, normalized size = 38.16

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((b*x + c*x^2)^3*(d + e*x)^(5/2)),x)

[Out]

log(14598144*A^3*b^9*c^27*d^32*e^4 - 884736*A^3*b^8*c^28*d^33*e^3 - ((d + e*x)^(1/2)*(589824*A^2*b^12*c^27*d^3
6*e^2 - 10616832*A^2*b^13*c^26*d^35*e^3 + 89518080*A^2*b^14*c^25*d^34*e^4 - 468971520*A^2*b^15*c^24*d^33*e^5 +
1707439360*A^2*b^16*c^23*d^32*e^6 - 4579446784*A^2*b^17*c^22*d^31*e^7 + 9364822016*A^2*b^18*c^21*d^30*e^8 - 1
4937190400*A^2*b^19*c^20*d^29*e^9 + 18936107520*A^2*b^20*c^19*d^28*e^10 - 19535324160*A^2*b^21*c^18*d^27*e^11
+ 17074641408*A^2*b^22*c^17*d^26*e^12 - 13484230656*A^2*b^23*c^16*d^25*e^13 + 10265639040*A^2*b^24*c^15*d^24*e
^14 - 7643066880*A^2*b^25*c^14*d^23*e^15 + 5421597440*A^2*b^26*c^13*d^22*e^16 - 3708136960*A^2*b^27*c^12*d^21*
e^17 + 2608529792*A^2*b^28*c^11*d^20*e^18 - 1894041600*A^2*b^29*c^10*d^19*e^19 + 1274465280*A^2*b^30*c^9*d^18*
e^20 - 707773440*A^2*b^31*c^8*d^17*e^21 + 301648512*A^2*b^32*c^7*d^16*e^22 - 93688320*A^2*b^33*c^6*d^15*e^23 +
19930880*A^2*b^34*c^5*d^14*e^24 - 2598400*A^2*b^35*c^4*d^13*e^25 + 156800*A^2*b^36*c^3*d^12*e^26 + 147456*B^2
*b^14*c^25*d^36*e^2 - 2777088*B^2*b^15*c^24*d^35*e^3 + 24555520*B^2*b^16*c^23*d^34*e^4 - 135055360*B^2*b^17*c^
22*d^33*e^5 + 515884160*B^2*b^18*c^21*d^32*e^6 - 1446258176*B^2*b^19*c^20*d^31*e^7 + 3062171904*B^2*b^20*c^19*
d^30*e^8 - 4951119360*B^2*b^21*c^18*d^29*e^9 + 6076371840*B^2*b^22*c^17*d^28*e^10 - 5478190080*B^2*b^23*c^16*d
^27*e^11 + 3273549312*B^2*b^24*c^15*d^26*e^12 - 766116864*B^2*b^25*c^14*d^25*e^13 - 668122240*B^2*b^26*c^13*d^
24*e^14 + 721318400*B^2*b^27*c^12*d^23*e^15 - 107134720*B^2*b^28*c^11*d^22*e^16 - 366558720*B^2*b^29*c^10*d^21
*e^17 + 437847168*B^2*b^30*c^9*d^20*e^18 - 282501120*B^2*b^31*c^8*d^19*e^19 + 121989120*B^2*b^32*c^7*d^18*e^20
- 36495360*B^2*b^33*c^6*d^17*e^21 + 7344128*B^2*b^34*c^5*d^16*e^22 - 901120*B^2*b^35*c^4*d^15*e^23 + 51200*B^
2*b^36*c^3*d^14*e^24 - 589824*A*B*b^13*c^26*d^36*e^2 + 10862592*A*B*b^14*c^25*d^35*e^3 - 93818880*A*B*b^15*c^2
4*d^34*e^4 + 503726080*A*B*b^16*c^23*d^33*e^5 - 1878764800*A*B*b^17*c^22*d^32*e^6 + 5151263744*A*B*b^18*c^21*d
^31*e^7 - 10713545216*A*B*b^19*c^20*d^30*e^8 + 17186104320*A*B*b^20*c^19*d^29*e^9 - 21406851840*A*B*b^21*c^18*
d^28*e^10 + 20693207040*A*B*b^22*c^17*d^27*e^11 - 15463523328*A*B*b^23*c^16*d^26*e^12 + 8955257856*A*B*b^24*c^
15*d^25*e^13 - 4111491840*A*B*b^25*c^14*d^24*e^14 + 1413002240*A*B*b^26*c^13*d^23*e^15 + 178449920*A*B*b^27*c^
12*d^22*e^16 - 1280942080*A*B*b^28*c^11*d^21*e^17 + 1742746368*A*B*b^29*c^10*d^20*e^18 - 1489551360*A*B*b^30*c
^9*d^19*e^19 + 892446720*A*B*b^31*c^8*d^18*e^20 - 383708160*A*B*b^32*c^7*d^17*e^21 + 117055488*A*B*b^33*c^6*d^
16*e^22 - 24217600*A*B*b^34*c^5*d^15*e^23 + 3061760*A*B*b^35*c^4*d^14*e^24 - 179200*A*B*b^36*c^3*d^13*e^25) -
((1225*A^2*b^4*e^4 + 2304*A^2*c^4*d^4 + 576*B^2*b^2*c^2*d^4 + 400*B^2*b^4*d^2*e^2 + 6960*A^2*b^2*c^2*d^2*e^2 +
5760*A^2*b*c^3*d^3*e + 4200*A^2*b^3*c*d*e^3 + 960*B^2*b^3*c*d^3*e - 2304*A*B*b*c^3*d^4 - 1400*A*B*b^4*d*e^3 -
4800*A*B*b^2*c^2*d^3*e - 4080*A*B*b^3*c*d^2*e^2)/(64*b^10*d^9))^(1/2)*((d + e*x)^(1/2)*((1225*A^2*b^4*e^4 + 2
304*A^2*c^4*d^4 + 576*B^2*b^2*c^2*d^4 + 400*B^2*b^4*d^2*e^2 + 6960*A^2*b^2*c^2*d^2*e^2 + 5760*A^2*b*c^3*d^3*e
+ 4200*A^2*b^3*c*d*e^3 + 960*B^2*b^3*c*d^3*e - 2304*A*B*b*c^3*d^4 - 1400*A*B*b^4*d*e^3 - 4800*A*B*b^2*c^2*d^3*
e - 4080*A*B*b^3*c*d^2*e^2)/(64*b^10*d^9))^(1/2)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 32768
00*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e
^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 378380288
0*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11
*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 2
06389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^2
2*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23) + 24576*A*b^18*c^24*d^38*e^3 - 466944*A*b^19*c^2
3*d^37*e^4 + 4185088*A*b^20*c^22*d^36*e^5 - 23500800*A*b^21*c^21*d^35*e^6 + 92710912*A*b^22*c^20*d^34*e^7 - 27
3566720*A*b^23*c^19*d^33*e^8 + 629578752*A*b^24*c^18*d^32*e^9 - 1169833984*A*b^25*c^17*d^31*e^10 + 1818910720*
A*b^26*c^16*d^30*e^11 - 2465058816*A*b^27*c^15*d^29*e^12 + 3031169024*A*b^28*c^14*d^28*e^13 - 3457871872*A*b^2
9*c^13*d^27*e^14 + 3626348544*A*b^30*c^12*d^26*e^15 - 3385559040*A*b^31*c^11*d^25*e^16 + 2714064896*A*b^32*c^1
0*d^24*e^17 - 1813512192*A*b^33*c^9*d^23*e^18 + 986251264*A*b^34*c^8*d^22*e^19 - 426815488*A*b^35*c^7*d^21*e^2
0 + 143109120*A*b^36*c^6*d^20*e^21 - 35796992*A*b^37*c^5*d^19*e^22 + 6285312*A*b^38*c^4*d^18*e^23 - 691200*A*b
^39*c^3*d^17*e^24 + 35840*A*b^40*c^2*d^16*e^25 - 12288*B*b^19*c^23*d^38*e^3 + 238592*B*b^20*c^22*d^37*e^4 - 21
87264*B*b^21*c^21*d^36*e^5 + 12492800*B*b^22*c^20*d^35*e^6 - 49401856*B*b^23*c^19*d^34*e^7 + 141926400*B*b^24*
c^18*d^33*e^8 - 300793856*B*b^25*c^17*d^32*e^9 + 460562432*B*b^26*c^16*d^31*e^10 - 455516160*B*b^27*c^15*d^30*
e^11 + 116267008*B*b^28*c^14*d^29*e^12 + 543981568*B*b^29*c^13*d^28*e^13 - 1250156544*B*b^30*c^12*d^27*e^14 +
1639292928*B*b^31*c^11*d^26*e^15 - 1547694080*B*b^32*c^10*d^25*e^16 + 1115799552*B*b^33*c^9*d^24*e^17 - 624861
184*B*b^34*c^8*d^23*e^18 + 271372288*B*b^35*c^7*d^22*e^19 - 89988096*B*b^36*c^6*d^21*e^20 + 22077440*B*b^37*c^
5*d^20*e^21 - 3784704*B*b^38*c^4*d^19*e^22 + 405504*B*b^39*c^3*d^18*e^23 - 20480*B*b^40*c^2*d^17*e^24))*((1225
*A^2*b^4*e^4 + 2304*A^2*c^4*d^4 + 576*B^2*b^2*c^2*d^4 + 400*B^2*b^4*d^2*e^2 + 6960*A^2*b^2*c^2*d^2*e^2 + 5760*
A^2*b*c^3*d^3*e + 4200*A^2*b^3*c*d*e^3 + 960*B^2*b^3*c*d^3*e - 2304*A*B*b*c^3*d^4 - 1400*A*B*b^4*d*e^3 - 4800*
A*B*b^2*c^2*d^3*e - 4080*A*B*b^3*c*d^2*e^2)/(64*b^10*d^9))^(1/2) - 111310848*A^3*b^10*c^26*d^31*e^5 + 51853824
0*A^3*b^11*c^25*d^30*e^6 - 1640557440*A^3*b^12*c^24*d^29*e^7 + 3692369088*A^3*b^13*c^23*d^28*e^8 - 5970365632*
A^3*b^14*c^22*d^27*e^9 + 6695810784*A^3*b^15*c^21*d^26*e^10 - 4411189120*A^3*b^16*c^20*d^25*e^11 - 87084400*A^
3*b^17*c^19*d^24*e^12 + 3954268032*A^3*b^18*c^18*d^23*e^13 - 5135394368*A^3*b^19*c^17*d^22*e^14 + 4434262976*A
^3*b^20*c^16*d^21*e^15 - 4011472080*A^3*b^21*c^15*d^20*e^16 + 4506553920*A^3*b^22*c^14*d^19*e^17 - 4740529184*
A^3*b^23*c^13*d^18*e^18 + 3806470656*A^3*b^24*c^12*d^17*e^19 - 2198096912*A^3*b^25*c^11*d^16*e^20 + 886408960*
A^3*b^26*c^10*d^15*e^21 - 237886080*A^3*b^27*c^9*d^14*e^22 + 38292800*A^3*b^28*c^8*d^13*e^23 - 2802800*A^3*b^2
9*c^7*d^12*e^24 + 110592*B^3*b^11*c^25*d^33*e^3 - 1963008*B^3*b^12*c^24*d^32*e^4 + 16183296*B^3*b^13*c^23*d^31
*e^5 - 82448000*B^3*b^14*c^22*d^30*e^6 + 291430080*B^3*b^15*c^21*d^29*e^7 - 760810496*B^3*b^16*c^20*d^28*e^8 +
1523208064*B^3*b^17*c^19*d^27*e^9 - 2387603328*B^3*b^18*c^18*d^26*e^10 + 2934367040*B^3*b^19*c^17*d^25*e^11 -
2735068160*B^3*b^20*c^16*d^24*e^12 + 1688898816*B^3*b^21*c^15*d^23*e^13 - 207986304*B^3*b^22*c^14*d^22*e^14 -
992919232*B^3*b^23*c^13*d^21*e^15 + 1419909120*B^3*b^24*c^12*d^20*e^16 - 1147707520*B^3*b^25*c^11*d^19*e^17 +
629449088*B^3*b^26*c^10*d^18*e^18 - 238930752*B^3*b^27*c^9*d^17*e^19 + 60427264*B^3*b^28*c^8*d^16*e^20 - 9180
160*B^3*b^29*c^7*d^15*e^21 + 633600*B^3*b^30*c^6*d^14*e^22 - 663552*A*B^2*b^10*c^26*d^33*e^3 + 11501568*A*B^2*
b^11*c^25*d^32*e^4 - 92445696*A*B^2*b^12*c^24*d^31*e^5 + 457608960*A*B^2*b^13*c^23*d^30*e^6 - 1561961280*A*B^2
*b^14*c^22*d^29*e^7 + 3897633456*A*B^2*b^15*c^21*d^28*e^8 - 7341910464*A*B^2*b^16*c^20*d^27*e^9 + 10584928608*
A*B^2*b^17*c^19*d^26*e^10 - 11615091840*A*B^2*b^18*c^18*d^25*e^11 + 9351305040*A*B^2*b^19*c^17*d^24*e^12 - 494
9763456*A*B^2*b^20*c^16*d^23*e^13 + 1152719424*A*B^2*b^21*c^15*d^22*e^14 + 35479872*A*B^2*b^22*c^14*d^21*e^15
+ 987243600*A*B^2*b^23*c^13*d^20*e^16 - 2238056640*A*B^2*b^24*c^12*d^19*e^17 + 2350093152*A*B^2*b^25*c^11*d^18
*e^18 - 1531638528*A*B^2*b^26*c^10*d^17*e^19 + 658359216*A*B^2*b^27*c^9*d^16*e^20 - 183198720*A*B^2*b^28*c^8*d
^15*e^21 + 30074880*A*B^2*b^29*c^7*d^14*e^22 - 2217600*A*B^2*b^30*c^6*d^13*e^23 + 1327104*A^2*B*b^9*c^27*d^33*
e^3 - 22450176*A^2*B*b^10*c^26*d^32*e^4 + 175813632*A^2*B*b^11*c^25*d^31*e^5 - 844727040*A^2*B*b^12*c^24*d^30*
e^6 + 2778960960*A^2*B*b^13*c^23*d^29*e^7 - 6601799472*A^2*B*b^14*c^22*d^28*e^8 + 11593951488*A^2*B*b^15*c^21*
d^27*e^9 - 15030223296*A^2*B*b^16*c^20*d^26*e^10 + 13855558080*A^2*B*b^17*c^19*d^25*e^11 - 7973238240*A^2*B*b^
18*c^18*d^24*e^12 + 1330213632*A^2*B*b^19*c^17*d^23*e^13 + 1474407552*A^2*B*b^20*c^16*d^22*e^14 + 280293696*A^
2*B*b^21*c^15*d^21*e^15 - 3189392640*A^2*B*b^22*c^14*d^20*e^16 + 3911942400*A^2*B*b^23*c^13*d^19*e^17 - 236024
0064*A^2*B*b^24*c^12*d^18*e^18 + 534716736*A^2*B*b^25*c^11*d^17*e^19 + 282511968*A^2*B*b^26*c^10*d^16*e^20 - 2
90822400*A^2*B*b^27*c^9*d^15*e^21 + 114170880*A^2*B*b^28*c^8*d^14*e^22 - 22915200*A^2*B*b^29*c^7*d^13*e^23 + 1
940400*A^2*B*b^30*c^6*d^12*e^24)*((1225*A^2*b^4*e^4 + 2304*A^2*c^4*d^4 + 576*B^2*b^2*c^2*d^4 + 400*B^2*b^4*d^2
*e^2 + 6960*A^2*b^2*c^2*d^2*e^2 + 5760*A^2*b*c^3*d^3*e + 4200*A^2*b^3*c*d*e^3 + 960*B^2*b^3*c*d^3*e - 2304*A*B
*b*c^3*d^4 - 1400*A*B*b^4*d*e^3 - 4800*A*B*b^2*c^2*d^3*e - 4080*A*B*b^3*c*d^2*e^2)/(64*b^10*d^9))^(1/2) - log(
((d + e*x)^(1/2)*(589824*A^2*b^12*c^27*d^36*e^2 - 10616832*A^2*b^13*c^26*d^35*e^3 + 89518080*A^2*b^14*c^25*d^3
4*e^4 - 468971520*A^2*b^15*c^24*d^33*e^5 + 1707439360*A^2*b^16*c^23*d^32*e^6 - 4579446784*A^2*b^17*c^22*d^31*e
^7 + 9364822016*A^2*b^18*c^21*d^30*e^8 - 14937190400*A^2*b^19*c^20*d^29*e^9 + 18936107520*A^2*b^20*c^19*d^28*e
^10 - 19535324160*A^2*b^21*c^18*d^27*e^11 + 17074641408*A^2*b^22*c^17*d^26*e^12 - 13484230656*A^2*b^23*c^16*d^
25*e^13 + 10265639040*A^2*b^24*c^15*d^24*e^14 - 7643066880*A^2*b^25*c^14*d^23*e^15 + 5421597440*A^2*b^26*c^13*
d^22*e^16 - 3708136960*A^2*b^27*c^12*d^21*e^17 + 2608529792*A^2*b^28*c^11*d^20*e^18 - 1894041600*A^2*b^29*c^10
*d^19*e^19 + 1274465280*A^2*b^30*c^9*d^18*e^20 - 707773440*A^2*b^31*c^8*d^17*e^21 + 301648512*A^2*b^32*c^7*d^1
6*e^22 - 93688320*A^2*b^33*c^6*d^15*e^23 + 19930880*A^2*b^34*c^5*d^14*e^24 - 2598400*A^2*b^35*c^4*d^13*e^25 +
156800*A^2*b^36*c^3*d^12*e^26 + 147456*B^2*b^14*c^25*d^36*e^2 - 2777088*B^2*b^15*c^24*d^35*e^3 + 24555520*B^2*
b^16*c^23*d^34*e^4 - 135055360*B^2*b^17*c^22*d^33*e^5 + 515884160*B^2*b^18*c^21*d^32*e^6 - 1446258176*B^2*b^19
*c^20*d^31*e^7 + 3062171904*B^2*b^20*c^19*d^30*e^8 - 4951119360*B^2*b^21*c^18*d^29*e^9 + 6076371840*B^2*b^22*c
^17*d^28*e^10 - 5478190080*B^2*b^23*c^16*d^27*e^11 + 3273549312*B^2*b^24*c^15*d^26*e^12 - 766116864*B^2*b^25*c
^14*d^25*e^13 - 668122240*B^2*b^26*c^13*d^24*e^14 + 721318400*B^2*b^27*c^12*d^23*e^15 - 107134720*B^2*b^28*c^1
1*d^22*e^16 - 366558720*B^2*b^29*c^10*d^21*e^17 + 437847168*B^2*b^30*c^9*d^20*e^18 - 282501120*B^2*b^31*c^8*d^
19*e^19 + 121989120*B^2*b^32*c^7*d^18*e^20 - 36495360*B^2*b^33*c^6*d^17*e^21 + 7344128*B^2*b^34*c^5*d^16*e^22
- 901120*B^2*b^35*c^4*d^15*e^23 + 51200*B^2*b^36*c^3*d^14*e^24 - 589824*A*B*b^13*c^26*d^36*e^2 + 10862592*A*B*
b^14*c^25*d^35*e^3 - 93818880*A*B*b^15*c^24*d^34*e^4 + 503726080*A*B*b^16*c^23*d^33*e^5 - 1878764800*A*B*b^17*
c^22*d^32*e^6 + 5151263744*A*B*b^18*c^21*d^31*e^7 - 10713545216*A*B*b^19*c^20*d^30*e^8 + 17186104320*A*B*b^20*
c^19*d^29*e^9 - 21406851840*A*B*b^21*c^18*d^28*e^10 + 20693207040*A*B*b^22*c^17*d^27*e^11 - 15463523328*A*B*b^
23*c^16*d^26*e^12 + 8955257856*A*B*b^24*c^15*d^25*e^13 - 4111491840*A*B*b^25*c^14*d^24*e^14 + 1413002240*A*B*b
^26*c^13*d^23*e^15 + 178449920*A*B*b^27*c^12*d^22*e^16 - 1280942080*A*B*b^28*c^11*d^21*e^17 + 1742746368*A*B*b
^29*c^10*d^20*e^18 - 1489551360*A*B*b^30*c^9*d^19*e^19 + 892446720*A*B*b^31*c^8*d^18*e^20 - 383708160*A*B*b^32
*c^7*d^17*e^21 + 117055488*A*B*b^33*c^6*d^16*e^22 - 24217600*A*B*b^34*c^5*d^15*e^23 + 3061760*A*B*b^35*c^4*d^1
4*e^24 - 179200*A*B*b^36*c^3*d^13*e^25) - (((1225*A^2*b^4*e^4)/64 + 36*A^2*c^4*d^4 + 9*B^2*b^2*c^2*d^4 + (25*B
^2*b^4*d^2*e^2)/4 + (435*A^2*b^2*c^2*d^2*e^2)/4 + 90*A^2*b*c^3*d^3*e + (525*A^2*b^3*c*d*e^3)/8 + 15*B^2*b^3*c*
d^3*e - 36*A*B*b*c^3*d^4 - (175*A*B*b^4*d*e^3)/8 - 75*A*B*b^2*c^2*d^3*e - (255*A*B*b^3*c*d^2*e^2)/4)/(b^10*d^9
))^(1/2)*((d + e*x)^(1/2)*(((1225*A^2*b^4*e^4)/64 + 36*A^2*c^4*d^4 + 9*B^2*b^2*c^2*d^4 + (25*B^2*b^4*d^2*e^2)/
4 + (435*A^2*b^2*c^2*d^2*e^2)/4 + 90*A^2*b*c^3*d^3*e + (525*A^2*b^3*c*d*e^3)/8 + 15*B^2*b^3*c*d^3*e - 36*A*B*b
*c^3*d^4 - (175*A*B*b^4*d*e^3)/8 - 75*A*B*b^2*c^2*d^3*e - (255*A*B*b^3*c*d^2*e^2)/4)/(b^10*d^9))^(1/2)*(16384*
b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88
719360*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16
*d^34*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 -
4265377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680
*b^36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^1
9 + 12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*
e^23) - 24576*A*b^18*c^24*d^38*e^3 + 466944*A*b^19*c^23*d^37*e^4 - 4185088*A*b^20*c^22*d^36*e^5 + 23500800*A*b
^21*c^21*d^35*e^6 - 92710912*A*b^22*c^20*d^34*e^7 + 273566720*A*b^23*c^19*d^33*e^8 - 629578752*A*b^24*c^18*d^3
2*e^9 + 1169833984*A*b^25*c^17*d^31*e^10 - 1818910720*A*b^26*c^16*d^30*e^11 + 2465058816*A*b^27*c^15*d^29*e^12
- 3031169024*A*b^28*c^14*d^28*e^13 + 3457871872*A*b^29*c^13*d^27*e^14 - 3626348544*A*b^30*c^12*d^26*e^15 + 33
85559040*A*b^31*c^11*d^25*e^16 - 2714064896*A*b^32*c^10*d^24*e^17 + 1813512192*A*b^33*c^9*d^23*e^18 - 98625126
4*A*b^34*c^8*d^22*e^19 + 426815488*A*b^35*c^7*d^21*e^20 - 143109120*A*b^36*c^6*d^20*e^21 + 35796992*A*b^37*c^5
*d^19*e^22 - 6285312*A*b^38*c^4*d^18*e^23 + 691200*A*b^39*c^3*d^17*e^24 - 35840*A*b^40*c^2*d^16*e^25 + 12288*B
*b^19*c^23*d^38*e^3 - 238592*B*b^20*c^22*d^37*e^4 + 2187264*B*b^21*c^21*d^36*e^5 - 12492800*B*b^22*c^20*d^35*e
^6 + 49401856*B*b^23*c^19*d^34*e^7 - 141926400*B*b^24*c^18*d^33*e^8 + 300793856*B*b^25*c^17*d^32*e^9 - 4605624
32*B*b^26*c^16*d^31*e^10 + 455516160*B*b^27*c^15*d^30*e^11 - 116267008*B*b^28*c^14*d^29*e^12 - 543981568*B*b^2
9*c^13*d^28*e^13 + 1250156544*B*b^30*c^12*d^27*e^14 - 1639292928*B*b^31*c^11*d^26*e^15 + 1547694080*B*b^32*c^1
0*d^25*e^16 - 1115799552*B*b^33*c^9*d^24*e^17 + 624861184*B*b^34*c^8*d^23*e^18 - 271372288*B*b^35*c^7*d^22*e^1
9 + 89988096*B*b^36*c^6*d^21*e^20 - 22077440*B*b^37*c^5*d^20*e^21 + 3784704*B*b^38*c^4*d^19*e^22 - 405504*B*b^
39*c^3*d^18*e^23 + 20480*B*b^40*c^2*d^17*e^24))*(((1225*A^2*b^4*e^4)/64 + 36*A^2*c^4*d^4 + 9*B^2*b^2*c^2*d^4 +
(25*B^2*b^4*d^2*e^2)/4 + (435*A^2*b^2*c^2*d^2*e^2)/4 + 90*A^2*b*c^3*d^3*e + (525*A^2*b^3*c*d*e^3)/8 + 15*B^2*
b^3*c*d^3*e - 36*A*B*b*c^3*d^4 - (175*A*B*b^4*d*e^3)/8 - 75*A*B*b^2*c^2*d^3*e - (255*A*B*b^3*c*d^2*e^2)/4)/(b^
10*d^9))^(1/2) - 884736*A^3*b^8*c^28*d^33*e^3 + 14598144*A^3*b^9*c^27*d^32*e^4 - 111310848*A^3*b^10*c^26*d^31*
e^5 + 518538240*A^3*b^11*c^25*d^30*e^6 - 1640557440*A^3*b^12*c^24*d^29*e^7 + 3692369088*A^3*b^13*c^23*d^28*e^8
- 5970365632*A^3*b^14*c^22*d^27*e^9 + 6695810784*A^3*b^15*c^21*d^26*e^10 - 4411189120*A^3*b^16*c^20*d^25*e^11
- 87084400*A^3*b^17*c^19*d^24*e^12 + 3954268032*A^3*b^18*c^18*d^23*e^13 - 5135394368*A^3*b^19*c^17*d^22*e^14
+ 4434262976*A^3*b^20*c^16*d^21*e^15 - 4011472080*A^3*b^21*c^15*d^20*e^16 + 4506553920*A^3*b^22*c^14*d^19*e^17
- 4740529184*A^3*b^23*c^13*d^18*e^18 + 3806470656*A^3*b^24*c^12*d^17*e^19 - 2198096912*A^3*b^25*c^11*d^16*e^2
0 + 886408960*A^3*b^26*c^10*d^15*e^21 - 237886080*A^3*b^27*c^9*d^14*e^22 + 38292800*A^3*b^28*c^8*d^13*e^23 - 2
802800*A^3*b^29*c^7*d^12*e^24 + 110592*B^3*b^11*c^25*d^33*e^3 - 1963008*B^3*b^12*c^24*d^32*e^4 + 16183296*B^3*
b^13*c^23*d^31*e^5 - 82448000*B^3*b^14*c^22*d^30*e^6 + 291430080*B^3*b^15*c^21*d^29*e^7 - 760810496*B^3*b^16*c
^20*d^28*e^8 + 1523208064*B^3*b^17*c^19*d^27*e^9 - 2387603328*B^3*b^18*c^18*d^26*e^10 + 2934367040*B^3*b^19*c^
17*d^25*e^11 - 2735068160*B^3*b^20*c^16*d^24*e^12 + 1688898816*B^3*b^21*c^15*d^23*e^13 - 207986304*B^3*b^22*c^
14*d^22*e^14 - 992919232*B^3*b^23*c^13*d^21*e^15 + 1419909120*B^3*b^24*c^12*d^20*e^16 - 1147707520*B^3*b^25*c^
11*d^19*e^17 + 629449088*B^3*b^26*c^10*d^18*e^18 - 238930752*B^3*b^27*c^9*d^17*e^19 + 60427264*B^3*b^28*c^8*d^
16*e^20 - 9180160*B^3*b^29*c^7*d^15*e^21 + 633600*B^3*b^30*c^6*d^14*e^22 - 663552*A*B^2*b^10*c^26*d^33*e^3 + 1
1501568*A*B^2*b^11*c^25*d^32*e^4 - 92445696*A*B^2*b^12*c^24*d^31*e^5 + 457608960*A*B^2*b^13*c^23*d^30*e^6 - 15
61961280*A*B^2*b^14*c^22*d^29*e^7 + 3897633456*A*B^2*b^15*c^21*d^28*e^8 - 7341910464*A*B^2*b^16*c^20*d^27*e^9
+ 10584928608*A*B^2*b^17*c^19*d^26*e^10 - 11615091840*A*B^2*b^18*c^18*d^25*e^11 + 9351305040*A*B^2*b^19*c^17*d
^24*e^12 - 4949763456*A*B^2*b^20*c^16*d^23*e^13 + 1152719424*A*B^2*b^21*c^15*d^22*e^14 + 35479872*A*B^2*b^22*c
^14*d^21*e^15 + 987243600*A*B^2*b^23*c^13*d^20*e^16 - 2238056640*A*B^2*b^24*c^12*d^19*e^17 + 2350093152*A*B^2*
b^25*c^11*d^18*e^18 - 1531638528*A*B^2*b^26*c^10*d^17*e^19 + 658359216*A*B^2*b^27*c^9*d^16*e^20 - 183198720*A*
B^2*b^28*c^8*d^15*e^21 + 30074880*A*B^2*b^29*c^7*d^14*e^22 - 2217600*A*B^2*b^30*c^6*d^13*e^23 + 1327104*A^2*B*
b^9*c^27*d^33*e^3 - 22450176*A^2*B*b^10*c^26*d^32*e^4 + 175813632*A^2*B*b^11*c^25*d^31*e^5 - 844727040*A^2*B*b
^12*c^24*d^30*e^6 + 2778960960*A^2*B*b^13*c^23*d^29*e^7 - 6601799472*A^2*B*b^14*c^22*d^28*e^8 + 11593951488*A^
2*B*b^15*c^21*d^27*e^9 - 15030223296*A^2*B*b^16*c^20*d^26*e^10 + 13855558080*A^2*B*b^17*c^19*d^25*e^11 - 79732
38240*A^2*B*b^18*c^18*d^24*e^12 + 1330213632*A^2*B*b^19*c^17*d^23*e^13 + 1474407552*A^2*B*b^20*c^16*d^22*e^14
+ 280293696*A^2*B*b^21*c^15*d^21*e^15 - 3189392640*A^2*B*b^22*c^14*d^20*e^16 + 3911942400*A^2*B*b^23*c^13*d^19
*e^17 - 2360240064*A^2*B*b^24*c^12*d^18*e^18 + 534716736*A^2*B*b^25*c^11*d^17*e^19 + 282511968*A^2*B*b^26*c^10
*d^16*e^20 - 290822400*A^2*B*b^27*c^9*d^15*e^21 + 114170880*A^2*B*b^28*c^8*d^14*e^22 - 22915200*A^2*B*b^29*c^7
*d^13*e^23 + 1940400*A^2*B*b^30*c^6*d^12*e^24)*(((1225*A^2*b^4*e^4)/64 + 36*A^2*c^4*d^4 + 9*B^2*b^2*c^2*d^4 +
(25*B^2*b^4*d^2*e^2)/4 + (435*A^2*b^2*c^2*d^2*e^2)/4 + 90*A^2*b*c^3*d^3*e + (525*A^2*b^3*c*d*e^3)/8 + 15*B^2*b
^3*c*d^3*e - 36*A*B*b*c^3*d^4 - (175*A*B*b^4*d*e^3)/8 - 75*A*B*b^2*c^2*d^3*e - (255*A*B*b^3*c*d^2*e^2)/4)/(b^1
0*d^9))^(1/2) - ((2*(A*e^5 - B*d*e^4))/(3*(c*d^2 - b*d*e)) - (2*(d + e*x)*(7*A*b*e^6 - 14*A*c*d*e^5 - 4*B*b*d*
e^5 + 11*B*c*d^2*e^4))/(3*(c*d^2 - b*d*e)^2) - ((d + e*x)^5*(24*A*c^7*d^5*e + 35*A*b^5*c^2*e^6 - 60*A*b*c^6*d^
4*e^2 - 80*A*b^4*c^3*d*e^5 - 20*B*b^5*c^2*d*e^5 + 28*A*b^2*c^5*d^3*e^3 + 18*A*b^3*c^4*d^2*e^4 + 35*B*b^2*c^5*d
^4*e^2 - 24*B*b^3*c^4*d^3*e^3 + 56*B*b^4*c^3*d^2*e^4 - 12*B*b*c^6*d^5*e))/(4*b^4*(c*d^2 - b*d*e)^4) + ((d + e*
x)^2*(72*A*c^6*d^6*e - 175*A*b^6*e^7 + 100*B*b^6*d*e^6 - 216*A*b*c^5*d^5*e^2 - 444*B*b^5*c*d^2*e^5 + 165*A*b^2
*c^4*d^4*e^3 + 30*A*b^3*c^3*d^3*e^4 - 738*A*b^4*c^2*d^2*e^5 + 123*B*b^2*c^4*d^5*e^2 - 120*B*b^3*c^3*d^4*e^3 +
552*B*b^4*c^2*d^3*e^4 + 687*A*b^5*c*d*e^6 - 36*B*b*c^5*d^6*e))/(12*b^4*(c*d^2 - b*d*e)^3) + ((d + e*x)^3*(60*B
*b^7*d*e^7 - 216*A*c^7*d^7*e - 105*A*b^7*e^8 + 756*A*b*c^6*d^6*e^2 - 488*B*b^6*c*d^2*e^6 - 822*A*b^2*c^5*d^5*e
^3 + 165*A*b^3*c^4*d^4*e^4 + 1372*A*b^4*c^3*d^3*e^5 - 1845*A*b^5*c^2*d^2*e^6 - 423*B*b^2*c^5*d^6*e^2 + 546*B*b
^3*c^4*d^5*e^3 - 1168*B*b^4*c^3*d^4*e^4 + 1260*B*b^5*c^2*d^3*e^5 + 800*A*b^6*c*d*e^7 + 108*B*b*c^6*d^7*e))/(12
*b^4*(c*d^2 - b*d*e)^4) + ((d + e*x)^4*(216*A*c^7*d^6*e - 210*A*b^6*c*e^7 - 648*A*b*c^6*d^5*e^2 + 865*A*b^5*c^
2*d*e^6 + 525*A*b^2*c^5*d^4*e^3 + 30*A*b^3*c^4*d^3*e^4 - 988*A*b^4*c^3*d^2*e^5 + 369*B*b^2*c^5*d^5*e^2 - 375*B
*b^3*c^4*d^4*e^3 + 760*B*b^4*c^3*d^3*e^4 - 556*B*b^5*c^2*d^2*e^5 - 108*B*b*c^6*d^6*e + 120*B*b^6*c*d*e^6))/(12
*b^4*(c*d^2 - b*d*e)^4))/(c^2*(d + e*x)^(11/2) - (4*c^2*d - 2*b*c*e)*(d + e*x)^(9/2) - (d + e*x)^(5/2)*(4*c^2*
d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) + (d + e*x)^(7/2)*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + (d + e*x)^(3/2)*(c^2*d^
4 + b^2*d^2*e^2 - 2*b*c*d^3*e)) - atan(-(((d + e*x)^(1/2)*(589824*A^2*b^12*c^27*d^36*e^2 - 10616832*A^2*b^13*c
^26*d^35*e^3 + 89518080*A^2*b^14*c^25*d^34*e^4 - 468971520*A^2*b^15*c^24*d^33*e^5 + 1707439360*A^2*b^16*c^23*d
^32*e^6 - 4579446784*A^2*b^17*c^22*d^31*e^7 + 9364822016*A^2*b^18*c^21*d^30*e^8 - 14937190400*A^2*b^19*c^20*d^
29*e^9 + 18936107520*A^2*b^20*c^19*d^28*e^10 - 19535324160*A^2*b^21*c^18*d^27*e^11 + 17074641408*A^2*b^22*c^17
*d^26*e^12 - 13484230656*A^2*b^23*c^16*d^25*e^13 + 10265639040*A^2*b^24*c^15*d^24*e^14 - 7643066880*A^2*b^25*c
^14*d^23*e^15 + 5421597440*A^2*b^26*c^13*d^22*e^16 - 3708136960*A^2*b^27*c^12*d^21*e^17 + 2608529792*A^2*b^28*
c^11*d^20*e^18 - 1894041600*A^2*b^29*c^10*d^19*e^19 + 1274465280*A^2*b^30*c^9*d^18*e^20 - 707773440*A^2*b^31*c
^8*d^17*e^21 + 301648512*A^2*b^32*c^7*d^16*e^22 - 93688320*A^2*b^33*c^6*d^15*e^23 + 19930880*A^2*b^34*c^5*d^14
*e^24 - 2598400*A^2*b^35*c^4*d^13*e^25 + 156800*A^2*b^36*c^3*d^12*e^26 + 147456*B^2*b^14*c^25*d^36*e^2 - 27770
88*B^2*b^15*c^24*d^35*e^3 + 24555520*B^2*b^16*c^23*d^34*e^4 - 135055360*B^2*b^17*c^22*d^33*e^5 + 515884160*B^2
*b^18*c^21*d^32*e^6 - 1446258176*B^2*b^19*c^20*d^31*e^7 + 3062171904*B^2*b^20*c^19*d^30*e^8 - 4951119360*B^2*b
^21*c^18*d^29*e^9 + 6076371840*B^2*b^22*c^17*d^28*e^10 - 5478190080*B^2*b^23*c^16*d^27*e^11 + 3273549312*B^2*b
^24*c^15*d^26*e^12 - 766116864*B^2*b^25*c^14*d^25*e^13 - 668122240*B^2*b^26*c^13*d^24*e^14 + 721318400*B^2*b^2
7*c^12*d^23*e^15 - 107134720*B^2*b^28*c^11*d^22*e^16 - 366558720*B^2*b^29*c^10*d^21*e^17 + 437847168*B^2*b^30*
c^9*d^20*e^18 - 282501120*B^2*b^31*c^8*d^19*e^19 + 121989120*B^2*b^32*c^7*d^18*e^20 - 36495360*B^2*b^33*c^6*d^
17*e^21 + 7344128*B^2*b^34*c^5*d^16*e^22 - 901120*B^2*b^35*c^4*d^15*e^23 + 51200*B^2*b^36*c^3*d^14*e^24 - 5898
24*A*B*b^13*c^26*d^36*e^2 + 10862592*A*B*b^14*c^25*d^35*e^3 - 93818880*A*B*b^15*c^24*d^34*e^4 + 503726080*A*B*
b^16*c^23*d^33*e^5 - 1878764800*A*B*b^17*c^22*d^32*e^6 + 5151263744*A*B*b^18*c^21*d^31*e^7 - 10713545216*A*B*b
^19*c^20*d^30*e^8 + 17186104320*A*B*b^20*c^19*d^29*e^9 - 21406851840*A*B*b^21*c^18*d^28*e^10 + 20693207040*A*B
*b^22*c^17*d^27*e^11 - 15463523328*A*B*b^23*c^16*d^26*e^12 + 8955257856*A*B*b^24*c^15*d^25*e^13 - 4111491840*A
*B*b^25*c^14*d^24*e^14 + 1413002240*A*B*b^26*c^13*d^23*e^15 + 178449920*A*B*b^27*c^12*d^22*e^16 - 1280942080*A
*B*b^28*c^11*d^21*e^17 + 1742746368*A*B*b^29*c^10*d^20*e^18 - 1489551360*A*B*b^30*c^9*d^19*e^19 + 892446720*A*
B*b^31*c^8*d^18*e^20 - 383708160*A*B*b^32*c^7*d^17*e^21 + 117055488*A*B*b^33*c^6*d^16*e^22 - 24217600*A*B*b^34
*c^5*d^15*e^23 + 3061760*A*B*b^35*c^4*d^14*e^24 - 179200*A*B*b^36*c^3*d^13*e^25) - (-(2304*A^2*c^13*d^4 + 2044
9*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d^4 + 9801*B^2*b^6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c
^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 14976*A^2*b*c^12*d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3
*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*B*b*c^12*d^4 + 15936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 4382
4*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9 - b^10*c^9*d^9 + 9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^
6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^15*c^4*d^4*e^5 - 84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e
^8)))^(1/2)*((d + e*x)^(1/2)*(-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d^4 + 9801*B^2*b^
6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 14976*A^2*b*c^12*
d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*B*b*c^12*d^4 + 1
5936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9 - b^10*c^9*d^9 +
9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^15*c^4*d^4*e^5 -
84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^
22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^37*e^6 - 2937077
76*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^
33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 3
439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16 - 571539456*b^3
7*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 1
884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23) - 24576*A*b^18*c^24*d^38*e^3
+ 466944*A*b^19*c^23*d^37*e^4 - 4185088*A*b^20*c^22*d^36*e^5 + 23500800*A*b^21*c^21*d^35*e^6 - 92710912*A*b^22
*c^20*d^34*e^7 + 273566720*A*b^23*c^19*d^33*e^8 - 629578752*A*b^24*c^18*d^32*e^9 + 1169833984*A*b^25*c^17*d^31
*e^10 - 1818910720*A*b^26*c^16*d^30*e^11 + 2465058816*A*b^27*c^15*d^29*e^12 - 3031169024*A*b^28*c^14*d^28*e^13
+ 3457871872*A*b^29*c^13*d^27*e^14 - 3626348544*A*b^30*c^12*d^26*e^15 + 3385559040*A*b^31*c^11*d^25*e^16 - 27
14064896*A*b^32*c^10*d^24*e^17 + 1813512192*A*b^33*c^9*d^23*e^18 - 986251264*A*b^34*c^8*d^22*e^19 + 426815488*
A*b^35*c^7*d^21*e^20 - 143109120*A*b^36*c^6*d^20*e^21 + 35796992*A*b^37*c^5*d^19*e^22 - 6285312*A*b^38*c^4*d^1
8*e^23 + 691200*A*b^39*c^3*d^17*e^24 - 35840*A*b^40*c^2*d^16*e^25 + 12288*B*b^19*c^23*d^38*e^3 - 238592*B*b^20
*c^22*d^37*e^4 + 2187264*B*b^21*c^21*d^36*e^5 - 12492800*B*b^22*c^20*d^35*e^6 + 49401856*B*b^23*c^19*d^34*e^7
- 141926400*B*b^24*c^18*d^33*e^8 + 300793856*B*b^25*c^17*d^32*e^9 - 460562432*B*b^26*c^16*d^31*e^10 + 45551616
0*B*b^27*c^15*d^30*e^11 - 116267008*B*b^28*c^14*d^29*e^12 - 543981568*B*b^29*c^13*d^28*e^13 + 1250156544*B*b^3
0*c^12*d^27*e^14 - 1639292928*B*b^31*c^11*d^26*e^15 + 1547694080*B*b^32*c^10*d^25*e^16 - 1115799552*B*b^33*c^9
*d^24*e^17 + 624861184*B*b^34*c^8*d^23*e^18 - 271372288*B*b^35*c^7*d^22*e^19 + 89988096*B*b^36*c^6*d^21*e^20 -
22077440*B*b^37*c^5*d^20*e^21 + 3784704*B*b^38*c^4*d^19*e^22 - 405504*B*b^39*c^3*d^18*e^23 + 20480*B*b^40*c^2
*d^17*e^24))*(-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d^4 + 9801*B^2*b^6*c^7*e^4 + 3806
4*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 14976*A^2*b*c^12*d^3*e - 44616*A^
2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*B*b*c^12*d^4 + 15936*A*B*b^2*c^1
1*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9 - b^10*c^9*d^9 + 9*b^11*c^8*d^8*
e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^15*c^4*d^4*e^5 - 84*b^16*c^3*d^3*
e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2)*1i + ((d + e*x)^(1/2)*(589824*A^2*b^12*c^27*d^36*e^2 - 106
16832*A^2*b^13*c^26*d^35*e^3 + 89518080*A^2*b^14*c^25*d^34*e^4 - 468971520*A^2*b^15*c^24*d^33*e^5 + 1707439360
*A^2*b^16*c^23*d^32*e^6 - 4579446784*A^2*b^17*c^22*d^31*e^7 + 9364822016*A^2*b^18*c^21*d^30*e^8 - 14937190400*
A^2*b^19*c^20*d^29*e^9 + 18936107520*A^2*b^20*c^19*d^28*e^10 - 19535324160*A^2*b^21*c^18*d^27*e^11 + 170746414
08*A^2*b^22*c^17*d^26*e^12 - 13484230656*A^2*b^23*c^16*d^25*e^13 + 10265639040*A^2*b^24*c^15*d^24*e^14 - 76430
66880*A^2*b^25*c^14*d^23*e^15 + 5421597440*A^2*b^26*c^13*d^22*e^16 - 3708136960*A^2*b^27*c^12*d^21*e^17 + 2608
529792*A^2*b^28*c^11*d^20*e^18 - 1894041600*A^2*b^29*c^10*d^19*e^19 + 1274465280*A^2*b^30*c^9*d^18*e^20 - 7077
73440*A^2*b^31*c^8*d^17*e^21 + 301648512*A^2*b^32*c^7*d^16*e^22 - 93688320*A^2*b^33*c^6*d^15*e^23 + 19930880*A
^2*b^34*c^5*d^14*e^24 - 2598400*A^2*b^35*c^4*d^13*e^25 + 156800*A^2*b^36*c^3*d^12*e^26 + 147456*B^2*b^14*c^25*
d^36*e^2 - 2777088*B^2*b^15*c^24*d^35*e^3 + 24555520*B^2*b^16*c^23*d^34*e^4 - 135055360*B^2*b^17*c^22*d^33*e^5
+ 515884160*B^2*b^18*c^21*d^32*e^6 - 1446258176*B^2*b^19*c^20*d^31*e^7 + 3062171904*B^2*b^20*c^19*d^30*e^8 -
4951119360*B^2*b^21*c^18*d^29*e^9 + 6076371840*B^2*b^22*c^17*d^28*e^10 - 5478190080*B^2*b^23*c^16*d^27*e^11 +
3273549312*B^2*b^24*c^15*d^26*e^12 - 766116864*B^2*b^25*c^14*d^25*e^13 - 668122240*B^2*b^26*c^13*d^24*e^14 + 7
21318400*B^2*b^27*c^12*d^23*e^15 - 107134720*B^2*b^28*c^11*d^22*e^16 - 366558720*B^2*b^29*c^10*d^21*e^17 + 437
847168*B^2*b^30*c^9*d^20*e^18 - 282501120*B^2*b^31*c^8*d^19*e^19 + 121989120*B^2*b^32*c^7*d^18*e^20 - 36495360
*B^2*b^33*c^6*d^17*e^21 + 7344128*B^2*b^34*c^5*d^16*e^22 - 901120*B^2*b^35*c^4*d^15*e^23 + 51200*B^2*b^36*c^3*
d^14*e^24 - 589824*A*B*b^13*c^26*d^36*e^2 + 10862592*A*B*b^14*c^25*d^35*e^3 - 93818880*A*B*b^15*c^24*d^34*e^4
+ 503726080*A*B*b^16*c^23*d^33*e^5 - 1878764800*A*B*b^17*c^22*d^32*e^6 + 5151263744*A*B*b^18*c^21*d^31*e^7 - 1
0713545216*A*B*b^19*c^20*d^30*e^8 + 17186104320*A*B*b^20*c^19*d^29*e^9 - 21406851840*A*B*b^21*c^18*d^28*e^10 +
20693207040*A*B*b^22*c^17*d^27*e^11 - 15463523328*A*B*b^23*c^16*d^26*e^12 + 8955257856*A*B*b^24*c^15*d^25*e^1
3 - 4111491840*A*B*b^25*c^14*d^24*e^14 + 1413002240*A*B*b^26*c^13*d^23*e^15 + 178449920*A*B*b^27*c^12*d^22*e^1
6 - 1280942080*A*B*b^28*c^11*d^21*e^17 + 1742746368*A*B*b^29*c^10*d^20*e^18 - 1489551360*A*B*b^30*c^9*d^19*e^1
9 + 892446720*A*B*b^31*c^8*d^18*e^20 - 383708160*A*B*b^32*c^7*d^17*e^21 + 117055488*A*B*b^33*c^6*d^16*e^22 - 2
4217600*A*B*b^34*c^5*d^15*e^23 + 3061760*A*B*b^35*c^4*d^14*e^24 - 179200*A*B*b^36*c^3*d^13*e^25) - (-(2304*A^2
*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d^4 + 9801*B^2*b^6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 +
12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 14976*A^2*b*c^12*d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*
B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*B*b*c^12*d^4 + 15936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*
c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9 - b^10*c^9*d^9 + 9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2
+ 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^15*c^4*d^4*e^5 - 84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^
7 - 9*b^18*c*d*e^8)))^(1/2)*((d + e*x)^(1/2)*(-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d
^4 + 9801*B^2*b^6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 1
4976*A^2*b*c^12*d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*
B*b*c^12*d^4 + 15936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9
- b^10*c^9*d^9 + 9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^1
5*c^4*d^4*e^5 - 84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2)*(16384*b^22*c^23*d^41*e^2
- 335872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^
37*e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 2698936
320*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^
12*d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16
- 571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c
^5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23) + 24576*A*b^1
8*c^24*d^38*e^3 - 466944*A*b^19*c^23*d^37*e^4 + 4185088*A*b^20*c^22*d^36*e^5 - 23500800*A*b^21*c^21*d^35*e^6 +
92710912*A*b^22*c^20*d^34*e^7 - 273566720*A*b^23*c^19*d^33*e^8 + 629578752*A*b^24*c^18*d^32*e^9 - 1169833984*
A*b^25*c^17*d^31*e^10 + 1818910720*A*b^26*c^16*d^30*e^11 - 2465058816*A*b^27*c^15*d^29*e^12 + 3031169024*A*b^2
8*c^14*d^28*e^13 - 3457871872*A*b^29*c^13*d^27*e^14 + 3626348544*A*b^30*c^12*d^26*e^15 - 3385559040*A*b^31*c^1
1*d^25*e^16 + 2714064896*A*b^32*c^10*d^24*e^17 - 1813512192*A*b^33*c^9*d^23*e^18 + 986251264*A*b^34*c^8*d^22*e
^19 - 426815488*A*b^35*c^7*d^21*e^20 + 143109120*A*b^36*c^6*d^20*e^21 - 35796992*A*b^37*c^5*d^19*e^22 + 628531
2*A*b^38*c^4*d^18*e^23 - 691200*A*b^39*c^3*d^17*e^24 + 35840*A*b^40*c^2*d^16*e^25 - 12288*B*b^19*c^23*d^38*e^3
+ 238592*B*b^20*c^22*d^37*e^4 - 2187264*B*b^21*c^21*d^36*e^5 + 12492800*B*b^22*c^20*d^35*e^6 - 49401856*B*b^2
3*c^19*d^34*e^7 + 141926400*B*b^24*c^18*d^33*e^8 - 300793856*B*b^25*c^17*d^32*e^9 + 460562432*B*b^26*c^16*d^31
*e^10 - 455516160*B*b^27*c^15*d^30*e^11 + 116267008*B*b^28*c^14*d^29*e^12 + 543981568*B*b^29*c^13*d^28*e^13 -
1250156544*B*b^30*c^12*d^27*e^14 + 1639292928*B*b^31*c^11*d^26*e^15 - 1547694080*B*b^32*c^10*d^25*e^16 + 11157
99552*B*b^33*c^9*d^24*e^17 - 624861184*B*b^34*c^8*d^23*e^18 + 271372288*B*b^35*c^7*d^22*e^19 - 89988096*B*b^36
*c^6*d^21*e^20 + 22077440*B*b^37*c^5*d^20*e^21 - 3784704*B*b^38*c^4*d^19*e^22 + 405504*B*b^39*c^3*d^18*e^23 -
20480*B*b^40*c^2*d^17*e^24))*(-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d^4 + 9801*B^2*b^
6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 14976*A^2*b*c^12*
d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*B*b*c^12*d^4 + 1
5936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9 - b^10*c^9*d^9 +
9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^15*c^4*d^4*e^5 -
84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2)*1i)/(((d + e*x)^(1/2)*(589824*A^2*b^12*c^2
7*d^36*e^2 - 10616832*A^2*b^13*c^26*d^35*e^3 + 89518080*A^2*b^14*c^25*d^34*e^4 - 468971520*A^2*b^15*c^24*d^33*
e^5 + 1707439360*A^2*b^16*c^23*d^32*e^6 - 4579446784*A^2*b^17*c^22*d^31*e^7 + 9364822016*A^2*b^18*c^21*d^30*e^
8 - 14937190400*A^2*b^19*c^20*d^29*e^9 + 18936107520*A^2*b^20*c^19*d^28*e^10 - 19535324160*A^2*b^21*c^18*d^27*
e^11 + 17074641408*A^2*b^22*c^17*d^26*e^12 - 13484230656*A^2*b^23*c^16*d^25*e^13 + 10265639040*A^2*b^24*c^15*d
^24*e^14 - 7643066880*A^2*b^25*c^14*d^23*e^15 + 5421597440*A^2*b^26*c^13*d^22*e^16 - 3708136960*A^2*b^27*c^12*
d^21*e^17 + 2608529792*A^2*b^28*c^11*d^20*e^18 - 1894041600*A^2*b^29*c^10*d^19*e^19 + 1274465280*A^2*b^30*c^9*
d^18*e^20 - 707773440*A^2*b^31*c^8*d^17*e^21 + 301648512*A^2*b^32*c^7*d^16*e^22 - 93688320*A^2*b^33*c^6*d^15*e
^23 + 19930880*A^2*b^34*c^5*d^14*e^24 - 2598400*A^2*b^35*c^4*d^13*e^25 + 156800*A^2*b^36*c^3*d^12*e^26 + 14745
6*B^2*b^14*c^25*d^36*e^2 - 2777088*B^2*b^15*c^24*d^35*e^3 + 24555520*B^2*b^16*c^23*d^34*e^4 - 135055360*B^2*b^
17*c^22*d^33*e^5 + 515884160*B^2*b^18*c^21*d^32*e^6 - 1446258176*B^2*b^19*c^20*d^31*e^7 + 3062171904*B^2*b^20*
c^19*d^30*e^8 - 4951119360*B^2*b^21*c^18*d^29*e^9 + 6076371840*B^2*b^22*c^17*d^28*e^10 - 5478190080*B^2*b^23*c
^16*d^27*e^11 + 3273549312*B^2*b^24*c^15*d^26*e^12 - 766116864*B^2*b^25*c^14*d^25*e^13 - 668122240*B^2*b^26*c^
13*d^24*e^14 + 721318400*B^2*b^27*c^12*d^23*e^15 - 107134720*B^2*b^28*c^11*d^22*e^16 - 366558720*B^2*b^29*c^10
*d^21*e^17 + 437847168*B^2*b^30*c^9*d^20*e^18 - 282501120*B^2*b^31*c^8*d^19*e^19 + 121989120*B^2*b^32*c^7*d^18
*e^20 - 36495360*B^2*b^33*c^6*d^17*e^21 + 7344128*B^2*b^34*c^5*d^16*e^22 - 901120*B^2*b^35*c^4*d^15*e^23 + 512
00*B^2*b^36*c^3*d^14*e^24 - 589824*A*B*b^13*c^26*d^36*e^2 + 10862592*A*B*b^14*c^25*d^35*e^3 - 93818880*A*B*b^1
5*c^24*d^34*e^4 + 503726080*A*B*b^16*c^23*d^33*e^5 - 1878764800*A*B*b^17*c^22*d^32*e^6 + 5151263744*A*B*b^18*c
^21*d^31*e^7 - 10713545216*A*B*b^19*c^20*d^30*e^8 + 17186104320*A*B*b^20*c^19*d^29*e^9 - 21406851840*A*B*b^21*
c^18*d^28*e^10 + 20693207040*A*B*b^22*c^17*d^27*e^11 - 15463523328*A*B*b^23*c^16*d^26*e^12 + 8955257856*A*B*b^
24*c^15*d^25*e^13 - 4111491840*A*B*b^25*c^14*d^24*e^14 + 1413002240*A*B*b^26*c^13*d^23*e^15 + 178449920*A*B*b^
27*c^12*d^22*e^16 - 1280942080*A*B*b^28*c^11*d^21*e^17 + 1742746368*A*B*b^29*c^10*d^20*e^18 - 1489551360*A*B*b
^30*c^9*d^19*e^19 + 892446720*A*B*b^31*c^8*d^18*e^20 - 383708160*A*B*b^32*c^7*d^17*e^21 + 117055488*A*B*b^33*c
^6*d^16*e^22 - 24217600*A*B*b^34*c^5*d^15*e^23 + 3061760*A*B*b^35*c^4*d^14*e^24 - 179200*A*B*b^36*c^3*d^13*e^2
5) - (-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d^4 + 9801*B^2*b^6*c^7*e^4 + 38064*A^2*b^
2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 14976*A^2*b*c^12*d^3*e - 44616*A^2*b^3*c^
10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*B*b*c^12*d^4 + 15936*A*B*b^2*c^11*d^3*e
+ 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9 - b^10*c^9*d^9 + 9*b^11*c^8*d^8*e - 36*b
^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^15*c^4*d^4*e^5 - 84*b^16*c^3*d^3*e^6 + 36
*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2)*((d + e*x)^(1/2)*(-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 57
6*B^2*b^2*c^11*d^4 + 9801*B^2*b^6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B
*b^5*c^8*e^4 - 14976*A^2*b*c^12*d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8
*d*e^3 - 2304*A*B*b*c^12*d^4 + 15936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2
)/(64*(b^19*e^9 - b^10*c^9*d^9 + 9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d
^5*e^4 + 126*b^15*c^4*d^4*e^5 - 84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2)*(16384*b^2
2*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719
360*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^
34*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 42
65377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^
36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 +
12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^2
3) - 24576*A*b^18*c^24*d^38*e^3 + 466944*A*b^19*c^23*d^37*e^4 - 4185088*A*b^20*c^22*d^36*e^5 + 23500800*A*b^21
*c^21*d^35*e^6 - 92710912*A*b^22*c^20*d^34*e^7 + 273566720*A*b^23*c^19*d^33*e^8 - 629578752*A*b^24*c^18*d^32*e
^9 + 1169833984*A*b^25*c^17*d^31*e^10 - 1818910720*A*b^26*c^16*d^30*e^11 + 2465058816*A*b^27*c^15*d^29*e^12 -
3031169024*A*b^28*c^14*d^28*e^13 + 3457871872*A*b^29*c^13*d^27*e^14 - 3626348544*A*b^30*c^12*d^26*e^15 + 33855
59040*A*b^31*c^11*d^25*e^16 - 2714064896*A*b^32*c^10*d^24*e^17 + 1813512192*A*b^33*c^9*d^23*e^18 - 986251264*A
*b^34*c^8*d^22*e^19 + 426815488*A*b^35*c^7*d^21*e^20 - 143109120*A*b^36*c^6*d^20*e^21 + 35796992*A*b^37*c^5*d^
19*e^22 - 6285312*A*b^38*c^4*d^18*e^23 + 691200*A*b^39*c^3*d^17*e^24 - 35840*A*b^40*c^2*d^16*e^25 + 12288*B*b^
19*c^23*d^38*e^3 - 238592*B*b^20*c^22*d^37*e^4 + 2187264*B*b^21*c^21*d^36*e^5 - 12492800*B*b^22*c^20*d^35*e^6
+ 49401856*B*b^23*c^19*d^34*e^7 - 141926400*B*b^24*c^18*d^33*e^8 + 300793856*B*b^25*c^17*d^32*e^9 - 460562432*
B*b^26*c^16*d^31*e^10 + 455516160*B*b^27*c^15*d^30*e^11 - 116267008*B*b^28*c^14*d^29*e^12 - 543981568*B*b^29*c
^13*d^28*e^13 + 1250156544*B*b^30*c^12*d^27*e^14 - 1639292928*B*b^31*c^11*d^26*e^15 + 1547694080*B*b^32*c^10*d
^25*e^16 - 1115799552*B*b^33*c^9*d^24*e^17 + 624861184*B*b^34*c^8*d^23*e^18 - 271372288*B*b^35*c^7*d^22*e^19 +
89988096*B*b^36*c^6*d^21*e^20 - 22077440*B*b^37*c^5*d^20*e^21 + 3784704*B*b^38*c^4*d^19*e^22 - 405504*B*b^39*
c^3*d^18*e^23 + 20480*B*b^40*c^2*d^17*e^24))*(-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d
^4 + 9801*B^2*b^6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 1
4976*A^2*b*c^12*d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*
B*b*c^12*d^4 + 15936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9
- b^10*c^9*d^9 + 9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^1
5*c^4*d^4*e^5 - 84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2) - ((d + e*x)^(1/2)*(589824
*A^2*b^12*c^27*d^36*e^2 - 10616832*A^2*b^13*c^26*d^35*e^3 + 89518080*A^2*b^14*c^25*d^34*e^4 - 468971520*A^2*b^
15*c^24*d^33*e^5 + 1707439360*A^2*b^16*c^23*d^32*e^6 - 4579446784*A^2*b^17*c^22*d^31*e^7 + 9364822016*A^2*b^18
*c^21*d^30*e^8 - 14937190400*A^2*b^19*c^20*d^29*e^9 + 18936107520*A^2*b^20*c^19*d^28*e^10 - 19535324160*A^2*b^
21*c^18*d^27*e^11 + 17074641408*A^2*b^22*c^17*d^26*e^12 - 13484230656*A^2*b^23*c^16*d^25*e^13 + 10265639040*A^
2*b^24*c^15*d^24*e^14 - 7643066880*A^2*b^25*c^14*d^23*e^15 + 5421597440*A^2*b^26*c^13*d^22*e^16 - 3708136960*A
^2*b^27*c^12*d^21*e^17 + 2608529792*A^2*b^28*c^11*d^20*e^18 - 1894041600*A^2*b^29*c^10*d^19*e^19 + 1274465280*
A^2*b^30*c^9*d^18*e^20 - 707773440*A^2*b^31*c^8*d^17*e^21 + 301648512*A^2*b^32*c^7*d^16*e^22 - 93688320*A^2*b^
33*c^6*d^15*e^23 + 19930880*A^2*b^34*c^5*d^14*e^24 - 2598400*A^2*b^35*c^4*d^13*e^25 + 156800*A^2*b^36*c^3*d^12
*e^26 + 147456*B^2*b^14*c^25*d^36*e^2 - 2777088*B^2*b^15*c^24*d^35*e^3 + 24555520*B^2*b^16*c^23*d^34*e^4 - 135
055360*B^2*b^17*c^22*d^33*e^5 + 515884160*B^2*b^18*c^21*d^32*e^6 - 1446258176*B^2*b^19*c^20*d^31*e^7 + 3062171
904*B^2*b^20*c^19*d^30*e^8 - 4951119360*B^2*b^21*c^18*d^29*e^9 + 6076371840*B^2*b^22*c^17*d^28*e^10 - 54781900
80*B^2*b^23*c^16*d^27*e^11 + 3273549312*B^2*b^24*c^15*d^26*e^12 - 766116864*B^2*b^25*c^14*d^25*e^13 - 66812224
0*B^2*b^26*c^13*d^24*e^14 + 721318400*B^2*b^27*c^12*d^23*e^15 - 107134720*B^2*b^28*c^11*d^22*e^16 - 366558720*
B^2*b^29*c^10*d^21*e^17 + 437847168*B^2*b^30*c^9*d^20*e^18 - 282501120*B^2*b^31*c^8*d^19*e^19 + 121989120*B^2*
b^32*c^7*d^18*e^20 - 36495360*B^2*b^33*c^6*d^17*e^21 + 7344128*B^2*b^34*c^5*d^16*e^22 - 901120*B^2*b^35*c^4*d^
15*e^23 + 51200*B^2*b^36*c^3*d^14*e^24 - 589824*A*B*b^13*c^26*d^36*e^2 + 10862592*A*B*b^14*c^25*d^35*e^3 - 938
18880*A*B*b^15*c^24*d^34*e^4 + 503726080*A*B*b^16*c^23*d^33*e^5 - 1878764800*A*B*b^17*c^22*d^32*e^6 + 51512637
44*A*B*b^18*c^21*d^31*e^7 - 10713545216*A*B*b^19*c^20*d^30*e^8 + 17186104320*A*B*b^20*c^19*d^29*e^9 - 21406851
840*A*B*b^21*c^18*d^28*e^10 + 20693207040*A*B*b^22*c^17*d^27*e^11 - 15463523328*A*B*b^23*c^16*d^26*e^12 + 8955
257856*A*B*b^24*c^15*d^25*e^13 - 4111491840*A*B*b^25*c^14*d^24*e^14 + 1413002240*A*B*b^26*c^13*d^23*e^15 + 178
449920*A*B*b^27*c^12*d^22*e^16 - 1280942080*A*B*b^28*c^11*d^21*e^17 + 1742746368*A*B*b^29*c^10*d^20*e^18 - 148
9551360*A*B*b^30*c^9*d^19*e^19 + 892446720*A*B*b^31*c^8*d^18*e^20 - 383708160*A*B*b^32*c^7*d^17*e^21 + 1170554
88*A*B*b^33*c^6*d^16*e^22 - 24217600*A*B*b^34*c^5*d^15*e^23 + 3061760*A*B*b^35*c^4*d^14*e^24 - 179200*A*B*b^36
*c^3*d^13*e^25) - (-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d^4 + 9801*B^2*b^6*c^7*e^4 +
38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 14976*A^2*b*c^12*d^3*e - 446
16*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*B*b*c^12*d^4 + 15936*A*B*b^
2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9 - b^10*c^9*d^9 + 9*b^11*c^8
*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^15*c^4*d^4*e^5 - 84*b^16*c^3
*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2)*((d + e*x)^(1/2)*(-(2304*A^2*c^13*d^4 + 20449*A^2*b^4
*c^9*e^4 + 576*B^2*b^2*c^11*d^4 + 9801*B^2*b^6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^
2 - 28314*A*B*b^5*c^8*e^4 - 14976*A^2*b*c^12*d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 1742
4*B^2*b^5*c^8*d*e^3 - 2304*A*B*b*c^12*d^4 + 15936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3
*c^10*d^2*e^2)/(64*(b^19*e^9 - b^10*c^9*d^9 + 9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 1
26*b^14*c^5*d^5*e^4 + 126*b^15*c^4*d^4*e^5 - 84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/
2)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^3
8*e^5 + 88719360*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600
*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d
^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 +
1270087680*b^36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^
6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43
*c^2*d^20*e^23) + 24576*A*b^18*c^24*d^38*e^3 - 466944*A*b^19*c^23*d^37*e^4 + 4185088*A*b^20*c^22*d^36*e^5 - 23
500800*A*b^21*c^21*d^35*e^6 + 92710912*A*b^22*c^20*d^34*e^7 - 273566720*A*b^23*c^19*d^33*e^8 + 629578752*A*b^2
4*c^18*d^32*e^9 - 1169833984*A*b^25*c^17*d^31*e^10 + 1818910720*A*b^26*c^16*d^30*e^11 - 2465058816*A*b^27*c^15
*d^29*e^12 + 3031169024*A*b^28*c^14*d^28*e^13 - 3457871872*A*b^29*c^13*d^27*e^14 + 3626348544*A*b^30*c^12*d^26
*e^15 - 3385559040*A*b^31*c^11*d^25*e^16 + 2714064896*A*b^32*c^10*d^24*e^17 - 1813512192*A*b^33*c^9*d^23*e^18
+ 986251264*A*b^34*c^8*d^22*e^19 - 426815488*A*b^35*c^7*d^21*e^20 + 143109120*A*b^36*c^6*d^20*e^21 - 35796992*
A*b^37*c^5*d^19*e^22 + 6285312*A*b^38*c^4*d^18*e^23 - 691200*A*b^39*c^3*d^17*e^24 + 35840*A*b^40*c^2*d^16*e^25
- 12288*B*b^19*c^23*d^38*e^3 + 238592*B*b^20*c^22*d^37*e^4 - 2187264*B*b^21*c^21*d^36*e^5 + 12492800*B*b^22*c
^20*d^35*e^6 - 49401856*B*b^23*c^19*d^34*e^7 + 141926400*B*b^24*c^18*d^33*e^8 - 300793856*B*b^25*c^17*d^32*e^9
+ 460562432*B*b^26*c^16*d^31*e^10 - 455516160*B*b^27*c^15*d^30*e^11 + 116267008*B*b^28*c^14*d^29*e^12 + 54398
1568*B*b^29*c^13*d^28*e^13 - 1250156544*B*b^30*c^12*d^27*e^14 + 1639292928*B*b^31*c^11*d^26*e^15 - 1547694080*
B*b^32*c^10*d^25*e^16 + 1115799552*B*b^33*c^9*d^24*e^17 - 624861184*B*b^34*c^8*d^23*e^18 + 271372288*B*b^35*c^
7*d^22*e^19 - 89988096*B*b^36*c^6*d^21*e^20 + 22077440*B*b^37*c^5*d^20*e^21 - 3784704*B*b^38*c^4*d^19*e^22 + 4
05504*B*b^39*c^3*d^18*e^23 - 20480*B*b^40*c^2*d^17*e^24))*(-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B
^2*b^2*c^11*d^4 + 9801*B^2*b^6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^
5*c^8*e^4 - 14976*A^2*b*c^12*d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*
e^3 - 2304*A*B*b*c^12*d^4 + 15936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(
64*(b^19*e^9 - b^10*c^9*d^9 + 9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*
e^4 + 126*b^15*c^4*d^4*e^5 - 84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2) - 1769472*A^3
*b^8*c^28*d^33*e^3 + 29196288*A^3*b^9*c^27*d^32*e^4 - 222621696*A^3*b^10*c^26*d^31*e^5 + 1037076480*A^3*b^11*c
^25*d^30*e^6 - 3281114880*A^3*b^12*c^24*d^29*e^7 + 7384738176*A^3*b^13*c^23*d^28*e^8 - 11940731264*A^3*b^14*c^
22*d^27*e^9 + 13391621568*A^3*b^15*c^21*d^26*e^10 - 8822378240*A^3*b^16*c^20*d^25*e^11 - 174168800*A^3*b^17*c^
19*d^24*e^12 + 7908536064*A^3*b^18*c^18*d^23*e^13 - 10270788736*A^3*b^19*c^17*d^22*e^14 + 8868525952*A^3*b^20*
c^16*d^21*e^15 - 8022944160*A^3*b^21*c^15*d^20*e^16 + 9013107840*A^3*b^22*c^14*d^19*e^17 - 9481058368*A^3*b^23
*c^13*d^18*e^18 + 7612941312*A^3*b^24*c^12*d^17*e^19 - 4396193824*A^3*b^25*c^11*d^16*e^20 + 1772817920*A^3*b^2
6*c^10*d^15*e^21 - 475772160*A^3*b^27*c^9*d^14*e^22 + 76585600*A^3*b^28*c^8*d^13*e^23 - 5605600*A^3*b^29*c^7*d
^12*e^24 + 221184*B^3*b^11*c^25*d^33*e^3 - 3926016*B^3*b^12*c^24*d^32*e^4 + 32366592*B^3*b^13*c^23*d^31*e^5 -
164896000*B^3*b^14*c^22*d^30*e^6 + 582860160*B^3*b^15*c^21*d^29*e^7 - 1521620992*B^3*b^16*c^20*d^28*e^8 + 3046
416128*B^3*b^17*c^19*d^27*e^9 - 4775206656*B^3*b^18*c^18*d^26*e^10 + 5868734080*B^3*b^19*c^17*d^25*e^11 - 5470
136320*B^3*b^20*c^16*d^24*e^12 + 3377797632*B^3*b^21*c^15*d^23*e^13 - 415972608*B^3*b^22*c^14*d^22*e^14 - 1985
838464*B^3*b^23*c^13*d^21*e^15 + 2839818240*B^3*b^24*c^12*d^20*e^16 - 2295415040*B^3*b^25*c^11*d^19*e^17 + 125
8898176*B^3*b^26*c^10*d^18*e^18 - 477861504*B^3*b^27*c^9*d^17*e^19 + 120854528*B^3*b^28*c^8*d^16*e^20 - 183603
20*B^3*b^29*c^7*d^15*e^21 + 1267200*B^3*b^30*c^6*d^14*e^22 - 1327104*A*B^2*b^10*c^26*d^33*e^3 + 23003136*A*B^2
*b^11*c^25*d^32*e^4 - 184891392*A*B^2*b^12*c^24*d^31*e^5 + 915217920*A*B^2*b^13*c^23*d^30*e^6 - 3123922560*A*B
^2*b^14*c^22*d^29*e^7 + 7795266912*A*B^2*b^15*c^21*d^28*e^8 - 14683820928*A*B^2*b^16*c^20*d^27*e^9 + 211698572
16*A*B^2*b^17*c^19*d^26*e^10 - 23230183680*A*B^2*b^18*c^18*d^25*e^11 + 18702610080*A*B^2*b^19*c^17*d^24*e^12 -
9899526912*A*B^2*b^20*c^16*d^23*e^13 + 2305438848*A*B^2*b^21*c^15*d^22*e^14 + 70959744*A*B^2*b^22*c^14*d^21*e
^15 + 1974487200*A*B^2*b^23*c^13*d^20*e^16 - 4476113280*A*B^2*b^24*c^12*d^19*e^17 + 4700186304*A*B^2*b^25*c^11
*d^18*e^18 - 3063277056*A*B^2*b^26*c^10*d^17*e^19 + 1316718432*A*B^2*b^27*c^9*d^16*e^20 - 366397440*A*B^2*b^28
*c^8*d^15*e^21 + 60149760*A*B^2*b^29*c^7*d^14*e^22 - 4435200*A*B^2*b^30*c^6*d^13*e^23 + 2654208*A^2*B*b^9*c^27
*d^33*e^3 - 44900352*A^2*B*b^10*c^26*d^32*e^4 + 351627264*A^2*B*b^11*c^25*d^31*e^5 - 1689454080*A^2*B*b^12*c^2
4*d^30*e^6 + 5557921920*A^2*B*b^13*c^23*d^29*e^7 - 13203598944*A^2*B*b^14*c^22*d^28*e^8 + 23187902976*A^2*B*b^
15*c^21*d^27*e^9 - 30060446592*A^2*B*b^16*c^20*d^26*e^10 + 27711116160*A^2*B*b^17*c^19*d^25*e^11 - 15946476480
*A^2*B*b^18*c^18*d^24*e^12 + 2660427264*A^2*B*b^19*c^17*d^23*e^13 + 2948815104*A^2*B*b^20*c^16*d^22*e^14 + 560
587392*A^2*B*b^21*c^15*d^21*e^15 - 6378785280*A^2*B*b^22*c^14*d^20*e^16 + 7823884800*A^2*B*b^23*c^13*d^19*e^17
- 4720480128*A^2*B*b^24*c^12*d^18*e^18 + 1069433472*A^2*B*b^25*c^11*d^17*e^19 + 565023936*A^2*B*b^26*c^10*d^1
6*e^20 - 581644800*A^2*B*b^27*c^9*d^15*e^21 + 228341760*A^2*B*b^28*c^8*d^14*e^22 - 45830400*A^2*B*b^29*c^7*d^1
3*e^23 + 3880800*A^2*B*b^30*c^6*d^12*e^24))*(-(2304*A^2*c^13*d^4 + 20449*A^2*b^4*c^9*e^4 + 576*B^2*b^2*c^11*d^
4 + 9801*B^2*b^6*c^7*e^4 + 38064*A^2*b^2*c^11*d^2*e^2 + 12496*B^2*b^4*c^9*d^2*e^2 - 28314*A*B*b^5*c^8*e^4 - 14
976*A^2*b*c^12*d^3*e - 44616*A^2*b^3*c^10*d*e^3 - 4224*B^2*b^3*c^10*d^3*e - 17424*B^2*b^5*c^8*d*e^3 - 2304*A*B
*b*c^12*d^4 + 15936*A*B*b^2*c^11*d^3*e + 56056*A*B*b^4*c^9*d*e^3 - 43824*A*B*b^3*c^10*d^2*e^2)/(64*(b^19*e^9 -
b^10*c^9*d^9 + 9*b^11*c^8*d^8*e - 36*b^12*c^7*d^7*e^2 + 84*b^13*c^6*d^6*e^3 - 126*b^14*c^5*d^5*e^4 + 126*b^15
*c^4*d^4*e^5 - 84*b^16*c^3*d^3*e^6 + 36*b^17*c^2*d^2*e^7 - 9*b^18*c*d*e^8)))^(1/2)*2i

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

________________________________________________________________________________________