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3.11.93 \(\int \frac {(A+B x) (d+e x)^{9/2}}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=386 \[ -\frac {d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (9 A b e-4 A c d+2 b B d)}{b^3}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac {e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 b^2 c^2}-\frac {(c d-b e)^{7/2} \left (-b c (2 B d-5 A e)+4 A c^2 d-7 b^2 B e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{9/2}}+\frac {e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 b^2 c^3}+\frac {e \sqrt {d+e x} \left (-b^3 c e^2 (5 A e+19 B d)+b^2 c^2 d e (11 A e+15 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{b^2 c^4} \]

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Rubi [A]  time = 1.19, antiderivative size = 386, 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 = {818, 824, 826, 1166, 208} \begin {gather*} \frac {e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 b^2 c^3}+\frac {e \sqrt {d+e x} \left (b^2 c^2 d e (11 A e+15 B d)-b^3 c e^2 (5 A e+19 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{b^2 c^4}-\frac {(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac {e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 b^2 c^2}-\frac {(c d-b e)^{7/2} \left (-b c (2 B d-5 A e)+4 A c^2 d-7 b^2 B e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{9/2}}-\frac {d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (9 A b e-4 A c d+2 b B d)}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^2,x]

[Out]

(e*(2*A*c^4*d^3 + 7*b^4*B*e^3 - b*c^3*d^2*(B*d + 3*A*e) - b^3*c*e^2*(19*B*d + 5*A*e) + b^2*c^2*d*e*(15*B*d + 1
1*A*e))*Sqrt[d + e*x])/(b^2*c^4) + (e*(6*A*c^3*d^2 - 7*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(12*B*d +
 5*A*e))*(d + e*x)^(3/2))/(3*b^2*c^3) + (e*(10*A*c^2*d + 7*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(5/2))/(5*b^
2*c^2) - ((d + e*x)^(7/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*c*(b*x + c*x^2)) - (d^(7
/2)*(2*b*B*d - 4*A*c*d + 9*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^(7/2)*(4*A*c^2*d - 7*b^2*
B*e - b*c*(2*B*d - 5*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(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 818

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

Rule 824

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

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 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)^{9/2}}{\left (b x+c x^2\right )^2} \, dx &=-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {(d+e x)^{5/2} \left (\frac {1}{2} c d (2 b B d-4 A c d+9 A b e)+\frac {1}{2} e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} c^2 d^2 (2 b B d-4 A c d+9 A b e)+\frac {1}{2} e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^2}\\ &=\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} c^3 d^3 (2 b B d-4 A c d+9 A b e)+\frac {1}{2} e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^3}\\ &=\frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{2} c^4 d^4 (2 b B d-4 A c d+9 A b e)-\frac {1}{2} e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^4}\\ &=\frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} c^4 d^4 e (2 b B d-4 A c d+9 A b e)+\frac {1}{2} d e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )-\frac {1}{2} e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 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^2 c^4}\\ &=\frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}+\frac {\left (c d^4 (2 b B d-4 A c d+9 A b e)\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 )}{b^3}-\frac {\left (2 \left (\frac {1}{4} e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )+\frac {\frac {1}{2} e (-2 c d+b e) \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )+2 c \left (\frac {1}{2} c^4 d^4 e (2 b B d-4 A c d+9 A b e)+\frac {1}{2} d e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )\right )}{2 b 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 )}{b^2 c^4}\\ &=\frac {e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt {d+e x}}{b^2 c^4}+\frac {e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac {e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\frac {(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \left (b x+c x^2\right )}-\frac {d^{7/2} (2 b B d-4 A c d+9 A b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3}+\frac {(c d-b e)^{7/2} \left (2 b B c d-4 A c^2 d+7 b^2 B e-5 A b c e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 2.36, size = 376, normalized size = 0.97 \begin {gather*} -\frac {-\frac {315 \left (\frac {2}{315} \sqrt {d+e x} \left (563 d^4+506 d^3 e x+408 d^2 e^2 x^2+185 d e^3 x^3+35 e^4 x^4\right )-2 d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right ) (9 A b e-4 A c d+2 b B d)-\frac {2 d \left (b c (2 B d-5 A e)-4 A c^2 d+7 b^2 B e\right ) \left (3 (c d-b e) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt {c} \sqrt {d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )+35 c^{9/2} (d+e x)^{9/2}\right )}{c^{9/2} (c d-b e)}}{630 b^2}+\frac {c (d+e x)^{11/2} (A b e-2 A c d+b B d)}{b (b+c x) (b e-c d)}+\frac {A (d+e x)^{11/2}}{x (b+c x)}}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^2,x]

[Out]

-(((c*(b*B*d - 2*A*c*d + A*b*e)*(d + e*x)^(11/2))/(b*(-(c*d) + b*e)*(b + c*x)) + (A*(d + e*x)^(11/2))/(x*(b +
c*x)) - (315*(2*b*B*d - 4*A*c*d + 9*A*b*e)*((2*Sqrt[d + e*x]*(563*d^4 + 506*d^3*e*x + 408*d^2*e^2*x^2 + 185*d*
e^3*x^3 + 35*e^4*x^4))/315 - 2*d^(9/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) - (2*d*(-4*A*c^2*d + 7*b^2*B*e + b*c*(2
*B*d - 5*A*e))*(35*c^(9/2)*(d + e*x)^(9/2) + 3*(c*d - b*e)*(15*c^(7/2)*(d + e*x)^(7/2) + 7*(c*d - b*e)*(3*c^(5
/2)*(d + e*x)^(5/2) + 5*(c*d - b*e)*(Sqrt[c]*Sqrt[d + e*x]*(4*c*d - 3*b*e + c*e*x) - 3*(c*d - b*e)^(3/2)*ArcTa
nh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])))))/(c^(9/2)*(c*d - b*e)))/(630*b^2))/(b*d))

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IntegrateAlgebraic [A]  time = 0.96, size = 669, normalized size = 1.73 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-9 A b d^{7/2} e+4 A c d^{9/2}-2 b B d^{9/2}\right )}{b^3}+\frac {\left (-4 A c^2 d (b e-c d)^{7/2}-5 A b c e (b e-c d)^{7/2}+7 b^2 B e (b e-c d)^{7/2}+2 b B c d (b e-c d)^{7/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 c^{9/2}}+\frac {\sqrt {d+e x} \left (-75 A b^4 c e^4 (d+e x)+75 A b^4 c d e^4-240 A b^3 c^2 d^2 e^3-50 A b^3 c^2 e^3 (d+e x)^2+290 A b^3 c^2 d e^3 (d+e x)+210 A b^2 c^3 d^3 e^2-320 A b^2 c^3 d^2 e^2 (d+e x)+10 A b^2 c^3 e^2 (d+e x)^3+100 A b^2 c^3 d e^2 (d+e x)^2-75 A b c^4 d^4 e+60 A b c^4 d^3 e (d+e x)+30 A c^5 d^5-30 A c^5 d^4 (d+e x)+105 b^5 B e^4 (d+e x)-105 b^5 B d e^4+390 b^4 B c d^2 e^3+70 b^4 B c e^3 (d+e x)^2-460 b^4 B c d e^3 (d+e x)-510 b^3 B c^2 d^3 e^2+700 b^3 B c^2 d^2 e^2 (d+e x)-14 b^3 B c^2 e^2 (d+e x)^3-176 b^3 B c^2 d e^2 (d+e x)^2+240 b^2 B c^3 d^4 e-390 b^2 B c^3 d^3 e (d+e x)+126 b^2 B c^3 d^2 e (d+e x)^2+6 b^2 B c^3 e (d+e x)^4+18 b^2 B c^3 d e (d+e x)^3-15 b B c^4 d^5+15 b B c^4 d^4 (d+e x)\right )}{15 b^2 c^4 x (b e+c (d+e x)-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^2,x]

[Out]

(Sqrt[d + e*x]*(-15*b*B*c^4*d^5 + 30*A*c^5*d^5 + 240*b^2*B*c^3*d^4*e - 75*A*b*c^4*d^4*e - 510*b^3*B*c^2*d^3*e^
2 + 210*A*b^2*c^3*d^3*e^2 + 390*b^4*B*c*d^2*e^3 - 240*A*b^3*c^2*d^2*e^3 - 105*b^5*B*d*e^4 + 75*A*b^4*c*d*e^4 +
 15*b*B*c^4*d^4*(d + e*x) - 30*A*c^5*d^4*(d + e*x) - 390*b^2*B*c^3*d^3*e*(d + e*x) + 60*A*b*c^4*d^3*e*(d + e*x
) + 700*b^3*B*c^2*d^2*e^2*(d + e*x) - 320*A*b^2*c^3*d^2*e^2*(d + e*x) - 460*b^4*B*c*d*e^3*(d + e*x) + 290*A*b^
3*c^2*d*e^3*(d + e*x) + 105*b^5*B*e^4*(d + e*x) - 75*A*b^4*c*e^4*(d + e*x) + 126*b^2*B*c^3*d^2*e*(d + e*x)^2 -
 176*b^3*B*c^2*d*e^2*(d + e*x)^2 + 100*A*b^2*c^3*d*e^2*(d + e*x)^2 + 70*b^4*B*c*e^3*(d + e*x)^2 - 50*A*b^3*c^2
*e^3*(d + e*x)^2 + 18*b^2*B*c^3*d*e*(d + e*x)^3 - 14*b^3*B*c^2*e^2*(d + e*x)^3 + 10*A*b^2*c^3*e^2*(d + e*x)^3
+ 6*b^2*B*c^3*e*(d + e*x)^4))/(15*b^2*c^4*x*(-(c*d) + b*e + c*(d + e*x))) + ((2*b*B*c*d*(-(c*d) + b*e)^(7/2) -
 4*A*c^2*d*(-(c*d) + b*e)^(7/2) + 7*b^2*B*e*(-(c*d) + b*e)^(7/2) - 5*A*b*c*e*(-(c*d) + b*e)^(7/2))*ArcTan[(Sqr
t[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(b^3*c^(9/2)) + ((-2*b*B*d^(9/2) + 4*A*c*d^(9/2) - 9*A*b*
d^(7/2)*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3

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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)^(9/2)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.31, size = 846, normalized size = 2.19 \begin {gather*} \frac {{\left (2 \, B b d^{5} - 4 \, A c d^{5} + 9 \, A b d^{4} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {{\left (2 \, B b c^{5} d^{5} - 4 \, A c^{6} d^{5} - B b^{2} c^{4} d^{4} e + 11 \, A b c^{5} d^{4} e - 16 \, B b^{3} c^{3} d^{3} e^{2} - 4 \, A b^{2} c^{4} d^{3} e^{2} + 34 \, B b^{4} c^{2} d^{2} e^{3} - 14 \, A b^{3} c^{3} d^{2} e^{3} - 26 \, B b^{5} c d e^{4} + 16 \, A b^{4} c^{2} d e^{4} + 7 \, B b^{6} e^{5} - 5 \, A b^{5} c e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{4}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} B b c^{4} d^{4} e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{5} d^{4} e - \sqrt {x e + d} B b c^{4} d^{5} e + 2 \, \sqrt {x e + d} A c^{5} d^{5} e - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c^{3} d^{3} e^{2} + 4 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{4} d^{3} e^{2} + 4 \, \sqrt {x e + d} B b^{2} c^{3} d^{4} e^{2} - 5 \, \sqrt {x e + d} A b c^{4} d^{4} e^{2} + 6 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} c^{2} d^{2} e^{3} - 6 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c^{3} d^{2} e^{3} - 6 \, \sqrt {x e + d} B b^{3} c^{2} d^{3} e^{3} + 6 \, \sqrt {x e + d} A b^{2} c^{3} d^{3} e^{3} - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} c d e^{4} + 4 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} c^{2} d e^{4} + 4 \, \sqrt {x e + d} B b^{4} c d^{2} e^{4} - 4 \, \sqrt {x e + d} A b^{3} c^{2} d^{2} e^{4} + {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} e^{5} - {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} c e^{5} - \sqrt {x e + d} B b^{5} d e^{5} + \sqrt {x e + d} A b^{4} c d e^{5}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )} b^{2} c^{4}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B c^{8} e^{2} + 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B c^{8} d e^{2} + 90 \, \sqrt {x e + d} B c^{8} d^{2} e^{2} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c^{7} e^{3} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{8} e^{3} - 120 \, \sqrt {x e + d} B b c^{7} d e^{3} + 60 \, \sqrt {x e + d} A c^{8} d e^{3} + 45 \, \sqrt {x e + d} B b^{2} c^{6} e^{4} - 30 \, \sqrt {x e + d} A b c^{7} e^{4}\right )}}{15 \, c^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

(2*B*b*d^5 - 4*A*c*d^5 + 9*A*b*d^4*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) - (2*B*b*c^5*d^5 - 4*A*c^6
*d^5 - B*b^2*c^4*d^4*e + 11*A*b*c^5*d^4*e - 16*B*b^3*c^3*d^3*e^2 - 4*A*b^2*c^4*d^3*e^2 + 34*B*b^4*c^2*d^2*e^3
- 14*A*b^3*c^3*d^2*e^3 - 26*B*b^5*c*d*e^4 + 16*A*b^4*c^2*d*e^4 + 7*B*b^6*e^5 - 5*A*b^5*c*e^5)*arctan(sqrt(x*e
+ d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3*c^4) + ((x*e + d)^(3/2)*B*b*c^4*d^4*e - 2*(x*e + d)^(3/
2)*A*c^5*d^4*e - sqrt(x*e + d)*B*b*c^4*d^5*e + 2*sqrt(x*e + d)*A*c^5*d^5*e - 4*(x*e + d)^(3/2)*B*b^2*c^3*d^3*e
^2 + 4*(x*e + d)^(3/2)*A*b*c^4*d^3*e^2 + 4*sqrt(x*e + d)*B*b^2*c^3*d^4*e^2 - 5*sqrt(x*e + d)*A*b*c^4*d^4*e^2 +
 6*(x*e + d)^(3/2)*B*b^3*c^2*d^2*e^3 - 6*(x*e + d)^(3/2)*A*b^2*c^3*d^2*e^3 - 6*sqrt(x*e + d)*B*b^3*c^2*d^3*e^3
 + 6*sqrt(x*e + d)*A*b^2*c^3*d^3*e^3 - 4*(x*e + d)^(3/2)*B*b^4*c*d*e^4 + 4*(x*e + d)^(3/2)*A*b^3*c^2*d*e^4 + 4
*sqrt(x*e + d)*B*b^4*c*d^2*e^4 - 4*sqrt(x*e + d)*A*b^3*c^2*d^2*e^4 + (x*e + d)^(3/2)*B*b^5*e^5 - (x*e + d)^(3/
2)*A*b^4*c*e^5 - sqrt(x*e + d)*B*b^5*d*e^5 + sqrt(x*e + d)*A*b^4*c*d*e^5)/(((x*e + d)^2*c - 2*(x*e + d)*c*d +
c*d^2 + (x*e + d)*b*e - b*d*e)*b^2*c^4) + 2/15*(3*(x*e + d)^(5/2)*B*c^8*e^2 + 15*(x*e + d)^(3/2)*B*c^8*d*e^2 +
 90*sqrt(x*e + d)*B*c^8*d^2*e^2 - 10*(x*e + d)^(3/2)*B*b*c^7*e^3 + 5*(x*e + d)^(3/2)*A*c^8*e^3 - 120*sqrt(x*e
+ d)*B*b*c^7*d*e^3 + 60*sqrt(x*e + d)*A*c^8*d*e^3 + 45*sqrt(x*e + d)*B*b^2*c^6*e^4 - 30*sqrt(x*e + d)*A*b*c^7*
e^4)/c^10

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maple [B]  time = 0.10, size = 1075, normalized size = 2.78

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^2,x)

[Out]

e^5*b^3/c^4*(e*x+d)^(1/2)/(c*e*x+b*e)*B+5*e^5*b^2/c^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(
1/2)*c)*A-7*e^5*b^3/c^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B+16*e^2/c/((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/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-
c*d)*c)^(1/2)*c)*B*d^4+e/b*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^4+14*e^3/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/(
(b*e-c*d)*c)^(1/2)*c)*A*d^2+4*e^2/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^3-4*e^
2/c*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^3-6*e^3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^2+4*e^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)*
A*d^3-2/b^2*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^5+4/b^3*c^2/((b*e-c*d)*c)^(1
/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^5+2/3*e^3/c^2*A*(e*x+d)^(3/2)+2/5*e^2/c^2*B*(e*x+d)^(5/2)-
2*d^(9/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*B-16*e^3/c^3*B*b*d*(e*x+d)^(1/2)-e^5*b^2/c^3*(e*x+d)^(1/2)/(c*e*x
+b*e)*A-d^4/b^2*A*(e*x+d)^(1/2)/x+6*e^4/c^4*B*b^2*(e*x+d)^(1/2)+4*d^(9/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A
*c+2*e^2/c^2*B*(e*x+d)^(3/2)*d+12*e^2/c^2*B*d^2*(e*x+d)^(1/2)-9*e*d^(7/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*A
-4/3*e^3/c^3*B*(e*x+d)^(3/2)*b-4*e^4/c^3*A*b*(e*x+d)^(1/2)+8*e^3/c^2*A*d*(e*x+d)^(1/2)-4*e^4*b^2/c^3*(e*x+d)^(
1/2)/(c*e*x+b*e)*B*d+26*e^4*b^2/c^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d-34*e^3
*b/c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^2-16*e^4*b/c^2/((b*e-c*d)*c)^(1/2)*
arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d-11*e/b^2*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)
*c)^(1/2)*c)*A*d^4+6*e^3*b/c^2*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^2+4*e^4*b/c^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d-e/b^2
*c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^4

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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)^(9/2)/(c*x^2+b*x)^2,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?

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mupad [B]  time = 6.71, size = 12636, normalized size = 32.74

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^2,x)

[Out]

atan(((((20*A*b^10*c^6*d*e^7 - 28*B*b^11*c^5*d*e^7 + 8*A*b^6*c^10*d^5*e^3 - 20*A*b^7*c^9*d^4*e^4 + 56*A*b^8*c^
8*d^3*e^5 - 64*A*b^9*c^7*d^2*e^6 - 4*B*b^7*c^9*d^5*e^3 + 64*B*b^8*c^8*d^4*e^4 - 136*B*b^9*c^7*d^3*e^5 + 104*B*
b^10*c^6*d^2*e^6)/(b^6*c^7) - (2*(4*b^7*c^9*e^3 - 8*b^6*c^10*d*e^2)*(d + e*x)^(1/2)*((16*A^2*c^11*d^9 - 49*B^2
*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^
2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c
^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^
3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*
b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*
A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^
8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2))/(b^4*c^7))*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2
*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^
4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 +
 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A
^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*
B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*
d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2) - (2
*(d + e*x)^(1/2)*(49*B^2*b^12*e^12 + 25*A^2*b^10*c^2*e^12 + 32*A^2*c^12*d^10*e^2 + 234*A^2*b^2*c^10*d^8*e^4 +
24*A^2*b^3*c^9*d^7*e^5 - 420*A^2*b^4*c^8*d^6*e^6 + 504*A^2*b^5*c^7*d^5*e^7 - 42*A^2*b^6*c^6*d^4*e^8 - 408*A^2*
b^7*c^5*d^3*e^9 + 396*A^2*b^8*c^4*d^2*e^10 + 8*B^2*b^2*c^10*d^10*e^2 - 4*B^2*b^3*c^9*d^9*e^3 - 63*B^2*b^4*c^8*
d^8*e^4 + 168*B^2*b^5*c^7*d^7*e^5 + 84*B^2*b^6*c^6*d^6*e^6 - 1008*B^2*b^7*c^5*d^5*e^7 + 1974*B^2*b^8*c^4*d^4*e
^8 - 1992*B^2*b^9*c^3*d^3*e^9 + 1152*B^2*b^10*c^2*d^2*e^10 - 364*B^2*b^11*c*d*e^11 - 160*A^2*b*c^11*d^9*e^3 -
160*A^2*b^9*c^3*d*e^11 - 70*A*B*b^11*c*e^12 - 32*A*B*b*c^11*d^10*e^2 + 484*A*B*b^10*c^2*d*e^11 + 88*A*B*b^2*c^
10*d^9*e^3 + 90*A*B*b^3*c^9*d^8*e^4 - 672*A*B*b^4*c^8*d^7*e^5 + 1176*A*B*b^5*c^7*d^6*e^6 - 504*A*B*b^6*c^6*d^5
*e^7 - 1092*A*B*b^7*c^5*d^4*e^8 + 1920*A*B*b^8*c^4*d^3*e^9 - 1368*A*B*b^9*c^3*d^2*e^10))/(b^4*c^7))*((16*A^2*c
^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*
d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^
7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 115
5*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3
*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^
8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*
e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2)*1i - (((20*A*b^10*c^6*d*e^7 - 28*B*b^11*c^5*d*e^7 + 8*A*b^6*
c^10*d^5*e^3 - 20*A*b^7*c^9*d^4*e^4 + 56*A*b^8*c^8*d^3*e^5 - 64*A*b^9*c^7*d^2*e^6 - 4*B*b^7*c^9*d^5*e^3 + 64*B
*b^8*c^8*d^4*e^4 - 136*B*b^9*c^7*d^3*e^5 + 104*B*b^10*c^6*d^2*e^6)/(b^6*c^7) + (2*(4*b^7*c^9*e^3 - 8*b^6*c^10*
d*e^2)*(d + e*x)^(1/2)*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b
^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5
*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^
4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B
^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A
*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c
^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2))/(b^4*c^7))*((16*A^2*c^11*d
^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e
^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 6
3*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2
*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^
8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7
*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 +
 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2) + (2*(d + e*x)^(1/2)*(49*B^2*b^12*e^12 + 25*A^2*b^10*c^2*e^12 + 3
2*A^2*c^12*d^10*e^2 + 234*A^2*b^2*c^10*d^8*e^4 + 24*A^2*b^3*c^9*d^7*e^5 - 420*A^2*b^4*c^8*d^6*e^6 + 504*A^2*b^
5*c^7*d^5*e^7 - 42*A^2*b^6*c^6*d^4*e^8 - 408*A^2*b^7*c^5*d^3*e^9 + 396*A^2*b^8*c^4*d^2*e^10 + 8*B^2*b^2*c^10*d
^10*e^2 - 4*B^2*b^3*c^9*d^9*e^3 - 63*B^2*b^4*c^8*d^8*e^4 + 168*B^2*b^5*c^7*d^7*e^5 + 84*B^2*b^6*c^6*d^6*e^6 -
1008*B^2*b^7*c^5*d^5*e^7 + 1974*B^2*b^8*c^4*d^4*e^8 - 1992*B^2*b^9*c^3*d^3*e^9 + 1152*B^2*b^10*c^2*d^2*e^10 -
364*B^2*b^11*c*d*e^11 - 160*A^2*b*c^11*d^9*e^3 - 160*A^2*b^9*c^3*d*e^11 - 70*A*B*b^11*c*e^12 - 32*A*B*b*c^11*d
^10*e^2 + 484*A*B*b^10*c^2*d*e^11 + 88*A*B*b^2*c^10*d^9*e^3 + 90*A*B*b^3*c^9*d^8*e^4 - 672*A*B*b^4*c^8*d^7*e^5
 + 1176*A*B*b^5*c^7*d^6*e^6 - 504*A*B*b^6*c^6*d^5*e^7 - 1092*A*B*b^7*c^5*d^4*e^8 + 1920*A*B*b^8*c^4*d^3*e^9 -
1368*A*B*b^9*c^3*d^2*e^10))/(b^4*c^7))*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^
9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 +
 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^
2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^
10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c
^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4
 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2)*1i)/((((20*
A*b^10*c^6*d*e^7 - 28*B*b^11*c^5*d*e^7 + 8*A*b^6*c^10*d^5*e^3 - 20*A*b^7*c^9*d^4*e^4 + 56*A*b^8*c^8*d^3*e^5 -
64*A*b^9*c^7*d^2*e^6 - 4*B*b^7*c^9*d^5*e^3 + 64*B*b^8*c^8*d^4*e^4 - 136*B*b^9*c^7*d^3*e^5 + 104*B*b^10*c^6*d^2
*e^6)/(b^6*c^7) - (2*(4*b^7*c^9*e^3 - 8*b^6*c^10*d*e^2)*(d + e*x)^(1/2)*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 -
25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^
5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 +
 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*
B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 +
 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*
d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^
7)/(4*b^6*c^9))^(1/2))/(b^4*c^7))*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9
 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*
A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6
*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^
8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^
8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 12
6*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2) - (2*(d + e*x)^(
1/2)*(49*B^2*b^12*e^12 + 25*A^2*b^10*c^2*e^12 + 32*A^2*c^12*d^10*e^2 + 234*A^2*b^2*c^10*d^8*e^4 + 24*A^2*b^3*c
^9*d^7*e^5 - 420*A^2*b^4*c^8*d^6*e^6 + 504*A^2*b^5*c^7*d^5*e^7 - 42*A^2*b^6*c^6*d^4*e^8 - 408*A^2*b^7*c^5*d^3*
e^9 + 396*A^2*b^8*c^4*d^2*e^10 + 8*B^2*b^2*c^10*d^10*e^2 - 4*B^2*b^3*c^9*d^9*e^3 - 63*B^2*b^4*c^8*d^8*e^4 + 16
8*B^2*b^5*c^7*d^7*e^5 + 84*B^2*b^6*c^6*d^6*e^6 - 1008*B^2*b^7*c^5*d^5*e^7 + 1974*B^2*b^8*c^4*d^4*e^8 - 1992*B^
2*b^9*c^3*d^3*e^9 + 1152*B^2*b^10*c^2*d^2*e^10 - 364*B^2*b^11*c*d*e^11 - 160*A^2*b*c^11*d^9*e^3 - 160*A^2*b^9*
c^3*d*e^11 - 70*A*B*b^11*c*e^12 - 32*A*B*b*c^11*d^10*e^2 + 484*A*B*b^10*c^2*d*e^11 + 88*A*B*b^2*c^10*d^9*e^3 +
 90*A*B*b^3*c^9*d^8*e^4 - 672*A*B*b^4*c^8*d^7*e^5 + 1176*A*B*b^5*c^7*d^6*e^6 - 504*A*B*b^6*c^6*d^5*e^7 - 1092*
A*B*b^7*c^5*d^4*e^8 + 1920*A*B*b^8*c^4*d^3*e^9 - 1368*A*B*b^9*c^3*d^2*e^10))/(b^4*c^7))*((16*A^2*c^11*d^9 - 49
*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 31
5*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b
^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^
3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*
A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 -
546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*
B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2) + (((20*A*b^10*c^6*d*e^7 - 28*B*b^11*c^5*d*e^7 + 8*A*b^6*c^10*d^5*e^3 -
20*A*b^7*c^9*d^4*e^4 + 56*A*b^8*c^8*d^3*e^5 - 64*A*b^9*c^7*d^2*e^6 - 4*B*b^7*c^9*d^5*e^3 + 64*B*b^8*c^8*d^4*e^
4 - 136*B*b^9*c^7*d^3*e^5 + 104*B*b^10*c^6*d^2*e^6)/(b^6*c^7) + (2*(4*b^7*c^9*e^3 - 8*b^6*c^10*d*e^2)*(d + e*x
)^(1/2)*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2
+ 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*
A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7
*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8
 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^
8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 96
6*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2))/(b^4*c^7))*((16*A^2*c^11*d^9 - 49*B^2*b^1
1*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^
4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d
^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^
6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^
10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*
b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^
3*d^2*e^7)/(4*b^6*c^9))^(1/2) + (2*(d + e*x)^(1/2)*(49*B^2*b^12*e^12 + 25*A^2*b^10*c^2*e^12 + 32*A^2*c^12*d^10
*e^2 + 234*A^2*b^2*c^10*d^8*e^4 + 24*A^2*b^3*c^9*d^7*e^5 - 420*A^2*b^4*c^8*d^6*e^6 + 504*A^2*b^5*c^7*d^5*e^7 -
 42*A^2*b^6*c^6*d^4*e^8 - 408*A^2*b^7*c^5*d^3*e^9 + 396*A^2*b^8*c^4*d^2*e^10 + 8*B^2*b^2*c^10*d^10*e^2 - 4*B^2
*b^3*c^9*d^9*e^3 - 63*B^2*b^4*c^8*d^8*e^4 + 168*B^2*b^5*c^7*d^7*e^5 + 84*B^2*b^6*c^6*d^6*e^6 - 1008*B^2*b^7*c^
5*d^5*e^7 + 1974*B^2*b^8*c^4*d^4*e^8 - 1992*B^2*b^9*c^3*d^3*e^9 + 1152*B^2*b^10*c^2*d^2*e^10 - 364*B^2*b^11*c*
d*e^11 - 160*A^2*b*c^11*d^9*e^3 - 160*A^2*b^9*c^3*d*e^11 - 70*A*B*b^11*c*e^12 - 32*A*B*b*c^11*d^10*e^2 + 484*A
*B*b^10*c^2*d*e^11 + 88*A*B*b^2*c^10*d^9*e^3 + 90*A*B*b^3*c^9*d^8*e^4 - 672*A*B*b^4*c^8*d^7*e^5 + 1176*A*B*b^5
*c^7*d^6*e^6 - 504*A*B*b^6*c^6*d^5*e^7 - 1092*A*B*b^7*c^5*d^4*e^8 + 1920*A*B*b^8*c^4*d^3*e^9 - 1368*A*B*b^9*c^
3*d^2*e^10))/(b^4*c^7))*((16*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*
b^2*c^9*d^7*e^2 + 105*A^2*b^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^
5*d^3*e^6 - 261*A^2*b^7*c^4*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e
^4 - 819*B^2*b^7*c^4*d^4*e^5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*
B^2*b^10*c*d*e^8 + 135*A^2*b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*
A*B*b^9*c^2*d*e^8 + 126*A*B*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*
c^5*d^4*e^5 - 966*A*B*b^7*c^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2) - (2*(32*A^3*c^12*d^14*e^3
 + 98*B^3*b^12*d^5*e^12 + 326*A^3*b^2*c^10*d^12*e^5 + 956*A^3*b^3*c^9*d^11*e^6 - 3430*A^3*b^4*c^8*d^10*e^7 + 3
048*A^3*b^5*c^7*d^9*e^8 + 1659*A^3*b^6*c^6*d^8*e^9 - 5256*A^3*b^7*c^5*d^7*e^10 + 4204*A^3*b^8*c^4*d^6*e^11 - 1
540*A^3*b^9*c^3*d^5*e^12 + 225*A^3*b^10*c^2*d^4*e^13 - 4*B^3*b^3*c^9*d^14*e^3 - 62*B^3*b^4*c^8*d^13*e^4 + 200*
B^3*b^5*c^7*d^12*e^5 + 272*B^3*b^6*c^6*d^11*e^6 - 2044*B^3*b^7*c^5*d^10*e^7 + 3948*B^3*b^8*c^4*d^9*e^8 - 3984*
B^3*b^9*c^3*d^8*e^9 + 2304*B^3*b^10*c^2*d^7*e^10 + 441*A*B^2*b^12*d^4*e^13 - 224*A^3*b*c^11*d^13*e^4 - 728*B^3
*b^11*c*d^6*e^11 + 24*A*B^2*b^2*c^10*d^14*e^3 + 192*A*B^2*b^3*c^9*d^13*e^4 - 1407*A*B^2*b^4*c^8*d^12*e^5 + 182
4*A*B^2*b^5*c^7*d^11*e^6 + 4848*A*B^2*b^6*c^6*d^10*e^7 - 19404*A*B^2*b^7*c^5*d^9*e^8 + 29574*A*B^2*b^8*c^4*d^8
*e^9 - 25272*A*B^2*b^9*c^3*d^7*e^10 + 12792*A*B^2*b^10*c^2*d^6*e^11 - 24*A^2*B*b^2*c^10*d^13*e^4 + 1851*A^2*B*
b^3*c^9*d^12*e^5 - 6672*A^2*B*b^4*c^8*d^11*e^6 + 8214*A^2*B*b^5*c^7*d^10*e^7 + 2538*A^2*B*b^6*c^6*d^9*e^8 - 18
891*A^2*B*b^7*c^5*d^8*e^9 + 23544*A^2*B*b^8*c^4*d^7*e^10 - 14568*A^2*B*b^9*c^3*d^6*e^11 + 4686*A^2*B*b^10*c^2*
d^5*e^12 - 3612*A*B^2*b^11*c*d^5*e^12 - 48*A^2*B*b*c^11*d^14*e^3 - 630*A^2*B*b^11*c*d^4*e^13))/(b^6*c^7)))*((1
6*A^2*c^11*d^9 - 49*B^2*b^11*e^9 - 25*A^2*b^9*c^2*e^9 + 4*B^2*b^2*c^9*d^9 + 81*A^2*b^2*c^9*d^7*e^2 + 105*A^2*b
^3*c^8*d^6*e^3 - 315*A^2*b^4*c^7*d^5*e^4 + 189*A^2*b^5*c^6*d^4*e^5 + 147*A^2*b^6*c^5*d^3*e^6 - 261*A^2*b^7*c^4
*d^2*e^7 - 63*B^2*b^4*c^7*d^7*e^2 + 105*B^2*b^5*c^6*d^6*e^3 + 189*B^2*b^6*c^5*d^5*e^4 - 819*B^2*b^7*c^4*d^4*e^
5 + 1155*B^2*b^8*c^3*d^3*e^6 - 837*B^2*b^9*c^2*d^2*e^7 - 72*A^2*b*c^10*d^8*e + 315*B^2*b^10*c*d*e^8 + 135*A^2*
b^8*c^3*d*e^8 - 16*A*B*b*c^10*d^9 + 70*A*B*b^10*c*e^9 + 36*A*B*b^2*c^9*d^8*e - 414*A*B*b^9*c^2*d*e^8 + 126*A*B
*b^3*c^8*d^7*e^2 - 546*A*B*b^4*c^7*d^6*e^3 + 630*A*B*b^5*c^6*d^5*e^4 + 126*A*B*b^6*c^5*d^4*e^5 - 966*A*B*b^7*c
^4*d^3*e^6 + 954*A*B*b^8*c^3*d^2*e^7)/(4*b^6*c^9))^(1/2)*2i - (((d + e*x)^(1/2)*(2*A*c^5*d^5*e - B*b^5*d*e^5 -
 5*A*b*c^4*d^4*e^2 + 4*B*b^4*c*d^2*e^4 + 6*A*b^2*c^3*d^3*e^3 - 4*A*b^3*c^2*d^2*e^4 + 4*B*b^2*c^3*d^4*e^2 - 6*B
*b^3*c^2*d^3*e^3 + A*b^4*c*d*e^5 - B*b*c^4*d^5*e))/b^2 + ((d + e*x)^(3/2)*(B*b^5*e^5 - A*b^4*c*e^5 - 2*A*c^5*d
^4*e + 4*A*b*c^4*d^3*e^2 + 4*A*b^3*c^2*d*e^4 - 6*A*b^2*c^3*d^2*e^3 - 4*B*b^2*c^3*d^3*e^2 + 6*B*b^3*c^2*d^2*e^3
 + B*b*c^4*d^4*e - 4*B*b^4*c*d*e^4))/b^2)/((2*c^5*d - b*c^4*e)*(d + e*x) - c^5*(d + e*x)^2 - c^5*d^2 + b*c^4*d
*e) - log((((((4*d*e^3*(b*e - c*d)*(2*A*c^4*d^3 + 7*B*b^4*e^3 - 5*A*b^3*c*e^3 - B*b*c^3*d^3 + 11*A*b^2*c^2*d*e
^2 + 15*B*b^2*c^2*d^2*e - 3*A*b*c^3*d^2*e - 19*B*b^3*c*d*e^2))/c^2 + 4*b^2*c^2*e^2*(b*e - 2*c*d)*(d + e*x)^(1/
2)*((d^7*(9*A*b*e - 4*A*c*d + 2*B*b*d)^2)/b^6)^(1/2))*((d^7*(9*A*b*e - 4*A*c*d + 2*B*b*d)^2)/b^6)^(1/2))/2 + (
2*(d + e*x)^(1/2)*(49*B^2*b^12*e^12 + 25*A^2*b^10*c^2*e^12 + 32*A^2*c^12*d^10*e^2 + 234*A^2*b^2*c^10*d^8*e^4 +
 24*A^2*b^3*c^9*d^7*e^5 - 420*A^2*b^4*c^8*d^6*e^6 + 504*A^2*b^5*c^7*d^5*e^7 - 42*A^2*b^6*c^6*d^4*e^8 - 408*A^2
*b^7*c^5*d^3*e^9 + 396*A^2*b^8*c^4*d^2*e^10 + 8*B^2*b^2*c^10*d^10*e^2 - 4*B^2*b^3*c^9*d^9*e^3 - 63*B^2*b^4*c^8
*d^8*e^4 + 168*B^2*b^5*c^7*d^7*e^5 + 84*B^2*b^6*c^6*d^6*e^6 - 1008*B^2*b^7*c^5*d^5*e^7 + 1974*B^2*b^8*c^4*d^4*
e^8 - 1992*B^2*b^9*c^3*d^3*e^9 + 1152*B^2*b^10*c^2*d^2*e^10 - 364*B^2*b^11*c*d*e^11 - 160*A^2*b*c^11*d^9*e^3 -
 160*A^2*b^9*c^3*d*e^11 - 70*A*B*b^11*c*e^12 - 32*A*B*b*c^11*d^10*e^2 + 484*A*B*b^10*c^2*d*e^11 + 88*A*B*b^2*c
^10*d^9*e^3 + 90*A*B*b^3*c^9*d^8*e^4 - 672*A*B*b^4*c^8*d^7*e^5 + 1176*A*B*b^5*c^7*d^6*e^6 - 504*A*B*b^6*c^6*d^
5*e^7 - 1092*A*B*b^7*c^5*d^4*e^8 + 1920*A*B*b^8*c^4*d^3*e^9 - 1368*A*B*b^9*c^3*d^2*e^10))/(b^4*c^7))*((d^7*(9*
A*b*e - 4*A*c*d + 2*B*b*d)^2)/b^6)^(1/2))/2 - (d^4*e^3*(b*e - c*d)^4*(4*B^3*b^3*c^5*d^6 - 225*A^3*b^6*c^2*e^6
- 32*A^3*c^8*d^6 - 441*A*B^2*b^8*e^6 - 98*B^3*b^8*d*e^5 + 250*A^3*b^2*c^6*d^4*e^2 - 660*A^3*b^3*c^5*d^3*e^3 -
294*A^3*b^4*c^4*d^2*e^4 + 88*B^3*b^5*c^3*d^4*e^2 - 372*B^3*b^6*c^2*d^3*e^3 + 48*A^2*B*b*c^7*d^6 + 630*A^2*B*b^
7*c*e^6 + 96*A^3*b*c^7*d^5*e - 24*A*B^2*b^2*c^6*d^6 + 640*A^3*b^5*c^3*d*e^5 + 78*B^3*b^4*c^4*d^5*e + 336*B^3*b
^7*c*d^2*e^4 + 399*A*B^2*b^4*c^4*d^4*e^2 + 1404*A*B^2*b^5*c^3*d^3*e^3 - 2754*A*B^2*b^6*c^2*d^2*e^4 - 1275*A^2*
B*b^3*c^5*d^4*e^2 + 468*A^2*B*b^4*c^4*d^3*e^3 + 2124*A^2*B*b^5*c^3*d^2*e^4 + 1848*A*B^2*b^7*c*d*e^5 - 288*A*B^
2*b^3*c^5*d^5*e + 216*A^2*B*b^2*c^6*d^5*e - 2166*A^2*B*b^6*c^2*d*e^5))/(b^6*c^7))*((4*A^2*c^2*d^9 + B^2*b^2*d^
9 + (81*A^2*b^2*d^7*e^2)/4 - 4*A*B*b*c*d^9 + 9*A*B*b^2*d^8*e - 18*A^2*b*c*d^8*e)/b^6)^(1/2) + log((((((4*d*e^3
*(b*e - c*d)*(2*A*c^4*d^3 + 7*B*b^4*e^3 - 5*A*b^3*c*e^3 - B*b*c^3*d^3 + 11*A*b^2*c^2*d*e^2 + 15*B*b^2*c^2*d^2*
e - 3*A*b*c^3*d^2*e - 19*B*b^3*c*d*e^2))/c^2 - 4*b^2*c^2*e^2*(b*e - 2*c*d)*(d + e*x)^(1/2)*((d^7*(9*A*b*e - 4*
A*c*d + 2*B*b*d)^2)/b^6)^(1/2))*((d^7*(9*A*b*e - 4*A*c*d + 2*B*b*d)^2)/b^6)^(1/2))/2 - (2*(d + e*x)^(1/2)*(49*
B^2*b^12*e^12 + 25*A^2*b^10*c^2*e^12 + 32*A^2*c^12*d^10*e^2 + 234*A^2*b^2*c^10*d^8*e^4 + 24*A^2*b^3*c^9*d^7*e^
5 - 420*A^2*b^4*c^8*d^6*e^6 + 504*A^2*b^5*c^7*d^5*e^7 - 42*A^2*b^6*c^6*d^4*e^8 - 408*A^2*b^7*c^5*d^3*e^9 + 396
*A^2*b^8*c^4*d^2*e^10 + 8*B^2*b^2*c^10*d^10*e^2 - 4*B^2*b^3*c^9*d^9*e^3 - 63*B^2*b^4*c^8*d^8*e^4 + 168*B^2*b^5
*c^7*d^7*e^5 + 84*B^2*b^6*c^6*d^6*e^6 - 1008*B^2*b^7*c^5*d^5*e^7 + 1974*B^2*b^8*c^4*d^4*e^8 - 1992*B^2*b^9*c^3
*d^3*e^9 + 1152*B^2*b^10*c^2*d^2*e^10 - 364*B^2*b^11*c*d*e^11 - 160*A^2*b*c^11*d^9*e^3 - 160*A^2*b^9*c^3*d*e^1
1 - 70*A*B*b^11*c*e^12 - 32*A*B*b*c^11*d^10*e^2 + 484*A*B*b^10*c^2*d*e^11 + 88*A*B*b^2*c^10*d^9*e^3 + 90*A*B*b
^3*c^9*d^8*e^4 - 672*A*B*b^4*c^8*d^7*e^5 + 1176*A*B*b^5*c^7*d^6*e^6 - 504*A*B*b^6*c^6*d^5*e^7 - 1092*A*B*b^7*c
^5*d^4*e^8 + 1920*A*B*b^8*c^4*d^3*e^9 - 1368*A*B*b^9*c^3*d^2*e^10))/(b^4*c^7))*((d^7*(9*A*b*e - 4*A*c*d + 2*B*
b*d)^2)/b^6)^(1/2))/2 - (d^4*e^3*(b*e - c*d)^4*(4*B^3*b^3*c^5*d^6 - 225*A^3*b^6*c^2*e^6 - 32*A^3*c^8*d^6 - 441
*A*B^2*b^8*e^6 - 98*B^3*b^8*d*e^5 + 250*A^3*b^2*c^6*d^4*e^2 - 660*A^3*b^3*c^5*d^3*e^3 - 294*A^3*b^4*c^4*d^2*e^
4 + 88*B^3*b^5*c^3*d^4*e^2 - 372*B^3*b^6*c^2*d^3*e^3 + 48*A^2*B*b*c^7*d^6 + 630*A^2*B*b^7*c*e^6 + 96*A^3*b*c^7
*d^5*e - 24*A*B^2*b^2*c^6*d^6 + 640*A^3*b^5*c^3*d*e^5 + 78*B^3*b^4*c^4*d^5*e + 336*B^3*b^7*c*d^2*e^4 + 399*A*B
^2*b^4*c^4*d^4*e^2 + 1404*A*B^2*b^5*c^3*d^3*e^3 - 2754*A*B^2*b^6*c^2*d^2*e^4 - 1275*A^2*B*b^3*c^5*d^4*e^2 + 46
8*A^2*B*b^4*c^4*d^3*e^3 + 2124*A^2*B*b^5*c^3*d^2*e^4 + 1848*A*B^2*b^7*c*d*e^5 - 288*A*B^2*b^3*c^5*d^5*e + 216*
A^2*B*b^2*c^6*d^5*e - 2166*A^2*B*b^6*c^2*d*e^5))/(b^6*c^7))*((16*A^2*c^2*d^9 + 4*B^2*b^2*d^9 + 81*A^2*b^2*d^7*
e^2 - 16*A*B*b*c*d^9 + 36*A*B*b^2*d^8*e - 72*A^2*b*c*d^8*e)/(4*b^6))^(1/2) + ((2*A*e^3 - 2*B*d*e^2)/(3*c^2) +
(2*B*e^2*(4*c^2*d - 2*b*c*e))/(3*c^4))*(d + e*x)^(3/2) + (((4*c^2*d - 2*b*c*e)*((2*A*e^3 - 2*B*d*e^2)/c^2 + (2
*B*e^2*(4*c^2*d - 2*b*c*e))/c^4))/c^2 - (2*B*e^2*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e))/c^4)*(d + e*x)^(1/2) + (2*
B*e^2*(d + e*x)^(5/2))/(5*c^2)

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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)**(9/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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