\(\int \frac {A+B x}{\sqrt {d+e x} (b x+c x^2)^3} \, dx\) [1252]

Optimal result

Integrand size = 26, antiderivative size = 394 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}-\frac {\left (48 A c^2 d^2-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{5/2}}+\frac {c^{3/2} \left (48 A c^3 d^2-35 b^3 B e^2-12 b c^2 d (2 B d+9 A e)+7 b^2 c e (8 B d+9 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}} \]

[Out]

-1/4*(48*A*c^2*d^2-b^2*e*(-3*A*e+4*B*d)-12*b*c*d*(-A*e+2*B*d))*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^(5/2)+1/4*
c^(3/2)*(48*A*c^3*d^2-35*b^3*B*e^2-12*b*c^2*d*(9*A*e+2*B*d)+7*b^2*c*e*(9*A*e+8*B*d))*arctanh(c^(1/2)*(e*x+d)^(
1/2)/(-b*e+c*d)^(1/2))/b^5/(-b*e+c*d)^(5/2)-1/2*(A*b*(-b*e+c*d)+c*(2*A*c*d-b*(A*e+B*d))*x)*(e*x+d)^(1/2)/b^2/d
/(-b*e+c*d)/(c*x^2+b*x)^2+1/4*(b*(-b*e+c*d)*(12*A*c^2*d^2+b^2*e*(-3*A*e+4*B*d)-b*c*d*(7*A*e+6*B*d))+c*(24*A*c^
3*d^3-b^3*e^2*(-3*A*e+4*B*d)-12*b*c^2*d^2*(3*A*e+B*d)+b^2*c*d*e*(6*A*e+19*B*d))*x)*(e*x+d)^(1/2)/b^4/d^2/(-b*e
+c*d)^2/(c*x^2+b*x)

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {836, 840, 1180, 214} \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 (-e) (4 B d-3 A e)-12 b c d (2 B d-A e)+48 A c^2 d^2\right )}{4 b^5 d^{5/2}}+\frac {c^{3/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2} \]

[In]

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

[Out]

-1/2*(Sqrt[d + e*x]*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(b*x + c*x^2)^2) + (
Sqrt[d + e*x]*(b*(c*d - b*e)*(12*A*c^2*d^2 + b^2*e*(4*B*d - 3*A*e) - b*c*d*(6*B*d + 7*A*e)) + c*(24*A*c^3*d^3
- b^3*e^2*(4*B*d - 3*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(19*B*d + 6*A*e))*x))/(4*b^4*d^2*(c*d - b*e
)^2*(b*x + c*x^2)) - ((48*A*c^2*d^2 - b^2*e*(4*B*d - 3*A*e) - 12*b*c*d*(2*B*d - A*e))*ArcTanh[Sqrt[d + e*x]/Sq
rt[d]])/(4*b^5*d^(5/2)) + (c^(3/2)*(48*A*c^3*d^2 - 35*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 9*A*e) + 7*b^2*c*e*(8*B*
d + 9*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(5/2))

Rule 214

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

Rule 836

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 840

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 1180

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*} \text {integral}& = -\frac {\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) \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-3 A e)-b c d (6 B d+7 A e)\right )-\frac {5}{2} c e (b B d-2 A c d+A b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)} \\ & = -\frac {\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) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x\right )}{4 b^4 d^2 (c d-b e)^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-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right )+\frac {1}{4} c e \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2} \\ & = -\frac {\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) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {1}{4} e (c d-b e)^2 \left (48 A c^2 d^2-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right )-\frac {1}{4} c d e \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right )+\frac {1}{4} c e \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 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^2 (c d-b e)^2} \\ & = -\frac {\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) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac {\left (c \left (48 A c^2 d^2-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right )\right ) \text {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^2}-\frac {\left (c^2 \left (48 A c^3 d^2-35 b^3 B e^2-12 b c^2 d (2 B d+9 A e)+7 b^2 c e (8 B d+9 A e)\right )\right ) \text {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)^2} \\ & = -\frac {\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) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right )+c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}-\frac {\left (48 A c^2 d^2-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{5/2}}+\frac {c^{3/2} \left (48 A c^3 d^2-35 b^3 B e^2-12 b c^2 d (2 B d+9 A e)+7 b^2 c e (8 B d+9 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)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.49 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=-\frac {\frac {b \sqrt {d+e x} \left (b B d x \left (4 b^4 e^2+12 c^4 d^2 x^2+b c^3 d x (18 d-19 e x)+8 b^3 c e (-d+e x)+b^2 c^2 \left (4 d^2-29 d e x+4 e^2 x^2\right )\right )+A \left (-24 c^5 d^3 x^3+b^5 e^2 (2 d-3 e x)-36 b c^4 d^2 x^2 (d-e x)+b^2 c^3 d x \left (-8 d^2+55 d e x-6 e^2 x^2\right )-2 b^4 c e \left (2 d^2+d e x+3 e^2 x^2\right )+b^3 c^2 \left (2 d^3+13 d^2 e x-10 d e^2 x^2-3 e^3 x^3\right )\right )\right )}{d^2 (c d-b e)^2 x^2 (b+c x)^2}+\frac {c^{3/2} \left (48 A c^3 d^2-35 b^3 B e^2-12 b c^2 d (2 B d+9 A e)+7 b^2 c e (8 B d+9 A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}+\frac {\left (48 A c^2 d^2+12 b c d (-2 B d+A e)+b^2 e (-4 B d+3 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2}}}{4 b^5} \]

[In]

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

[Out]

-1/4*((b*Sqrt[d + e*x]*(b*B*d*x*(4*b^4*e^2 + 12*c^4*d^2*x^2 + b*c^3*d*x*(18*d - 19*e*x) + 8*b^3*c*e*(-d + e*x)
+ b^2*c^2*(4*d^2 - 29*d*e*x + 4*e^2*x^2)) + A*(-24*c^5*d^3*x^3 + b^5*e^2*(2*d - 3*e*x) - 36*b*c^4*d^2*x^2*(d
- e*x) + b^2*c^3*d*x*(-8*d^2 + 55*d*e*x - 6*e^2*x^2) - 2*b^4*c*e*(2*d^2 + d*e*x + 3*e^2*x^2) + b^3*c^2*(2*d^3
+ 13*d^2*e*x - 10*d*e^2*x^2 - 3*e^3*x^3))))/(d^2*(c*d - b*e)^2*x^2*(b + c*x)^2) + (c^(3/2)*(48*A*c^3*d^2 - 35*
b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 9*A*e) + 7*b^2*c*e*(8*B*d + 9*A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d)
+ b*e]])/(-(c*d) + b*e)^(5/2) + ((48*A*c^2*d^2 + 12*b*c*d*(-2*B*d + A*e) + b^2*e*(-4*B*d + 3*A*e))*ArcTanh[Sq
rt[d + e*x]/Sqrt[d]])/d^(5/2))/b^5

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.96

method	result	size
risch	\(-\frac {\sqrt {e x +d}\, \left (-3 A b e x -12 A c d x +4 B b d x +2 A b d \right )}{4 d^{2} b^{4} x^{2}}+\frac {e \left (-\frac {8 c^{2} d^{2} \left (\frac {\frac {b c e \left (15 A b c e -12 A \,c^{2} d -11 b^{2} B e +8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b^{2} e^{2}-16 b c d e +8 c^{2} d^{2}}+\frac {\left (17 A b c e -12 A \,c^{2} d -13 b^{2} B e +8 B b c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (63 A \,b^{2} c \,e^{2}-108 A b \,c^{2} d e +48 A \,c^{3} d^{2}-35 b^{3} B \,e^{2}+56 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}}\right )}{e b}-\frac {\left (3 A \,b^{2} e^{2}+12 A b c d e +48 A \,c^{2} d^{2}-4 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}\right )}{4 b^{4} d^{2}}\)	\(379\)
derivativedivides	\(2 e^{4} \left (-\frac {c^{2} \left (\frac {\frac {b c e \left (15 A b c e -12 A \,c^{2} d -11 b^{2} B e +8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b^{2} e^{2}-16 b c d e +8 c^{2} d^{2}}+\frac {\left (17 A b c e -12 A \,c^{2} d -13 b^{2} B e +8 B b c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (63 A \,b^{2} c \,e^{2}-108 A b \,c^{2} d e +48 A \,c^{3} d^{2}-35 b^{3} B \,e^{2}+56 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{4}}-\frac {\frac {-\frac {b e \left (3 A b e +12 A c d -4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d^{2}}+\frac {b e \left (5 A b e +12 A c d -4 B b d \right ) \sqrt {e x +d}}{8 d}}{e^{2} x^{2}}+\frac {\left (3 A \,b^{2} e^{2}+12 A b c d e +48 A \,c^{2} d^{2}-4 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {5}{2}}}}{e^{4} b^{5}}\right )\)	\(400\)
default	\(2 e^{4} \left (-\frac {c^{2} \left (\frac {\frac {b c e \left (15 A b c e -12 A \,c^{2} d -11 b^{2} B e +8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b^{2} e^{2}-16 b c d e +8 c^{2} d^{2}}+\frac {\left (17 A b c e -12 A \,c^{2} d -13 b^{2} B e +8 B b c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (63 A \,b^{2} c \,e^{2}-108 A b \,c^{2} d e +48 A \,c^{3} d^{2}-35 b^{3} B \,e^{2}+56 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{4}}-\frac {\frac {-\frac {b e \left (3 A b e +12 A c d -4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d^{2}}+\frac {b e \left (5 A b e +12 A c d -4 B b d \right ) \sqrt {e x +d}}{8 d}}{e^{2} x^{2}}+\frac {\left (3 A \,b^{2} e^{2}+12 A b c d e +48 A \,c^{2} d^{2}-4 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {5}{2}}}}{e^{4} b^{5}}\right )\)	\(400\)
pseudoelliptic	\(\frac {-c^{2} \left (63 A \,b^{2} c \,e^{2}-108 A b \,c^{2} d e +48 A \,c^{3} d^{2}-35 b^{3} B \,e^{2}+56 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) x^{2} \left (c x +b \right )^{2} d^{\frac {9}{2}} \left (b e -c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+\sqrt {\left (b e -c d \right ) c}\, \left (-\left (3 A \,b^{2} e^{2}+12 A b c d e +48 A \,c^{2} d^{2}-4 B \,b^{2} d e -24 c \,d^{2} B b \right ) x^{2} \left (c x +b \right )^{2} \left (b e -c d \right )^{3} d^{2} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )-\sqrt {e x +d}\, d^{\frac {5}{2}} \left (-3 A \,b^{3} c^{2} e^{3} x^{3}-6 A \,b^{2} c^{3} d \,e^{2} x^{3}+36 A b \,c^{4} d^{2} e \,x^{3}-24 A \,c^{5} d^{3} x^{3}+4 B \,b^{3} c^{2} d \,e^{2} x^{3}-19 B \,b^{2} c^{3} d^{2} e \,x^{3}+12 B b \,c^{4} d^{3} x^{3}-6 A \,b^{4} c \,e^{3} x^{2}-10 A \,b^{3} c^{2} d \,e^{2} x^{2}+55 A \,b^{2} c^{3} d^{2} e \,x^{2}-36 A b \,c^{4} d^{3} x^{2}+8 B \,b^{4} c d \,e^{2} x^{2}-29 B \,b^{3} c^{2} d^{2} e \,x^{2}+18 B \,b^{2} c^{3} d^{3} x^{2}-3 A \,b^{5} e^{3} x -2 A \,b^{4} c d \,e^{2} x +13 A \,b^{3} c^{2} d^{2} e x -8 A \,b^{2} c^{3} d^{3} x +4 B \,b^{5} d \,e^{2} x -8 B \,b^{4} c \,d^{2} e x +4 B \,b^{3} c^{2} d^{3} x +2 A \,b^{5} d \,e^{2}-4 A \,b^{4} c \,d^{2} e +2 A \,b^{3} c^{2} d^{3}\right ) \left (b e -c d \right ) b \right )}{4 \sqrt {\left (b e -c d \right ) c}\, x^{2} \left (c x +b \right )^{2} d^{\frac {9}{2}} \left (b e -c d \right )^{3} b^{5}}\)	\(585\)

[In]

int((B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/4*(e*x+d)^(1/2)*(-3*A*b*e*x-12*A*c*d*x+4*B*b*d*x+2*A*b*d)/d^2/b^4/x^2+1/4/b^4/d^2*e*(-8*c^2*d^2/e/b*((1/8*b
*c*e*(15*A*b*c*e-12*A*c^2*d-11*B*b^2*e+8*B*b*c*d)/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)+1/8*(17*A*b*c*e-12
*A*c^2*d-13*B*b^2*e+8*B*b*c*d)*b*e/(b*e-c*d)*(e*x+d)^(1/2))/(c*(e*x+d)+b*e-c*d)^2+1/8*(63*A*b^2*c*e^2-108*A*b*
c^2*d*e+48*A*c^3*d^2-35*B*b^3*e^2+56*B*b^2*c*d*e-24*B*b*c^2*d^2)/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/
2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))-1/b/e*(3*A*b^2*e^2+12*A*b*c*d*e+48*A*c^2*d^2-4*B*b^2*d*e-24*B*
b*c*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1069 vs. \(2 (366) = 732\).

Time = 24.58 (sec) , antiderivative size = 4310, normalized size of antiderivative = 10.94 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[-1/8*(((24*(B*b*c^5 - 2*A*c^6)*d^5 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^4*e + 7*(5*B*b^3*c^3 - 9*A*b^2*c^4)*d^3*
e^2)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - 4*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^4*e + 7*(5*B*b^4*c^2 - 9*A*b^
3*c^3)*d^3*e^2)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^4*e + 7*(5*B*b^5*c
- 9*A*b^4*c^2)*d^3*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(
c/(c*d - b*e)))/(c*x + b)) - ((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^
3*e - (16*B*b^3*c^3 - 27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24
*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 -
2*(2*B*b^5*c - 3*A*b^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 -
21*A*b^3*c^3)*d^3*e - (16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*sqrt(d)*log((e
*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c^2*d^4 - 4*A*b^5*c*d^3*e + 2*A*b^6*d^2*e^2 - (3*A*b^4*c^2
*d*e^3 - 12*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + (19*B*b^3*c^3 - 36*A*b^2*c^4)*d^3*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*
d^2*e^2)*x^3 - (6*A*b^5*c*d*e^3 - 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^3*e - 2*(
4*B*b^5*c - 5*A*b^4*c^2)*d^2*e^2)*x^2 - (3*A*b^6*d*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4 + (8*B*b^5*c - 13*A*b
^4*c^2)*d^3*e - 2*(2*B*b^6 - A*b^5*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3
*e^2)*x^4 + 2*(b^6*c^3*d^5 - 2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2
)*x^2), -1/8*(2*((24*(B*b*c^5 - 2*A*c^6)*d^5 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^4*e + 7*(5*B*b^3*c^3 - 9*A*b^2*
c^4)*d^3*e^2)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - 4*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^4*e + 7*(5*B*b^4*c^2
- 9*A*b^3*c^3)*d^3*e^2)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^4*e + 7*(
5*B*b^5*c - 9*A*b^4*c^2)*d^3*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b
*e))/(c*e*x + c*d)) - ((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^3*e - (
16*B*b^3*c^3 - 27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24*(B*b^2
*c^4 - 2*A*b*c^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 - 2*(2*B
*b^5*c - 3*A*b^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 - 21*A*b^
3*c^3)*d^3*e - (16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*sqrt(d)*log((e*x - 2*
sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c^2*d^4 - 4*A*b^5*c*d^3*e + 2*A*b^6*d^2*e^2 - (3*A*b^4*c^2*d*e^3
- 12*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + (19*B*b^3*c^3 - 36*A*b^2*c^4)*d^3*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^2
)*x^3 - (6*A*b^5*c*d*e^3 - 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^3*e - 2*(4*B*b^5
*c - 5*A*b^4*c^2)*d^2*e^2)*x^2 - (3*A*b^6*d*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4 + (8*B*b^5*c - 13*A*b^4*c^2)
*d^3*e - 2*(2*B*b^6 - A*b^5*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3*e^2)*x
^4 + 2*(b^6*c^3*d^5 - 2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2),
1/8*(2*((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^3*e - (16*B*b^3*c^3 -
27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24*(B*b^2*c^4 - 2*A*b*c
^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 - 2*(2*B*b^5*c - 3*A*b
^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 - 21*A*b^3*c^3)*d^3*e -
(16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(
-d)/d) - ((24*(B*b*c^5 - 2*A*c^6)*d^5 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^4*e + 7*(5*B*b^3*c^3 - 9*A*b^2*c^4)*d^
3*e^2)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - 4*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^4*e + 7*(5*B*b^4*c^2 - 9*A*
b^3*c^3)*d^3*e^2)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^4*e + 7*(5*B*b^5
*c - 9*A*b^4*c^2)*d^3*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqr
t(c/(c*d - b*e)))/(c*x + b)) - 2*(2*A*b^4*c^2*d^4 - 4*A*b^5*c*d^3*e + 2*A*b^6*d^2*e^2 - (3*A*b^4*c^2*d*e^3 - 1
2*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + (19*B*b^3*c^3 - 36*A*b^2*c^4)*d^3*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^2)*x
^3 - (6*A*b^5*c*d*e^3 - 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^3*e - 2*(4*B*b^5*c
- 5*A*b^4*c^2)*d^2*e^2)*x^2 - (3*A*b^6*d*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4 + (8*B*b^5*c - 13*A*b^4*c^2)*d^
3*e - 2*(2*B*b^6 - A*b^5*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3*e^2)*x^4
+ 2*(b^6*c^3*d^5 - 2*b^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2), -1
/4*(((24*(B*b*c^5 - 2*A*c^6)*d^5 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^4*e + 7*(5*B*b^3*c^3 - 9*A*b^2*c^4)*d^3*e^2
)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - 4*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^4*e + 7*(5*B*b^4*c^2 - 9*A*b^3*c
^3)*d^3*e^2)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^4*e + 7*(5*B*b^5*c -
9*A*b^4*c^2)*d^3*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x
+ c*d)) - ((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^3*e - (16*B*b^3*c^3
- 27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24*(B*b^2*c^4 - 2*A*b
*c^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 - 2*(2*B*b^5*c - 3*A
*b^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 - 21*A*b^3*c^3)*d^3*e
- (16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqr
t(-d)/d) + (2*A*b^4*c^2*d^4 - 4*A*b^5*c*d^3*e + 2*A*b^6*d^2*e^2 - (3*A*b^4*c^2*d*e^3 - 12*(B*b^2*c^4 - 2*A*b*c
^5)*d^4 + (19*B*b^3*c^3 - 36*A*b^2*c^4)*d^3*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e^2)*x^3 - (6*A*b^5*c*d*e^3
- 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^3*e - 2*(4*B*b^5*c - 5*A*b^4*c^2)*d^2*e^2
)*x^2 - (3*A*b^6*d*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4 + (8*B*b^5*c - 13*A*b^4*c^2)*d^3*e - 2*(2*B*b^6 - A*b
^5*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^5*c^4*d^5 - 2*b^6*c^3*d^4*e + b^7*c^2*d^3*e^2)*x^4 + 2*(b^6*c^3*d^5 - 2*b
^7*c^2*d^4*e + b^8*c*d^3*e^2)*x^3 + (b^7*c^2*d^5 - 2*b^8*c*d^4*e + b^9*d^3*e^2)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)/(e*x+d)**(1/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)/(e*x+d)^(1/2)/(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 detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1020 vs. \(2 (366) = 732\).

Time = 0.30 (sec) , antiderivative size = 1020, normalized size of antiderivative = 2.59 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/4*(24*B*b*c^4*d^2 - 48*A*c^5*d^2 - 56*B*b^2*c^3*d*e + 108*A*b*c^4*d*e + 35*B*b^3*c^2*e^2 - 63*A*b^2*c^3*e^2)
*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^2*d^2 - 2*b^6*c*d*e + b^7*e^2)*sqrt(-c^2*d + b*c*e)) - 1
/4*(12*(e*x + d)^(7/2)*B*b*c^4*d^3*e - 24*(e*x + d)^(7/2)*A*c^5*d^3*e - 36*(e*x + d)^(5/2)*B*b*c^4*d^4*e + 72*
(e*x + d)^(5/2)*A*c^5*d^4*e + 36*(e*x + d)^(3/2)*B*b*c^4*d^5*e - 72*(e*x + d)^(3/2)*A*c^5*d^5*e - 12*sqrt(e*x
+ d)*B*b*c^4*d^6*e + 24*sqrt(e*x + d)*A*c^5*d^6*e - 19*(e*x + d)^(7/2)*B*b^2*c^3*d^2*e^2 + 36*(e*x + d)^(7/2)*
A*b*c^4*d^2*e^2 + 75*(e*x + d)^(5/2)*B*b^2*c^3*d^3*e^2 - 144*(e*x + d)^(5/2)*A*b*c^4*d^3*e^2 - 93*(e*x + d)^(3
/2)*B*b^2*c^3*d^4*e^2 + 180*(e*x + d)^(3/2)*A*b*c^4*d^4*e^2 + 37*sqrt(e*x + d)*B*b^2*c^3*d^5*e^2 - 72*sqrt(e*x
+ d)*A*b*c^4*d^5*e^2 + 4*(e*x + d)^(7/2)*B*b^3*c^2*d*e^3 - 6*(e*x + d)^(7/2)*A*b^2*c^3*d*e^3 - 41*(e*x + d)^(
5/2)*B*b^3*c^2*d^2*e^3 + 73*(e*x + d)^(5/2)*A*b^2*c^3*d^2*e^3 + 74*(e*x + d)^(3/2)*B*b^3*c^2*d^3*e^3 - 136*(e*
x + d)^(3/2)*A*b^2*c^3*d^3*e^3 - 37*sqrt(e*x + d)*B*b^3*c^2*d^4*e^3 + 69*sqrt(e*x + d)*A*b^2*c^3*d^4*e^3 - 3*(
e*x + d)^(7/2)*A*b^3*c^2*e^4 + 8*(e*x + d)^(5/2)*B*b^4*c*d*e^4 - (e*x + d)^(5/2)*A*b^3*c^2*d*e^4 - 24*(e*x + d
)^(3/2)*B*b^4*c*d^2*e^4 + 24*(e*x + d)^(3/2)*A*b^3*c^2*d^2*e^4 + 16*sqrt(e*x + d)*B*b^4*c*d^3*e^4 - 18*sqrt(e*
x + d)*A*b^3*c^2*d^3*e^4 - 6*(e*x + d)^(5/2)*A*b^4*c*e^5 + 4*(e*x + d)^(3/2)*B*b^5*d*e^5 + 10*(e*x + d)^(3/2)*
A*b^4*c*d*e^5 - 4*sqrt(e*x + d)*B*b^5*d^2*e^5 - 8*sqrt(e*x + d)*A*b^4*c*d^2*e^5 - 3*(e*x + d)^(3/2)*A*b^5*e^6
+ 5*sqrt(e*x + d)*A*b^5*d*e^6)/((b^4*c^2*d^4 - 2*b^5*c*d^3*e + b^6*d^2*e^2)*((e*x + d)^2*c - 2*(e*x + d)*c*d +
c*d^2 + (e*x + d)*b*e - b*d*e)^2) - 1/4*(24*B*b*c*d^2 - 48*A*c^2*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e - 3*A*b^2*e
^2)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^5*sqrt(-d)*d^2)

Mupad [B] (verification not implemented)

Time = 16.63 (sec) , antiderivative size = 11338, normalized size of antiderivative = 28.78 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

int((A + B*x)/((b*x + c*x^2)^3*(d + e*x)^(1/2)),x)

[Out]

log((((((c^2*e^3*(3*A*b^4*e^4 + 24*A*c^4*d^4 - 12*B*b*c^3*d^4 - 4*B*b^4*d*e^3 + 25*B*b^2*c^2*d^3*e - 12*B*b^3*
c*d^2*e^2 + 21*A*b^2*c^2*d^2*e^2 - 48*A*b*c^3*d^3*e + 3*A*b^3*c*d*e^3))/(b^2*d^2*(b*e - c*d)^2) - b^2*c^2*e^2*
(b*e - 2*c*d)*(d + e*x)^(1/2)*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^1
0*d^5))^(1/2))*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))/
8 - ((d + e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4608*A^2*c^11*d^8*e^2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c
^8*d^5*e^5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7 + 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d
^8*e^2 - 4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2*b^4*c^7*d^6*e^4 - 5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4
*e^6 + 128*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*c^3*d^2*e^8 - 18432*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 46
08*A*B*b*c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9 + 18816*A*B*b^2*c^9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A
*B*b^4*c^7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*b^8*d^4
*(b*e - c*d)^4))*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2)
)/8 - (567*A^3*b^7*c^5*e^10 + 55296*A^3*c^12*d^7*e^3 + 224640*A^3*b^2*c^10*d^5*e^5 - 77760*A^3*b^3*c^9*d^4*e^6
- 13608*A^3*b^4*c^8*d^3*e^7 + 1404*A^3*b^5*c^7*d^2*e^8 - 6912*B^3*b^3*c^9*d^7*e^3 + 25920*B^3*b^4*c^8*d^6*e^4
- 33408*B^3*b^5*c^7*d^5*e^5 + 15016*B^3*b^6*c^6*d^4*e^6 + 196*B^3*b^7*c^5*d^3*e^7 - 560*B^3*b^8*c^4*d^2*e^8 -
315*A^2*B*b^8*c^4*e^10 - 193536*A^3*b*c^11*d^6*e^4 + 2430*A^3*b^6*c^6*d*e^9 + 41472*A*B^2*b^2*c^10*d^7*e^3 -
152064*A*B^2*b^3*c^9*d^6*e^4 + 189504*A*B^2*b^4*c^8*d^5*e^5 - 78768*A*B^2*b^5*c^7*d^4*e^6 - 4764*A*B^2*b^6*c^6
*d^3*e^7 + 2709*A*B^2*b^7*c^5*d^2*e^8 + 297216*A^2*B*b^2*c^10*d^6*e^4 - 357696*A^2*B*b^3*c^9*d^5*e^5 + 136368*
A^2*B*b^4*c^8*d^4*e^6 + 15516*A^2*B*b^5*c^7*d^3*e^7 - 3861*A^2*B*b^6*c^6*d^2*e^8 + 840*A*B^2*b^8*c^4*d*e^9 - 8
2944*A^2*B*b*c^11*d^7*e^3 - 2898*A^2*B*b^7*c^5*d*e^9)/(64*b^12*d^4*(b*e - c*d)^4))*((9*A^2*b^4*e^4 + 2304*A^2*
c^4*d^4 + 576*B^2*b^2*c^2*d^4 + 16*B^2*b^4*d^2*e^2 + 432*A^2*b^2*c^2*d^2*e^2 + 1152*A^2*b*c^3*d^3*e + 72*A^2*b
^3*c*d*e^3 + 192*B^2*b^3*c*d^3*e - 2304*A*B*b*c^3*d^4 - 24*A*B*b^4*d*e^3 - 960*A*B*b^2*c^2*d^3*e - 240*A*B*b^3
*c*d^2*e^2)/(64*b^10*d^5))^(1/2) - log((((((c^2*e^3*(3*A*b^4*e^4 + 24*A*c^4*d^4 - 12*B*b*c^3*d^4 - 4*B*b^4*d*e
^3 + 25*B*b^2*c^2*d^3*e - 12*B*b^3*c*d^2*e^2 + 21*A*b^2*c^2*d^2*e^2 - 48*A*b*c^3*d^3*e + 3*A*b^3*c*d*e^3))/(b^
2*d^2*(b*e - c*d)^2) + b^2*c^2*e^2*(b*e - 2*c*d)*(d + e*x)^(1/2)*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 -
4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e +
12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))/8 + ((d + e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4608*A^2*c^11*d^8*e^2 + 27360*A^
2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7 + 198*A^2*b
^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 - 4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2*b^4*c^7*d^6*e^4 - 5136*B^2*b^5
*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 128*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*c^3*d^2*e^8 - 18432*A^2*b*c^10*
d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9 + 18816*A*B*b^2*c^9*d^7*e^3 -
28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5*d^3*e^7 - 1
44*A*B*b^7*c^4*d^2*e^8))/(8*b^8*d^4*(b*e - c*d)^4))*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e
+ 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))/8 - (567*A^3*b^7*c^5*e^10 + 55296*A^3*c^12*d^7*e^3 + 224640*A^3*b^2*c^10*
d^5*e^5 - 77760*A^3*b^3*c^9*d^4*e^6 - 13608*A^3*b^4*c^8*d^3*e^7 + 1404*A^3*b^5*c^7*d^2*e^8 - 6912*B^3*b^3*c^9*
d^7*e^3 + 25920*B^3*b^4*c^8*d^6*e^4 - 33408*B^3*b^5*c^7*d^5*e^5 + 15016*B^3*b^6*c^6*d^4*e^6 + 196*B^3*b^7*c^5*
d^3*e^7 - 560*B^3*b^8*c^4*d^2*e^8 - 315*A^2*B*b^8*c^4*e^10 - 193536*A^3*b*c^11*d^6*e^4 + 2430*A^3*b^6*c^6*d*e^
9 + 41472*A*B^2*b^2*c^10*d^7*e^3 - 152064*A*B^2*b^3*c^9*d^6*e^4 + 189504*A*B^2*b^4*c^8*d^5*e^5 - 78768*A*B^2*b
^5*c^7*d^4*e^6 - 4764*A*B^2*b^6*c^6*d^3*e^7 + 2709*A*B^2*b^7*c^5*d^2*e^8 + 297216*A^2*B*b^2*c^10*d^6*e^4 - 357
696*A^2*B*b^3*c^9*d^5*e^5 + 136368*A^2*B*b^4*c^8*d^4*e^6 + 15516*A^2*B*b^5*c^7*d^3*e^7 - 3861*A^2*B*b^6*c^6*d^
2*e^8 + 840*A*B^2*b^8*c^4*d*e^9 - 82944*A^2*B*b*c^11*d^7*e^3 - 2898*A^2*B*b^7*c^5*d*e^9)/(64*b^12*d^4*(b*e - c
*d)^4))*(((9*A^2*b^4*e^4)/64 + 36*A^2*c^4*d^4 + 9*B^2*b^2*c^2*d^4 + (B^2*b^4*d^2*e^2)/4 + (27*A^2*b^2*c^2*d^2*
e^2)/4 + 18*A^2*b*c^3*d^3*e + (9*A^2*b^3*c*d*e^3)/8 + 3*B^2*b^3*c*d^3*e - 36*A*B*b*c^3*d^4 - (3*A*B*b^4*d*e^3)
/8 - 15*A*B*b^2*c^2*d^3*e - (15*A*B*b^3*c*d^2*e^2)/4)/(b^10*d^5))^(1/2) - atan(((((6144*A*b^11*c^7*d^7*e^4 - 1
536*A*b^10*c^8*d^8*e^3 - 9024*A*b^12*c^6*d^6*e^5 + 5568*A*b^13*c^5*d^5*e^6 - 1152*A*b^14*c^4*d^4*e^7 + 192*A*b
^15*c^3*d^3*e^8 - 192*A*b^16*c^2*d^2*e^9 + 768*B*b^11*c^7*d^8*e^3 - 3136*B*b^12*c^6*d^7*e^4 + 4736*B*b^13*c^5*
d^6*e^5 - 2880*B*b^14*c^4*d^5*e^6 + 256*B*b^15*c^3*d^4*e^7 + 256*B*b^16*c^2*d^3*e^8)/(64*(b^12*c^4*d^8 + b^16*
d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) - ((d + e*x)^(1/2)*(-(2304*A^2*c^9*d^4 +
3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c
^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e -
3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b
^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 -
5*b^14*c*d*e^4)))^(1/2)*(128*b^10*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7*e^4 - 896*b^13*c^4*d^
6*e^5 + 384*b^14*c^3*d^5*e^6 - 64*b^15*c^2*d^4*e^7))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11
*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2
*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d
^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*
A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11
*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) + ((d + e*x)^(1/2)*(9*A^2*b^8
*c^3*e^10 + 4608*A^2*c^11*d^8*e^2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^5 + 3978*A^2*b^4*c^7*d
^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7 + 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 - 4800*B^2*b^3*c^8*d^7*e
^3 + 7520*B^2*b^4*c^7*d^6*e^4 - 5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 128*B^2*b^7*c^4*d^3*e^7
+ 16*B^2*b^8*c^3*d^2*e^8 - 18432*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*c^10*d^8*e^2 - 24*A*B*
b^8*c^3*d*e^9 + 18816*A*B*b^2*c^9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^7*d^5*e^5 - 4218*A*B*b
^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^
3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^
7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 1
0368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*
c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*
c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2)*1i - (((6144*
A*b^11*c^7*d^7*e^4 - 1536*A*b^10*c^8*d^8*e^3 - 9024*A*b^12*c^6*d^6*e^5 + 5568*A*b^13*c^5*d^5*e^6 - 1152*A*b^14
*c^4*d^4*e^7 + 192*A*b^15*c^3*d^3*e^8 - 192*A*b^16*c^2*d^2*e^9 + 768*B*b^11*c^7*d^8*e^3 - 3136*B*b^12*c^6*d^7*
e^4 + 4736*B*b^13*c^5*d^6*e^5 - 2880*B*b^14*c^4*d^5*e^6 + 256*B*b^15*c^3*d^4*e^7 + 256*B*b^16*c^2*d^3*e^8)/(64
*(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) + ((d + e*x)^(1/2)*
(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^
2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 26
88*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c
^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 +
10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2)*(128*b^10*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7
*e^4 - 896*b^13*c^4*d^6*e^5 + 384*b^14*c^3*d^5*e^6 - 64*b^15*c^2*d^4*e^7))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*
b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*
b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e
^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*
A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 -
b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) - ((d +
e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4608*A^2*c^11*d^8*e^2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^
5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7 + 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 -
4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2*b^4*c^7*d^6*e^4 - 5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 12
8*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*c^3*d^2*e^8 - 18432*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*
c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9 + 18816*A*B*b^2*c^9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^
7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*(b^8*c^4*d^8 + b
^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^
5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 44
10*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c
^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)
/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)
))^(1/2)*1i)/((567*A^3*b^7*c^5*e^10 + 55296*A^3*c^12*d^7*e^3 + 224640*A^3*b^2*c^10*d^5*e^5 - 77760*A^3*b^3*c^9
*d^4*e^6 - 13608*A^3*b^4*c^8*d^3*e^7 + 1404*A^3*b^5*c^7*d^2*e^8 - 6912*B^3*b^3*c^9*d^7*e^3 + 25920*B^3*b^4*c^8
*d^6*e^4 - 33408*B^3*b^5*c^7*d^5*e^5 + 15016*B^3*b^6*c^6*d^4*e^6 + 196*B^3*b^7*c^5*d^3*e^7 - 560*B^3*b^8*c^4*d
^2*e^8 - 315*A^2*B*b^8*c^4*e^10 - 193536*A^3*b*c^11*d^6*e^4 + 2430*A^3*b^6*c^6*d*e^9 + 41472*A*B^2*b^2*c^10*d^
7*e^3 - 152064*A*B^2*b^3*c^9*d^6*e^4 + 189504*A*B^2*b^4*c^8*d^5*e^5 - 78768*A*B^2*b^5*c^7*d^4*e^6 - 4764*A*B^2
*b^6*c^6*d^3*e^7 + 2709*A*B^2*b^7*c^5*d^2*e^8 + 297216*A^2*B*b^2*c^10*d^6*e^4 - 357696*A^2*B*b^3*c^9*d^5*e^5 +
136368*A^2*B*b^4*c^8*d^4*e^6 + 15516*A^2*B*b^5*c^7*d^3*e^7 - 3861*A^2*B*b^6*c^6*d^2*e^8 + 840*A*B^2*b^8*c^4*d
*e^9 - 82944*A^2*B*b*c^11*d^7*e^3 - 2898*A^2*B*b^7*c^5*d*e^9)/(32*(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b^13*c^3*d^
7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) + (((6144*A*b^11*c^7*d^7*e^4 - 1536*A*b^10*c^8*d^8*e^3 - 9024*A*
b^12*c^6*d^6*e^5 + 5568*A*b^13*c^5*d^5*e^6 - 1152*A*b^14*c^4*d^4*e^7 + 192*A*b^15*c^3*d^3*e^8 - 192*A*b^16*c^2
*d^2*e^9 + 768*B*b^11*c^7*d^8*e^3 - 3136*B*b^12*c^6*d^7*e^4 + 4736*B*b^13*c^5*d^6*e^5 - 2880*B*b^14*c^4*d^5*e^
6 + 256*B*b^15*c^3*d^4*e^7 + 256*B*b^16*c^2*d^3*e^8)/(64*(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b
^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) - ((d + e*x)^(1/2)*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b
^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^
4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A
*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 -
b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2)*(128*b^1
0*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7*e^4 - 896*b^13*c^4*d^6*e^5 + 384*b^14*c^3*d^5*e^6 - 6
4*b^15*c^2*d^4*e^7))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)
))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7
*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 -
2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^
4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2
+ 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) + ((d + e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4608*A^2*c^11*d^8*e^
2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*A^2*b^5*c^6*d^3*e^7
+ 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 - 4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2*b^4*c^7*d^6*e^4 -
5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 128*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*c^3*d^2*e^8 - 18432
*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9 + 18816*A*B*b^2*c^
9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^6 - 144*A*B*b^6*c^5
*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b
^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 1
7712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2
*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*
e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*
b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) + (((6144*A*b^11*c^7*d^7*e^4 - 1536*A*b^10*c^
8*d^8*e^3 - 9024*A*b^12*c^6*d^6*e^5 + 5568*A*b^13*c^5*d^5*e^6 - 1152*A*b^14*c^4*d^4*e^7 + 192*A*b^15*c^3*d^3*e
^8 - 192*A*b^16*c^2*d^2*e^9 + 768*B*b^11*c^7*d^8*e^3 - 3136*B*b^12*c^6*d^7*e^4 + 4736*B*b^13*c^5*d^6*e^5 - 288
0*B*b^14*c^4*d^5*e^6 + 256*B*b^15*c^3*d^4*e^7 + 256*B*b^16*c^2*d^3*e^8)/(64*(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b
^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2)) + ((d + e*x)^(1/2)*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*
c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 -
4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5
*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^
2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^
4)))^(1/2)*(128*b^10*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7*e^4 - 896*b^13*c^4*d^6*e^5 + 384*b
^14*c^3*d^5*e^6 - 64*b^15*c^2*d^4*e^7))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 +
6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4
+ 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*
A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d
^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e -
10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4
608*A^2*c^11*d^8*e^2 + 27360*A^2*b^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^5 + 3978*A^2*b^4*c^7*d^4*e^6 - 180*
A^2*b^5*c^6*d^3*e^7 + 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 - 4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2
*b^4*c^7*d^6*e^4 - 5136*B^2*b^5*c^6*d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 128*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8*
c^3*d^2*e^8 - 18432*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b*c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9
+ 18816*A*B*b^2*c^9*d^7*e^3 - 28704*A*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^
6 - 144*A*B*b^6*c^5*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b
^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)))*(-(2304*A^2*c^9*d^4 + 3969*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*
B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^
8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 105
60*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b
^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*b^14*c*d*e^4)))^(1/2)))*(-(2304*A^2*c^9*d^4 + 39
69*A^2*b^4*c^5*e^4 + 576*B^2*b^2*c^7*d^4 + 1225*B^2*b^6*c^3*e^4 + 17712*A^2*b^2*c^7*d^2*e^2 + 4816*B^2*b^4*c^5
*d^2*e^2 - 4410*A*B*b^5*c^4*e^4 - 10368*A^2*b*c^8*d^3*e - 13608*A^2*b^3*c^6*d*e^3 - 2688*B^2*b^3*c^6*d^3*e - 3
920*B^2*b^5*c^4*d*e^3 - 2304*A*B*b*c^8*d^4 + 10560*A*B*b^2*c^7*d^3*e + 14616*A*B*b^4*c^5*d*e^3 - 18480*A*B*b^3
*c^6*d^2*e^2)/(64*(b^15*e^5 - b^10*c^5*d^5 + 5*b^11*c^4*d^4*e - 10*b^12*c^3*d^3*e^2 + 10*b^13*c^2*d^2*e^3 - 5*
b^14*c*d*e^4)))^(1/2)*2i - (((d + e*x)^(3/2)*(4*B*b^5*d*e^5 - 72*A*c^5*d^5*e - 3*A*b^5*e^6 + 180*A*b*c^4*d^4*e
^2 - 24*B*b^4*c*d^2*e^4 - 136*A*b^2*c^3*d^3*e^3 + 24*A*b^3*c^2*d^2*e^4 - 93*B*b^2*c^3*d^4*e^2 + 74*B*b^3*c^2*d
^3*e^3 + 10*A*b^4*c*d*e^5 + 36*B*b*c^4*d^5*e))/(4*b^4*(c*d^2 - b*d*e)^2) - ((d + e*x)^(5/2)*(6*A*b^4*c*e^5 - 7
2*A*c^5*d^4*e + 144*A*b*c^4*d^3*e^2 + A*b^3*c^2*d*e^4 - 73*A*b^2*c^3*d^2*e^3 - 75*B*b^2*c^3*d^3*e^2 + 41*B*b^3
*c^2*d^2*e^3 + 36*B*b*c^4*d^4*e - 8*B*b^4*c*d*e^4))/(4*b^4*(c*d^2 - b*d*e)^2) + ((d + e*x)^(1/2)*(24*A*c^4*d^4
*e - 5*A*b^4*e^5 + 4*B*b^4*d*e^4 - 48*A*b*c^3*d^3*e^2 - 12*B*b^3*c*d^2*e^3 + 21*A*b^2*c^2*d^2*e^3 + 25*B*b^2*c
^2*d^3*e^2 + 3*A*b^3*c*d*e^4 - 12*B*b*c^3*d^4*e))/(4*b^4*(c*d^2 - b*d*e)) - (c*(d + e*x)^(7/2)*(3*A*b^3*c*e^4
+ 24*A*c^4*d^3*e - 36*A*b*c^3*d^2*e^2 + 6*A*b^2*c^2*d*e^3 + 19*B*b^2*c^2*d^2*e^2 - 12*B*b*c^3*d^3*e - 4*B*b^3*
c*d*e^3))/(4*b^4*(c*d^2 - b*d*e)^2))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4
*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*
d^3*e)