3.1252 \(\int \frac{A+B x}{\sqrt{d+e x} (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=394 \[ \frac{\sqrt{d+e x} \left (c x \left (b^2 c d e (6 A e+19 B d)+b^3 \left (-e^2\right ) (4 B d-3 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2}+\frac{c^{3/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 (-e) (4 B d-3 A e)-12 b c d (2 B d-A e)+48 A c^2 d^2\right )}{4 b^5 d^{5/2}}-\frac{\sqrt{d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)} \]

[Out]

-(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)*(b*x + c*x^2)^2) + (Sq
rt[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]/Sqrt
[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))

________________________________________________________________________________________

Rubi [A]  time = 0.900164, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {822, 826, 1166, 208} \[ \frac{\sqrt{d+e x} \left (c x \left (b^2 c d e (6 A e+19 B d)+b^3 \left (-e^2\right ) (4 B d-3 A e)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2}+\frac{c^{3/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 (-e) (4 B d-3 A e)-12 b c d (2 B d-A e)+48 A c^2 d^2\right )}{4 b^5 d^{5/2}}-\frac{\sqrt{d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)*(b*x + c*x^2)^2) + (Sq
rt[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]/Sqrt
[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 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 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]

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]

Rubi steps

\begin{align*} \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{\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{\operatorname{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 ) \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^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 ) \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)^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]  time = 1.64775, size = 408, normalized size = 1.04 \[ \frac{\frac{-\frac{b c \sqrt{d} \sqrt{d+e x} \left (-b^2 c d e (6 A e+19 B d)+b^3 e^2 (4 B d-3 A e)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )}{(b+c x) (c d-b e)^2}+\frac{c^{3/2} d^{5/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{(c d-b e)^{5/2}}-\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 e (3 A e-4 B d)+12 b c d (A e-2 B d)+48 A c^2 d^2\right )}{b^4 d^{3/2}}+\frac{c \sqrt{d+e x} \left (b^2 e (3 A e-4 B d)+b c d (7 A e+6 B d)-12 A c^2 d^2\right )}{b^2 d (b+c x)^2 (b e-c d)}+\frac{\sqrt{d+e x} (3 A b e+8 A c d-4 b B d)}{b d x (b+c x)^2}-\frac{2 A \sqrt{d+e x}}{x^2 (b+c x)^2}}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*(-12*A*c^2*d^2 + b^2*e*(-4*B*d + 3*A*e) + b*c*d*(6*B*d + 7*A*e))*Sqrt[d + e*x])/(b^2*d*(-(c*d) + b*e)*(b +
c*x)^2) - (2*A*Sqrt[d + e*x])/(x^2*(b + c*x)^2) + ((-4*b*B*d + 8*A*c*d + 3*A*b*e)*Sqrt[d + e*x])/(b*d*x*(b +
c*x)^2) + (-((b*c*Sqrt[d]*(-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*(1
9*B*d + 6*A*e))*Sqrt[d + e*x])/((c*d - b*e)^2*(b + c*x))) - (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[Sqrt[d + e*x]/Sqrt[d]] + (c^(3/2)*d^(5/2)*(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]])/(c*d - b*e)^(5/2)
)/(b^4*d^(3/2)))/(4*b*d)

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Maple [B]  time = 0.027, size = 1009, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13/4*e^2*c^2/b^2/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*B-63/4*e^2*c^3/b^3/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*
d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A+3/e/b^4/x^2/d*(e*x+d)^(3/2)*A*c-15/4*e^2*c^4/b^3/(c*
e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*A+11/4*e^2*c^3/b^2/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d
^2)*(e*x+d)^(3/2)*B-17/4*e^2*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*A-12*c^5/b^5/(b^2*e^2-2*b*c*d*e+c^2
*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^2+6*c^4/b^4/(b^2*e^2-2*b*c*d*e+c^2*d
^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^2+35/4*e^2*c^2/b^2/(b^2*e^2-2*b*c*d*e+
c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B-3/e/b^4/x^2*(e*x+d)^(1/2)*A*c-3*e/b
^4/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-1/e/b^3/x^2/d*(e*x+d)^(3/2)*B-12/b^5/d^(1/2)*arctanh((e*x+d)^(1/
2)/d^(1/2))*A*c^2+6/b^4/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c+1/e/b^3/x^2*(e*x+d)^(1/2)*B-3/4*e^2/b^3/d^(
5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A+e/b^3/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B+3/4/b^3/x^2/d^2*(e*x+d)^(
3/2)*A-5/4/b^3/x^2/d*(e*x+d)^(1/2)*A+3*e*c^5/b^4/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*A*d-2
*e*c^4/b^3/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*B*d+3*e*c^4/b^4/(c*e*x+b*e)^2/(b*e-c*d)*(e*
x+d)^(1/2)*A*d-2*e*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*B*d+27*e*c^4/b^4/(b^2*e^2-2*b*c*d*e+c^2*d^2)/
((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d-14*e*c^3/b^3/(b^2*e^2-2*b*c*d*e+c^2*d^2)/(
(b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.5122, size = 1412, normalized size = 3.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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