3.1845 \(\int (A+B x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=308 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)} \]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) - (2*(b*d - a*e)
^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) + (2*b*(b*d -
a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)) - (2*b^2*(4*b*
B*d - A*b*e - 3*a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) + (2*b^3*B*(d + e*x)
^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.143681, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {770, 77} \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5 (a+b x)}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) - (2*(b*d - a*e)
^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) + (2*b*(b*d -
a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)) - (2*b^2*(4*b*
B*d - A*b*e - 3*a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) + (2*b^3*B*(d + e*x)
^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^{3/2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^3 (b d-a e)^3 (-B d+A e) (d+e x)^{3/2}}{e^4}+\frac{b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{5/2}}{e^4}-\frac{3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^4}+\frac{b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{9/2}}{e^4}+\frac{b^6 B (d+e x)^{11/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{2 (b d-a e)^3 (B d-A e) (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac{2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac{2 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac{2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac{2 b^3 B (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.211159, size = 163, normalized size = 0.53 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{5/2} \left (-1365 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+5005 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-2145 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+3003 (b d-a e)^3 (B d-A e)+1155 b^3 B (d+e x)^4\right )}{15015 e^5 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a + b*x)^2)^(3/2)*(d + e*x)^(5/2)*(3003*(b*d - a*e)^3*(B*d - A*e) - 2145*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e
- a*B*e)*(d + e*x) + 5005*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 1365*b^2*(4*b*B*d - A*b*e - 3
*a*B*e)*(d + e*x)^3 + 1155*b^3*B*(d + e*x)^4))/(15015*e^5*(a + b*x)^3)

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 317, normalized size = 1. \begin{align*}{\frac{2310\,B{x}^{4}{b}^{3}{e}^{4}+2730\,A{x}^{3}{b}^{3}{e}^{4}+8190\,B{x}^{3}a{b}^{2}{e}^{4}-1680\,B{x}^{3}{b}^{3}d{e}^{3}+10010\,A{x}^{2}a{b}^{2}{e}^{4}-1820\,A{x}^{2}{b}^{3}d{e}^{3}+10010\,B{x}^{2}{a}^{2}b{e}^{4}-5460\,B{x}^{2}a{b}^{2}d{e}^{3}+1120\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+12870\,Ax{a}^{2}b{e}^{4}-5720\,Axa{b}^{2}d{e}^{3}+1040\,Ax{b}^{3}{d}^{2}{e}^{2}+4290\,Bx{a}^{3}{e}^{4}-5720\,Bx{a}^{2}bd{e}^{3}+3120\,Bxa{b}^{2}{d}^{2}{e}^{2}-640\,Bx{b}^{3}{d}^{3}e+6006\,A{a}^{3}{e}^{4}-5148\,Ad{e}^{3}{a}^{2}b+2288\,Aa{b}^{2}{d}^{2}{e}^{2}-416\,A{b}^{3}{d}^{3}e-1716\,Bd{e}^{3}{a}^{3}+2288\,B{a}^{2}b{d}^{2}{e}^{2}-1248\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{15015\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(e*x+d)^(5/2)*(1155*B*b^3*e^4*x^4+1365*A*b^3*e^4*x^3+4095*B*a*b^2*e^4*x^3-840*B*b^3*d*e^3*x^3+5005*A*a
*b^2*e^4*x^2-910*A*b^3*d*e^3*x^2+5005*B*a^2*b*e^4*x^2-2730*B*a*b^2*d*e^3*x^2+560*B*b^3*d^2*e^2*x^2+6435*A*a^2*
b*e^4*x-2860*A*a*b^2*d*e^3*x+520*A*b^3*d^2*e^2*x+2145*B*a^3*e^4*x-2860*B*a^2*b*d*e^3*x+1560*B*a*b^2*d^2*e^2*x-
320*B*b^3*d^3*e*x+3003*A*a^3*e^4-2574*A*a^2*b*d*e^3+1144*A*a*b^2*d^2*e^2-208*A*b^3*d^3*e-858*B*a^3*d*e^3+1144*
B*a^2*b*d^2*e^2-624*B*a*b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

________________________________________________________________________________________

Maxima [B]  time = 1.03378, size = 659, normalized size = 2.14 \begin{align*} \frac{2 \,{\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \,{\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \,{\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d} A}{1155 \, e^{4}} + \frac{2 \,{\left (1155 \, b^{3} e^{6} x^{6} + 128 \, b^{3} d^{6} - 624 \, a b^{2} d^{5} e + 1144 \, a^{2} b d^{4} e^{2} - 858 \, a^{3} d^{3} e^{3} + 105 \,{\left (14 \, b^{3} d e^{5} + 39 \, a b^{2} e^{6}\right )} x^{5} + 35 \,{\left (b^{3} d^{2} e^{4} + 156 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{3} e^{3} - 39 \, a b^{2} d^{2} e^{4} - 1430 \, a^{2} b d e^{5} - 429 \, a^{3} e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{4} e^{2} - 78 \, a b^{2} d^{3} e^{3} + 143 \, a^{2} b d^{2} e^{4} + 1144 \, a^{3} d e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{5} e - 312 \, a b^{2} d^{4} e^{2} + 572 \, a^{2} b d^{3} e^{3} - 429 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d} B}{15015 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*b^3*e^5*x^5 - 16*b^3*d^5 + 88*a*b^2*d^4*e - 198*a^2*b*d^3*e^2 + 231*a^3*d^2*e^3 + 35*(4*b^3*d*e^4
+ 11*a*b^2*e^5)*x^4 + 5*(b^3*d^2*e^3 + 110*a*b^2*d*e^4 + 99*a^2*b*e^5)*x^3 - 3*(2*b^3*d^3*e^2 - 11*a*b^2*d^2*e
^3 - 264*a^2*b*d*e^4 - 77*a^3*e^5)*x^2 + (8*b^3*d^4*e - 44*a*b^2*d^3*e^2 + 99*a^2*b*d^2*e^3 + 462*a^3*d*e^4)*x
)*sqrt(e*x + d)*A/e^4 + 2/15015*(1155*b^3*e^6*x^6 + 128*b^3*d^6 - 624*a*b^2*d^5*e + 1144*a^2*b*d^4*e^2 - 858*a
^3*d^3*e^3 + 105*(14*b^3*d*e^5 + 39*a*b^2*e^6)*x^5 + 35*(b^3*d^2*e^4 + 156*a*b^2*d*e^5 + 143*a^2*b*e^6)*x^4 -
5*(8*b^3*d^3*e^3 - 39*a*b^2*d^2*e^4 - 1430*a^2*b*d*e^5 - 429*a^3*e^6)*x^3 + 3*(16*b^3*d^4*e^2 - 78*a*b^2*d^3*e
^3 + 143*a^2*b*d^2*e^4 + 1144*a^3*d*e^5)*x^2 - (64*b^3*d^5*e - 312*a*b^2*d^4*e^2 + 572*a^2*b*d^3*e^3 - 429*a^3
*d^2*e^4)*x)*sqrt(e*x + d)*B/e^5

________________________________________________________________________________________

Fricas [A]  time = 1.26392, size = 994, normalized size = 3.23 \begin{align*} \frac{2 \,{\left (1155 \, B b^{3} e^{6} x^{6} + 128 \, B b^{3} d^{6} + 3003 \, A a^{3} d^{2} e^{4} - 208 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e + 1144 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{2} - 858 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{3} + 105 \,{\left (14 \, B b^{3} d e^{5} + 13 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{6}\right )} x^{5} + 35 \,{\left (B b^{3} d^{2} e^{4} + 52 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{5} + 143 \,{\left (B a^{2} b + A a b^{2}\right )} e^{6}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{3} e^{3} - 13 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{4} - 1430 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{5} - 429 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{4} e^{2} + 1001 \, A a^{3} e^{6} - 26 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{3} + 143 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{4} + 1144 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{5} e - 6006 \, A a^{3} d e^{5} - 104 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{2} + 572 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{3} - 429 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d}}{15015 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(1155*B*b^3*e^6*x^6 + 128*B*b^3*d^6 + 3003*A*a^3*d^2*e^4 - 208*(3*B*a*b^2 + A*b^3)*d^5*e + 1144*(B*a^2
*b + A*a*b^2)*d^4*e^2 - 858*(B*a^3 + 3*A*a^2*b)*d^3*e^3 + 105*(14*B*b^3*d*e^5 + 13*(3*B*a*b^2 + A*b^3)*e^6)*x^
5 + 35*(B*b^3*d^2*e^4 + 52*(3*B*a*b^2 + A*b^3)*d*e^5 + 143*(B*a^2*b + A*a*b^2)*e^6)*x^4 - 5*(8*B*b^3*d^3*e^3 -
13*(3*B*a*b^2 + A*b^3)*d^2*e^4 - 1430*(B*a^2*b + A*a*b^2)*d*e^5 - 429*(B*a^3 + 3*A*a^2*b)*e^6)*x^3 + 3*(16*B*
b^3*d^4*e^2 + 1001*A*a^3*e^6 - 26*(3*B*a*b^2 + A*b^3)*d^3*e^3 + 143*(B*a^2*b + A*a*b^2)*d^2*e^4 + 1144*(B*a^3
+ 3*A*a^2*b)*d*e^5)*x^2 - (64*B*b^3*d^5*e - 6006*A*a^3*d*e^5 - 104*(3*B*a*b^2 + A*b^3)*d^4*e^2 + 572*(B*a^2*b
+ A*a*b^2)*d^3*e^3 - 429*(B*a^3 + 3*A*a^2*b)*d^2*e^4)*x)*sqrt(e*x + d)/e^5

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*x)*(d + e*x)**(3/2)*((a + b*x)**2)**(3/2), x)

________________________________________________________________________________________

Giac [B]  time = 1.2452, size = 1220, normalized size = 3.96 \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)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^3*d*e^(-1)*sgn(b*x + a) + 9009*(3*(x*e + d)^(5/2)
- 5*(x*e + d)^(3/2)*d)*A*a^2*b*d*e^(-1)*sgn(b*x + a) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x
*e + d)^(3/2)*d^2)*B*a^2*b*d*e^(-2)*sgn(b*x + a) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e +
d)^(3/2)*d^2)*A*a*b^2*d*e^(-2)*sgn(b*x + a) + 429*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)
^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a*b^2*d*e^(-3)*sgn(b*x + a) + 143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^
(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*b^3*d*e^(-3)*sgn(b*x + a) + 13*(315*(x*e + d)^(
11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^
4)*B*b^3*d*e^(-4)*sgn(b*x + a) + 15015*(x*e + d)^(3/2)*A*a^3*d*sgn(b*x + a) + 429*(15*(x*e + d)^(7/2) - 42*(x*
e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^3*e^(-1)*sgn(b*x + a) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(
5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^2*b*e^(-1)*sgn(b*x + a) + 429*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d
+ 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a^2*b*e^(-2)*sgn(b*x + a) + 429*(35*(x*e + d)^(9/2) -
135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*A*a*b^2*e^(-2)*sgn(b*x + a) + 39*(3
15*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e
+ d)^(3/2)*d^4)*B*a*b^2*e^(-3)*sgn(b*x + a) + 13*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e +
d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*b^3*e^(-3)*sgn(b*x + a) + 5*(693*(x*e +
d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^
(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*b^3*e^(-4)*sgn(b*x + a) + 3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*
d)*A*a^3*sgn(b*x + a))*e^(-1)