3.1844 \(\int (A+B x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=308 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^5 (a+b x)} \]

[Out]

(2*(b*d - a*e)^3*(B*d - A*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*d - a*e)
^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (6*b*(b*d -
a*e)*(2*b*B*d - A*b*e - 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^2*(4*
b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x)) + (2*b^3*B*(d + e*
x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*e^5*(a + b*x))

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Rubi [A]  time = 0.138917, 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} \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (-3 a B e-A b e+4 b B d)}{13 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e) (-a B e-A b e+2 b B d)}{11 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^3 (B d-A e)}{7 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(5/2)*(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)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*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)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)) + (6*b*(b*d -
a*e)*(2*b*B*d - A*b*e - 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^2*(4*
b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x)) + (2*b^3*B*(d + e*
x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(15*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)^{5/2} \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)^{5/2} \, 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)^{5/2}}{e^4}+\frac{b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{7/2}}{e^4}-\frac{3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{9/2}}{e^4}+\frac{b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{11/2}}{e^4}+\frac{b^6 B (d+e x)^{13/2}}{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)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 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)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac{6 b (b d-a e) (2 b B d-A b e-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^2 (4 b B d-A b e-3 a B e) (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}+\frac{2 b^3 B (d+e x)^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.221232, size = 163, normalized size = 0.53 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{7/2} \left (-3465 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+12285 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-5005 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+6435 (b d-a e)^3 (B d-A e)+3003 b^3 B (d+e x)^4\right )}{45045 e^5 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a + b*x)^2)^(3/2)*(d + e*x)^(7/2)*(6435*(b*d - a*e)^3*(B*d - A*e) - 5005*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e
- a*B*e)*(d + e*x) + 12285*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 3465*b^2*(4*b*B*d - A*b*e -
3*a*B*e)*(d + e*x)^3 + 3003*b^3*B*(d + e*x)^4))/(45045*e^5*(a + b*x)^3)

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Maple [A]  time = 0.007, size = 317, normalized size = 1. \begin{align*}{\frac{6006\,B{x}^{4}{b}^{3}{e}^{4}+6930\,A{x}^{3}{b}^{3}{e}^{4}+20790\,B{x}^{3}a{b}^{2}{e}^{4}-3696\,B{x}^{3}{b}^{3}d{e}^{3}+24570\,A{x}^{2}a{b}^{2}{e}^{4}-3780\,A{x}^{2}{b}^{3}d{e}^{3}+24570\,B{x}^{2}{a}^{2}b{e}^{4}-11340\,B{x}^{2}a{b}^{2}d{e}^{3}+2016\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+30030\,Ax{a}^{2}b{e}^{4}-10920\,Axa{b}^{2}d{e}^{3}+1680\,Ax{b}^{3}{d}^{2}{e}^{2}+10010\,Bx{a}^{3}{e}^{4}-10920\,Bx{a}^{2}bd{e}^{3}+5040\,Bxa{b}^{2}{d}^{2}{e}^{2}-896\,Bx{b}^{3}{d}^{3}e+12870\,A{a}^{3}{e}^{4}-8580\,Ad{e}^{3}{a}^{2}b+3120\,Aa{b}^{2}{d}^{2}{e}^{2}-480\,A{b}^{3}{d}^{3}e-2860\,Bd{e}^{3}{a}^{3}+3120\,B{a}^{2}b{d}^{2}{e}^{2}-1440\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{45045\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{7}{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)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

2/45045*(e*x+d)^(7/2)*(3003*B*b^3*e^4*x^4+3465*A*b^3*e^4*x^3+10395*B*a*b^2*e^4*x^3-1848*B*b^3*d*e^3*x^3+12285*
A*a*b^2*e^4*x^2-1890*A*b^3*d*e^3*x^2+12285*B*a^2*b*e^4*x^2-5670*B*a*b^2*d*e^3*x^2+1008*B*b^3*d^2*e^2*x^2+15015
*A*a^2*b*e^4*x-5460*A*a*b^2*d*e^3*x+840*A*b^3*d^2*e^2*x+5005*B*a^3*e^4*x-5460*B*a^2*b*d*e^3*x+2520*B*a*b^2*d^2
*e^2*x-448*B*b^3*d^3*e*x+6435*A*a^3*e^4-4290*A*a^2*b*d*e^3+1560*A*a*b^2*d^2*e^2-240*A*b^3*d^3*e-1430*B*a^3*d*e
^3+1560*B*a^2*b*d^2*e^2-720*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

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Maxima [B]  time = 1.04376, size = 799, normalized size = 2.59 \begin{align*} \frac{2 \,{\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \,{\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \,{\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} +{\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} +{\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{e x + d} A}{3003 \, e^{4}} + \frac{2 \,{\left (3003 \, b^{3} e^{7} x^{7} + 128 \, b^{3} d^{7} - 720 \, a b^{2} d^{6} e + 1560 \, a^{2} b d^{5} e^{2} - 1430 \, a^{3} d^{4} e^{3} + 231 \,{\left (31 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 63 \,{\left (71 \, b^{3} d^{2} e^{5} + 405 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 35 \,{\left (b^{3} d^{3} e^{4} + 477 \, a b^{2} d^{2} e^{5} + 897 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{4} e^{3} - 45 \, a b^{2} d^{3} e^{4} - 4407 \, a^{2} b d^{2} e^{5} - 2717 \, a^{3} d e^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{5} e^{2} - 90 \, a b^{2} d^{4} e^{3} + 195 \, a^{2} b d^{3} e^{4} + 3575 \, a^{3} d^{2} e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{6} e - 360 \, a b^{2} d^{5} e^{2} + 780 \, a^{2} b d^{4} e^{3} - 715 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d} B}{45045 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(231*b^3*e^6*x^6 - 16*b^3*d^6 + 104*a*b^2*d^5*e - 286*a^2*b*d^4*e^2 + 429*a^3*d^3*e^3 + 63*(9*b^3*d*e^5
+ 13*a*b^2*e^6)*x^5 + 7*(53*b^3*d^2*e^4 + 299*a*b^2*d*e^5 + 143*a^2*b*e^6)*x^4 + (5*b^3*d^3*e^3 + 1469*a*b^2*
d^2*e^4 + 2717*a^2*b*d*e^5 + 429*a^3*e^6)*x^3 - 3*(2*b^3*d^4*e^2 - 13*a*b^2*d^3*e^3 - 715*a^2*b*d^2*e^4 - 429*
a^3*d*e^5)*x^2 + (8*b^3*d^5*e - 52*a*b^2*d^4*e^2 + 143*a^2*b*d^3*e^3 + 1287*a^3*d^2*e^4)*x)*sqrt(e*x + d)*A/e^
4 + 2/45045*(3003*b^3*e^7*x^7 + 128*b^3*d^7 - 720*a*b^2*d^6*e + 1560*a^2*b*d^5*e^2 - 1430*a^3*d^4*e^3 + 231*(3
1*b^3*d*e^6 + 45*a*b^2*e^7)*x^6 + 63*(71*b^3*d^2*e^5 + 405*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 35*(b^3*d^3*e^4
+ 477*a*b^2*d^2*e^5 + 897*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 - 5*(8*b^3*d^4*e^3 - 45*a*b^2*d^3*e^4 - 4407*a^2*b*d^
2*e^5 - 2717*a^3*d*e^6)*x^3 + 3*(16*b^3*d^5*e^2 - 90*a*b^2*d^4*e^3 + 195*a^2*b*d^3*e^4 + 3575*a^3*d^2*e^5)*x^2
- (64*b^3*d^6*e - 360*a*b^2*d^5*e^2 + 780*a^2*b*d^4*e^3 - 715*a^3*d^3*e^4)*x)*sqrt(e*x + d)*B/e^5

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Fricas [B]  time = 1.35742, size = 1203, normalized size = 3.91 \begin{align*} \frac{2 \,{\left (3003 \, B b^{3} e^{7} x^{7} + 128 \, B b^{3} d^{7} + 6435 \, A a^{3} d^{3} e^{4} - 240 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e + 1560 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{2} - 1430 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{3} + 231 \,{\left (31 \, B b^{3} d e^{6} + 15 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{7}\right )} x^{6} + 63 \,{\left (71 \, B b^{3} d^{2} e^{5} + 135 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{6} + 195 \,{\left (B a^{2} b + A a b^{2}\right )} e^{7}\right )} x^{5} + 35 \,{\left (B b^{3} d^{3} e^{4} + 159 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{5} + 897 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{6} + 143 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{4} e^{3} - 1287 \, A a^{3} e^{7} - 15 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{4} - 4407 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{5} - 2717 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{5} e^{2} + 6435 \, A a^{3} d e^{6} - 30 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{3} + 195 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{4} + 3575 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{6} e - 19305 \, A a^{3} d^{2} e^{5} - 120 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{2} + 780 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{3} - 715 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{4}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*B*b^3*e^7*x^7 + 128*B*b^3*d^7 + 6435*A*a^3*d^3*e^4 - 240*(3*B*a*b^2 + A*b^3)*d^6*e + 1560*(B*a^2
*b + A*a*b^2)*d^5*e^2 - 1430*(B*a^3 + 3*A*a^2*b)*d^4*e^3 + 231*(31*B*b^3*d*e^6 + 15*(3*B*a*b^2 + A*b^3)*e^7)*x
^6 + 63*(71*B*b^3*d^2*e^5 + 135*(3*B*a*b^2 + A*b^3)*d*e^6 + 195*(B*a^2*b + A*a*b^2)*e^7)*x^5 + 35*(B*b^3*d^3*e
^4 + 159*(3*B*a*b^2 + A*b^3)*d^2*e^5 + 897*(B*a^2*b + A*a*b^2)*d*e^6 + 143*(B*a^3 + 3*A*a^2*b)*e^7)*x^4 - 5*(8
*B*b^3*d^4*e^3 - 1287*A*a^3*e^7 - 15*(3*B*a*b^2 + A*b^3)*d^3*e^4 - 4407*(B*a^2*b + A*a*b^2)*d^2*e^5 - 2717*(B*
a^3 + 3*A*a^2*b)*d*e^6)*x^3 + 3*(16*B*b^3*d^5*e^2 + 6435*A*a^3*d*e^6 - 30*(3*B*a*b^2 + A*b^3)*d^4*e^3 + 195*(B
*a^2*b + A*a*b^2)*d^3*e^4 + 3575*(B*a^3 + 3*A*a^2*b)*d^2*e^5)*x^2 - (64*B*b^3*d^6*e - 19305*A*a^3*d^2*e^5 - 12
0*(3*B*a*b^2 + A*b^3)*d^5*e^2 + 780*(B*a^2*b + A*a*b^2)*d^4*e^3 - 715*(B*a^3 + 3*A*a^2*b)*d^3*e^4)*x)*sqrt(e*x
+ d)/e^5

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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)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.33316, size = 2064, normalized size = 6.7 \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)^(5/2)*(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^2*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^2*e^(-1)*sgn(b*x + a) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 3
5*(x*e + d)^(3/2)*d^2)*B*a^2*b*d^2*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^2*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^2*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^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)*B*b^3*d^2*e^(-4)*sgn(b*x + a) + 15015*(x*e + d)^(3/2)*A*a^3*d^2*sgn(b*x + a) + 858*(15*(x*e + d
)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a^3*d*e^(-1)*sgn(b*x + a) + 2574*(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*d*e^(-1)*sgn(b*x + a) + 858*(35*(x*e + d)^(9/2) - 1
35*(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*d*e^(-2)*sgn(b*x + a) + 858*
(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*d*e^(
-2)*sgn(b*x + a) + 78*(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*a*b^2*d*e^(-3)*sgn(b*x + a) + 26*(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*d*e^(-3)*
sgn(b*x + a) + 10*(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*d*e^(-4)*sgn(b*x + a) + 6006*(3*(x*e
+ d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^3*d*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)*B*a^3*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)*A*a^2*b*e^(-1)*sgn(b*x + a) + 39*(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*a^2*b*e^(-2)*sgn(b*x + a) + 39*(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*a*b^2*e^(-2)*sgn(b*x + a) + 15*(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*a*b^2*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)*A*b^3*e^(-3)*sgn(b*x + a) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2
- 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*
d^6)*B*b^3*e^(-4)*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)*A*a^
3*sgn(b*x + a))*e^(-1)