3.1846 \(\int (A+B x) \sqrt{d+e x} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

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

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*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)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) + (6*b*(b*d -
a*e)*(2*b*B*d - 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^2*(4*b*
B*d - A*b*e - 3*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)) + (2*b^3*B*(d + e*x)^(
11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.142207, 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} \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.200699, size = 163, normalized size = 0.53 \[ \frac{2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{3/2} \left (-385 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+1485 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-693 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+1155 (b d-a e)^3 (B d-A e)+315 b^3 B (d+e x)^4\right )}{3465 e^5 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a + b*x)^2)^(3/2)*(d + e*x)^(3/2)*(1155*(b*d - a*e)^3*(B*d - A*e) - 693*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e
- a*B*e)*(d + e*x) + 1485*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 385*b^2*(4*b*B*d - A*b*e - 3*a
*B*e)*(d + e*x)^3 + 315*b^3*B*(d + e*x)^4))/(3465*e^5*(a + b*x)^3)

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 317, normalized size = 1. \begin{align*}{\frac{630\,B{x}^{4}{b}^{3}{e}^{4}+770\,A{x}^{3}{b}^{3}{e}^{4}+2310\,B{x}^{3}a{b}^{2}{e}^{4}-560\,B{x}^{3}{b}^{3}d{e}^{3}+2970\,A{x}^{2}a{b}^{2}{e}^{4}-660\,A{x}^{2}{b}^{3}d{e}^{3}+2970\,B{x}^{2}{a}^{2}b{e}^{4}-1980\,B{x}^{2}a{b}^{2}d{e}^{3}+480\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+4158\,Ax{a}^{2}b{e}^{4}-2376\,Axa{b}^{2}d{e}^{3}+528\,Ax{b}^{3}{d}^{2}{e}^{2}+1386\,Bx{a}^{3}{e}^{4}-2376\,Bx{a}^{2}bd{e}^{3}+1584\,Bxa{b}^{2}{d}^{2}{e}^{2}-384\,Bx{b}^{3}{d}^{3}e+2310\,A{a}^{3}{e}^{4}-2772\,Ad{e}^{3}{a}^{2}b+1584\,Aa{b}^{2}{d}^{2}{e}^{2}-352\,A{b}^{3}{d}^{3}e-924\,Bd{e}^{3}{a}^{3}+1584\,B{a}^{2}b{d}^{2}{e}^{2}-1056\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{3465\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{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)*(b^2*x^2+2*a*b*x+a^2)^(3/2)*(e*x+d)^(1/2),x)

[Out]

2/3465*(e*x+d)^(3/2)*(315*B*b^3*e^4*x^4+385*A*b^3*e^4*x^3+1155*B*a*b^2*e^4*x^3-280*B*b^3*d*e^3*x^3+1485*A*a*b^
2*e^4*x^2-330*A*b^3*d*e^3*x^2+1485*B*a^2*b*e^4*x^2-990*B*a*b^2*d*e^3*x^2+240*B*b^3*d^2*e^2*x^2+2079*A*a^2*b*e^
4*x-1188*A*a*b^2*d*e^3*x+264*A*b^3*d^2*e^2*x+693*B*a^3*e^4*x-1188*B*a^2*b*d*e^3*x+792*B*a*b^2*d^2*e^2*x-192*B*
b^3*d^3*e*x+1155*A*a^3*e^4-1386*A*a^2*b*d*e^3+792*A*a*b^2*d^2*e^2-176*A*b^3*d^3*e-462*B*a^3*d*e^3+792*B*a^2*b*
d^2*e^2-528*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 [A]  time = 1.03046, size = 518, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \,{\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} +{\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt{e x + d} A}{315 \, e^{4}} + \frac{2 \,{\left (315 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 528 \, a b^{2} d^{4} e + 792 \, a^{2} b d^{3} e^{2} - 462 \, a^{3} d^{2} e^{3} + 35 \,{\left (b^{3} d e^{4} + 33 \, a b^{2} e^{5}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{2} e^{3} - 33 \, a b^{2} d e^{4} - 297 \, a^{2} b e^{5}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{3} e^{2} - 66 \, a b^{2} d^{2} e^{3} + 99 \, a^{2} b d e^{4} + 231 \, a^{3} e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{4} e - 264 \, a b^{2} d^{3} e^{2} + 396 \, a^{2} b d^{2} e^{3} - 231 \, a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d} B}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*b^3*e^4*x^4 - 16*b^3*d^4 + 72*a*b^2*d^3*e - 126*a^2*b*d^2*e^2 + 105*a^3*d*e^3 + 5*(b^3*d*e^3 + 27*a*
b^2*e^4)*x^3 - 3*(2*b^3*d^2*e^2 - 9*a*b^2*d*e^3 - 63*a^2*b*e^4)*x^2 + (8*b^3*d^3*e - 36*a*b^2*d^2*e^2 + 63*a^2
*b*d*e^3 + 105*a^3*e^4)*x)*sqrt(e*x + d)*A/e^4 + 2/3465*(315*b^3*e^5*x^5 + 128*b^3*d^5 - 528*a*b^2*d^4*e + 792
*a^2*b*d^3*e^2 - 462*a^3*d^2*e^3 + 35*(b^3*d*e^4 + 33*a*b^2*e^5)*x^4 - 5*(8*b^3*d^2*e^3 - 33*a*b^2*d*e^4 - 297
*a^2*b*e^5)*x^3 + 3*(16*b^3*d^3*e^2 - 66*a*b^2*d^2*e^3 + 99*a^2*b*d*e^4 + 231*a^3*e^5)*x^2 - (64*b^3*d^4*e - 2
64*a*b^2*d^3*e^2 + 396*a^2*b*d^2*e^3 - 231*a^3*d*e^4)*x)*sqrt(e*x + d)*B/e^5

________________________________________________________________________________________

Fricas [A]  time = 1.48092, size = 780, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (315 \, B b^{3} e^{5} x^{5} + 128 \, B b^{3} d^{5} + 1155 \, A a^{3} d e^{4} - 176 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 792 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} - 462 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3} + 35 \,{\left (B b^{3} d e^{4} + 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{2} e^{3} - 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} - 297 \,{\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{3} e^{2} - 22 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 99 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{4} + 231 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{4} e - 1155 \, A a^{3} e^{5} - 88 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 396 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} - 231 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*B*b^3*e^5*x^5 + 128*B*b^3*d^5 + 1155*A*a^3*d*e^4 - 176*(3*B*a*b^2 + A*b^3)*d^4*e + 792*(B*a^2*b +
A*a*b^2)*d^3*e^2 - 462*(B*a^3 + 3*A*a^2*b)*d^2*e^3 + 35*(B*b^3*d*e^4 + 11*(3*B*a*b^2 + A*b^3)*e^5)*x^4 - 5*(8*
B*b^3*d^2*e^3 - 11*(3*B*a*b^2 + A*b^3)*d*e^4 - 297*(B*a^2*b + A*a*b^2)*e^5)*x^3 + 3*(16*B*b^3*d^3*e^2 - 22*(3*
B*a*b^2 + A*b^3)*d^2*e^3 + 99*(B*a^2*b + A*a*b^2)*d*e^4 + 231*(B*a^3 + 3*A*a^2*b)*e^5)*x^2 - (64*B*b^3*d^4*e -
1155*A*a^3*e^5 - 88*(3*B*a*b^2 + A*b^3)*d^3*e^2 + 396*(B*a^2*b + A*a*b^2)*d^2*e^3 - 231*(B*a^3 + 3*A*a^2*b)*d
*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 ) \sqrt{d + e x} \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)*(b**2*x**2+2*a*b*x+a**2)**(3/2)*(e*x+d)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.21298, size = 536, normalized size = 1.74 \begin{align*} \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a^{3} e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) + 693 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A a^{2} b e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} B a^{2} b e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} A a b^{2} e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) + 33 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} B a b^{2} e^{\left (-3\right )} \mathrm{sgn}\left (b x + a\right ) + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} A b^{3} e^{\left (-3\right )} \mathrm{sgn}\left (b x + a\right ) +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} B b^{3} e^{\left (-4\right )} \mathrm{sgn}\left (b x + a\right ) + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^3*e^(-1)*sgn(b*x + a) + 693*(3*(x*e + d)^(5/2) - 5*(
x*e + d)^(3/2)*d)*A*a^2*b*e^(-1)*sgn(b*x + a) + 99*(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*e^(-2)*sgn(b*x + a) + 99*(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*e^(-2)*sgn(b*x + a) + 33*(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*e^(-3)*sgn(b*x + a) + 11*(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*e^(-3)*sgn(b*x + a) + (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*e^(-4)*sgn(b*x
+ a) + 1155*(x*e + d)^(3/2)*A*a^3*sgn(b*x + a))*e^(-1)