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

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)*Sqr
t[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))

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Rubi [A]  time = 0.467985, 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 \[ -\frac{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5 (a+b x)}+\frac{2 b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^5 (a+b x)} \]

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)*Sqr
t[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))

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Rubi in Sympy [A]  time = 46.8314, size = 311, normalized size = 1.01 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{11 b e} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (11 A b e - 3 B a e - 8 B b d\right )}{99 b e^{2}} + \frac{4 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (11 A b e - 3 B a e - 8 B b d\right )}{693 b e^{3}} + \frac{16 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (11 A b e - 3 B a e - 8 B b d\right )}{1155 b e^{4}} + \frac{32 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (11 A b e - 3 B a e - 8 B b d\right )}{3465 b e^{5} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)*(e*x+d)**(1/2),x)

[Out]

B*(2*a + 2*b*x)*(d + e*x)**(3/2)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(11*b*e) +
2*(d + e*x)**(3/2)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(11*A*b*e - 3*B*a*e - 8*B
*b*d)/(99*b*e**2) + 4*(3*a + 3*b*x)*(d + e*x)**(3/2)*(a*e - b*d)*sqrt(a**2 + 2*a
*b*x + b**2*x**2)*(11*A*b*e - 3*B*a*e - 8*B*b*d)/(693*b*e**3) + 16*(d + e*x)**(3
/2)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(11*A*b*e - 3*B*a*e - 8*B*b*
d)/(1155*b*e**4) + 32*(d + e*x)**(3/2)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2
*x**2)*(11*A*b*e - 3*B*a*e - 8*B*b*d)/(3465*b*e**5*(a + b*x))

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Mathematica [A]  time = 0.274537, size = 245, normalized size = 0.8 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (231 a^3 e^3 (5 A e-2 B d+3 B e x)+99 a^2 b e^2 \left (7 A e (3 e x-2 d)+B \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )-33 a b^2 e \left (B \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )-3 A e \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+b^3 \left (11 A e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+B \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )\right )}{3465 e^5 (a+b x)} \]

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*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(231*a^3*e^3*(-2*B*d + 5*A*e + 3*B*e*x) + 9
9*a^2*b*e^2*(7*A*e*(-2*d + 3*e*x) + B*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 33*a*b^
2*e*(-3*A*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + B*(16*d^3 - 24*d^2*e*x + 30*d*e^2*
x^2 - 35*e^3*x^3)) + b^3*(11*A*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x
^3) + B*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4))
))/(3465*e^5*(a + b*x))

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Maple [A]  time = 0.013, size = 317, normalized size = 1. \[{\frac{630\,B{x}^{4}{b}^{3}{e}^{4}+770\,A{x}^{3}{b}^{3}{e}^{4}+2310\,B{x}^{3}a{b}^{2}{e}^{4}-560\,B{x}^{3}{b}^{3}d{e}^{3}+2970\,A{x}^{2}a{b}^{2}{e}^{4}-660\,A{x}^{2}{b}^{3}d{e}^{3}+2970\,B{x}^{2}{a}^{2}b{e}^{4}-1980\,B{x}^{2}a{b}^{2}d{e}^{3}+480\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+4158\,Ax{a}^{2}b{e}^{4}-2376\,Axa{b}^{2}d{e}^{3}+528\,Ax{b}^{3}{d}^{2}{e}^{2}+1386\,Bx{a}^{3}{e}^{4}-2376\,Bx{a}^{2}bd{e}^{3}+1584\,Bxa{b}^{2}{d}^{2}{e}^{2}-384\,Bx{b}^{3}{d}^{3}e+2310\,A{a}^{3}{e}^{4}-2772\,Ad{e}^{3}{a}^{2}b+1584\,Aa{b}^{2}{d}^{2}{e}^{2}-352\,A{b}^{3}{d}^{3}e-924\,Bd{e}^{3}{a}^{3}+1584\,B{a}^{2}b{d}^{2}{e}^{2}-1056\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{3465\,{e}^{5} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^3-2
80*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

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Maxima [A]  time = 0.739583, size = 518, normalized size = 1.68 \[ \frac{2 \,{\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \,{\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} +{\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt{e x + d} A}{315 \, e^{4}} + \frac{2 \,{\left (315 \, b^{3} e^{5} x^{5} + 128 \, b^{3} d^{5} - 528 \, a b^{2} d^{4} e + 792 \, a^{2} b d^{3} e^{2} - 462 \, a^{3} d^{2} e^{3} + 35 \,{\left (b^{3} d e^{4} + 33 \, a b^{2} e^{5}\right )} x^{4} - 5 \,{\left (8 \, b^{3} d^{2} e^{3} - 33 \, a b^{2} d e^{4} - 297 \, a^{2} b e^{5}\right )} x^{3} + 3 \,{\left (16 \, b^{3} d^{3} e^{2} - 66 \, a b^{2} d^{2} e^{3} + 99 \, a^{2} b d e^{4} + 231 \, a^{3} e^{5}\right )} x^{2} -{\left (64 \, b^{3} d^{4} e - 264 \, a b^{2} d^{3} e^{2} + 396 \, a^{2} b d^{2} e^{3} - 231 \, a^{3} d e^{4}\right )} x\right )} \sqrt{e x + d} B}{3465 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*sqrt(e*x + d),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 - 264*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

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Fricas [A]  time = 0.279167, size = 477, normalized size = 1.55 \[ \frac{2 \,{\left (315 \, B b^{3} e^{5} x^{5} + 128 \, B b^{3} d^{5} + 1155 \, A a^{3} d e^{4} - 176 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 792 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} - 462 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3} + 35 \,{\left (B b^{3} d e^{4} + 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{2} e^{3} - 11 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} - 297 \,{\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{3} e^{2} - 22 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 99 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{4} + 231 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{4} e - 1155 \, A a^{3} e^{5} - 88 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 396 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} - 231 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*sqrt(e*x + d),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)*s
qrt(e*x + d)/e^5

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

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)

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GIAC/XCAS [A]  time = 0.29284, size = 587, normalized size = 1.91 \[ \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 )}{\rm sign}\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 )}{\rm sign}\left (b x + a\right ) + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} B a^{2} b e^{\left (-14\right )}{\rm sign}\left (b x + a\right ) + 99 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} A a b^{2} e^{\left (-14\right )}{\rm sign}\left (b x + a\right ) + 33 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} B a b^{2} e^{\left (-27\right )}{\rm sign}\left (b x + a\right ) + 11 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} A b^{3} e^{\left (-27\right )}{\rm sign}\left (b x + a\right ) +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} B b^{3} e^{\left (-44\right )}{\rm sign}\left (b x + a\right ) + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)*sqrt(e*x + d),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)*sign(b*x + a)
+ 693*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^2*b*e^(-1)*sign(b*x + a) +
99*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2
*e^12)*B*a^2*b*e^(-14)*sign(b*x + a) + 99*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d
)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)*A*a*b^2*e^(-14)*sign(b*x + a) + 33
*(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2
*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*B*a*b^2*e^(-27)*sign(b*x + a) + 11*(35*(x*
e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 -
105*(x*e + d)^(3/2)*d^3*e^24)*A*b^3*e^(-27)*sign(b*x + a) + (315*(x*e + d)^(11/2
)*e^40 - 1540*(x*e + d)^(9/2)*d*e^40 + 2970*(x*e + d)^(7/2)*d^2*e^40 - 2772*(x*e
+ d)^(5/2)*d^3*e^40 + 1155*(x*e + d)^(3/2)*d^4*e^40)*B*b^3*e^(-44)*sign(b*x + a
) + 1155*(x*e + d)^(3/2)*A*a^3*sign(b*x + a))*e^(-1)