3.2117 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=368 \[ -\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{3/2}}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^7 (a+b x)}-\frac{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^7 (a+b x)}+\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) \sqrt{d+e x}} \]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2
)) + (12*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*
x)^(5/2)) - (10*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*
(d + e*x)^(3/2)) + (40*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a
+ b*x)*Sqrt[d + e*x]) + (30*b^4*(b*d - a*e)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(e^7*(a + b*x)) - (4*b^5*(b*d - a*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(e^7*(a + b*x)) + (2*b^6*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(5*e^7*(a + b*x))

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Rubi [A]  time = 0.397082, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{3/2}}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2}}{5 e^7 (a+b x)}-\frac{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}{e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^2}{e^7 (a+b x)}+\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(9/2),x]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^(7/2
)) + (12*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*
x)^(5/2)) - (10*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*
(d + e*x)^(3/2)) + (40*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a
+ b*x)*Sqrt[d + e*x]) + (30*b^4*(b*d - a*e)^2*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(e^7*(a + b*x)) - (4*b^5*(b*d - a*e)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(e^7*(a + b*x)) + (2*b^6*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(5*e^7*(a + b*x))

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Rubi in Sympy [A]  time = 52.857, size = 308, normalized size = 0.84 \[ \frac{256 b^{4} \left (3 a + 3 b x\right ) \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{5}} + \frac{1024 b^{4} \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{6}} + \frac{2048 b^{4} \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{7} \left (a + b x\right )} - \frac{128 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 e^{4} \sqrt{d + e x}} - \frac{16 b^{2} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{35 e^{3} \left (d + e x\right )^{\frac{3}{2}}} - \frac{24 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{35 e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{7 e \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(5/2)/(e*x+d)**(9/2),x)

[Out]

256*b**4*(3*a + 3*b*x)*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**5)
+ 1024*b**4*sqrt(d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**6)
+ 2048*b**4*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e
**7*(a + b*x)) - 128*b**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(7*e**4*sqrt(d + e
*x)) - 16*b**2*(5*a + 5*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(35*e**3*(d + e
*x)**(3/2)) - 24*b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(35*e**2*(d + e*x)**(5/2)
) - 2*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(7*e*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.330782, size = 173, normalized size = 0.47 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (7 b^4 \left (75 a^2 e^2-140 a b d e+66 b^2 d^2\right )-14 b^5 e x (4 b d-5 a e)+\frac{700 b^3 (b d-a e)^3}{d+e x}-\frac{175 b^2 (b d-a e)^4}{(d+e x)^2}+\frac{42 b (b d-a e)^5}{(d+e x)^3}-\frac{5 (b d-a e)^6}{(d+e x)^4}+7 b^6 e^2 x^2\right )}{35 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(9/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(7*b^4*(66*b^2*d^2 - 140*a*b*d*e + 75*a^2*e^2
) - 14*b^5*e*(4*b*d - 5*a*e)*x + 7*b^6*e^2*x^2 - (5*(b*d - a*e)^6)/(d + e*x)^4 +
(42*b*(b*d - a*e)^5)/(d + e*x)^3 - (175*b^2*(b*d - a*e)^4)/(d + e*x)^2 + (700*b
^3*(b*d - a*e)^3)/(d + e*x)))/(35*e^7*(a + b*x))

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Maple [A]  time = 0.013, size = 393, normalized size = 1.1 \[ -{\frac{-14\,{x}^{6}{b}^{6}{e}^{6}-140\,{x}^{5}a{b}^{5}{e}^{6}+56\,{x}^{5}{b}^{6}d{e}^{5}-1050\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+1400\,{x}^{4}a{b}^{5}d{e}^{5}-560\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+1400\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-8400\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+11200\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-4480\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+350\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+2800\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-16800\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+22400\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-8960\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+84\,x{a}^{5}b{e}^{6}+280\,x{a}^{4}{b}^{2}d{e}^{5}+2240\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-13440\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+17920\,xa{b}^{5}{d}^{4}{e}^{2}-7168\,x{b}^{6}{d}^{5}e+10\,{a}^{6}{e}^{6}+24\,{a}^{5}bd{e}^{5}+80\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+640\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-3840\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+5120\,{d}^{5}a{b}^{5}e-2048\,{b}^{6}{d}^{6}}{35\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \]

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)^(5/2)/(e*x+d)^(9/2),x)

[Out]

-2/35/(e*x+d)^(7/2)*(-7*b^6*e^6*x^6-70*a*b^5*e^6*x^5+28*b^6*d*e^5*x^5-525*a^2*b^
4*e^6*x^4+700*a*b^5*d*e^5*x^4-280*b^6*d^2*e^4*x^4+700*a^3*b^3*e^6*x^3-4200*a^2*b
^4*d*e^5*x^3+5600*a*b^5*d^2*e^4*x^3-2240*b^6*d^3*e^3*x^3+175*a^4*b^2*e^6*x^2+140
0*a^3*b^3*d*e^5*x^2-8400*a^2*b^4*d^2*e^4*x^2+11200*a*b^5*d^3*e^3*x^2-4480*b^6*d^
4*e^2*x^2+42*a^5*b*e^6*x+140*a^4*b^2*d*e^5*x+1120*a^3*b^3*d^2*e^4*x-6720*a^2*b^4
*d^3*e^3*x+8960*a*b^5*d^4*e^2*x-3584*b^6*d^5*e*x+5*a^6*e^6+12*a^5*b*d*e^5+40*a^4
*b^2*d^2*e^4+320*a^3*b^3*d^3*e^3-1920*a^2*b^4*d^4*e^2+2560*a*b^5*d^5*e-1024*b^6*
d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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Maxima [A]  time = 0.752028, size = 902, normalized size = 2.45 \[ \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \,{\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \,{\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \,{\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \,{\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )} a}{21 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (21 \, b^{5} e^{6} x^{6} + 3072 \, b^{5} d^{6} - 6400 \, a b^{4} d^{5} e + 3840 \, a^{2} b^{3} d^{4} e^{2} - 480 \, a^{3} b^{2} d^{3} e^{3} - 40 \, a^{4} b d^{2} e^{4} - 6 \, a^{5} d e^{5} - 7 \,{\left (12 \, b^{5} d e^{5} - 25 \, a b^{4} e^{6}\right )} x^{5} + 70 \,{\left (12 \, b^{5} d^{2} e^{4} - 25 \, a b^{4} d e^{5} + 15 \, a^{2} b^{3} e^{6}\right )} x^{4} + 70 \,{\left (96 \, b^{5} d^{3} e^{3} - 200 \, a b^{4} d^{2} e^{4} + 120 \, a^{2} b^{3} d e^{5} - 15 \, a^{3} b^{2} e^{6}\right )} x^{3} + 35 \,{\left (384 \, b^{5} d^{4} e^{2} - 800 \, a b^{4} d^{3} e^{3} + 480 \, a^{2} b^{3} d^{2} e^{4} - 60 \, a^{3} b^{2} d e^{5} - 5 \, a^{4} b e^{6}\right )} x^{2} + 7 \,{\left (1536 \, b^{5} d^{5} e - 3200 \, a b^{4} d^{4} e^{2} + 1920 \, a^{2} b^{3} d^{3} e^{3} - 240 \, a^{3} b^{2} d^{2} e^{4} - 20 \, a^{4} b d e^{5} - 3 \, a^{5} e^{6}\right )} x\right )} b}{105 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

2/21*(7*b^5*e^5*x^5 - 256*b^5*d^5 + 384*a*b^4*d^4*e - 96*a^2*b^3*d^3*e^2 - 16*a^
3*b^2*d^2*e^3 - 6*a^4*b*d*e^4 - 3*a^5*e^5 - 35*(2*b^5*d*e^4 - 3*a*b^4*e^5)*x^4 -
70*(8*b^5*d^2*e^3 - 12*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 - 70*(16*b^5*d^3*e^2 -
24*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 - 7*(128*b^5*d^4*e - 192*a
*b^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 8*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x)*a/((e^9*x^
3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)*sqrt(e*x + d)) + 2/105*(21*b^5*e^6*x^6
+ 3072*b^5*d^6 - 6400*a*b^4*d^5*e + 3840*a^2*b^3*d^4*e^2 - 480*a^3*b^2*d^3*e^3 -
40*a^4*b*d^2*e^4 - 6*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 25*a*b^4*e^6)*x^5 + 70*(12*b
^5*d^2*e^4 - 25*a*b^4*d*e^5 + 15*a^2*b^3*e^6)*x^4 + 70*(96*b^5*d^3*e^3 - 200*a*b
^4*d^2*e^4 + 120*a^2*b^3*d*e^5 - 15*a^3*b^2*e^6)*x^3 + 35*(384*b^5*d^4*e^2 - 800
*a*b^4*d^3*e^3 + 480*a^2*b^3*d^2*e^4 - 60*a^3*b^2*d*e^5 - 5*a^4*b*e^6)*x^2 + 7*(
1536*b^5*d^5*e - 3200*a*b^4*d^4*e^2 + 1920*a^2*b^3*d^3*e^3 - 240*a^3*b^2*d^2*e^4
- 20*a^4*b*d*e^5 - 3*a^5*e^6)*x)*b/((e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3
*e^7)*sqrt(e*x + d))

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Fricas [A]  time = 0.295607, size = 524, normalized size = 1.42 \[ \frac{2 \,{\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 2560 \, a b^{5} d^{5} e + 1920 \, a^{2} b^{4} d^{4} e^{2} - 320 \, a^{3} b^{3} d^{3} e^{3} - 40 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 5 \, a^{6} e^{6} - 14 \,{\left (2 \, b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{2} e^{4} - 20 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 140 \,{\left (16 \, b^{6} d^{3} e^{3} - 40 \, a b^{5} d^{2} e^{4} + 30 \, a^{2} b^{4} d e^{5} - 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 35 \,{\left (128 \, b^{6} d^{4} e^{2} - 320 \, a b^{5} d^{3} e^{3} + 240 \, a^{2} b^{4} d^{2} e^{4} - 40 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \,{\left (256 \, b^{6} d^{5} e - 640 \, a b^{5} d^{4} e^{2} + 480 \, a^{2} b^{4} d^{3} e^{3} - 80 \, a^{3} b^{3} d^{2} e^{4} - 10 \, a^{4} b^{2} d e^{5} - 3 \, a^{5} b e^{6}\right )} x\right )}}{35 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(9/2),x, algorithm="fricas")

[Out]

2/35*(7*b^6*e^6*x^6 + 1024*b^6*d^6 - 2560*a*b^5*d^5*e + 1920*a^2*b^4*d^4*e^2 - 3
20*a^3*b^3*d^3*e^3 - 40*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 5*a^6*e^6 - 14*(2*b^6
*d*e^5 - 5*a*b^5*e^6)*x^5 + 35*(8*b^6*d^2*e^4 - 20*a*b^5*d*e^5 + 15*a^2*b^4*e^6)
*x^4 + 140*(16*b^6*d^3*e^3 - 40*a*b^5*d^2*e^4 + 30*a^2*b^4*d*e^5 - 5*a^3*b^3*e^6
)*x^3 + 35*(128*b^6*d^4*e^2 - 320*a*b^5*d^3*e^3 + 240*a^2*b^4*d^2*e^4 - 40*a^3*b
^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 14*(256*b^6*d^5*e - 640*a*b^5*d^4*e^2 + 480*a^2*
b^4*d^3*e^3 - 80*a^3*b^3*d^2*e^4 - 10*a^4*b^2*d*e^5 - 3*a^5*b*e^6)*x)/((e^10*x^3
+ 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)*sqrt(e*x + d))

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

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)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.319695, size = 844, normalized size = 2.29 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

2/5*((x*e + d)^(5/2)*b^6*e^28*sign(b*x + a) - 10*(x*e + d)^(3/2)*b^6*d*e^28*sign
(b*x + a) + 75*sqrt(x*e + d)*b^6*d^2*e^28*sign(b*x + a) + 10*(x*e + d)^(3/2)*a*b
^5*e^29*sign(b*x + a) - 150*sqrt(x*e + d)*a*b^5*d*e^29*sign(b*x + a) + 75*sqrt(x
*e + d)*a^2*b^4*e^30*sign(b*x + a))*e^(-35) + 2/35*(700*(x*e + d)^3*b^6*d^3*sign
(b*x + a) - 175*(x*e + d)^2*b^6*d^4*sign(b*x + a) + 42*(x*e + d)*b^6*d^5*sign(b*
x + a) - 5*b^6*d^6*sign(b*x + a) - 2100*(x*e + d)^3*a*b^5*d^2*e*sign(b*x + a) +
700*(x*e + d)^2*a*b^5*d^3*e*sign(b*x + a) - 210*(x*e + d)*a*b^5*d^4*e*sign(b*x +
a) + 30*a*b^5*d^5*e*sign(b*x + a) + 2100*(x*e + d)^3*a^2*b^4*d*e^2*sign(b*x + a
) - 1050*(x*e + d)^2*a^2*b^4*d^2*e^2*sign(b*x + a) + 420*(x*e + d)*a^2*b^4*d^3*e
^2*sign(b*x + a) - 75*a^2*b^4*d^4*e^2*sign(b*x + a) - 700*(x*e + d)^3*a^3*b^3*e^
3*sign(b*x + a) + 700*(x*e + d)^2*a^3*b^3*d*e^3*sign(b*x + a) - 420*(x*e + d)*a^
3*b^3*d^2*e^3*sign(b*x + a) + 100*a^3*b^3*d^3*e^3*sign(b*x + a) - 175*(x*e + d)^
2*a^4*b^2*e^4*sign(b*x + a) + 210*(x*e + d)*a^4*b^2*d*e^4*sign(b*x + a) - 75*a^4
*b^2*d^2*e^4*sign(b*x + a) - 42*(x*e + d)*a^5*b*e^5*sign(b*x + a) + 30*a^5*b*d*e
^5*sign(b*x + a) - 5*a^6*e^6*sign(b*x + a))*e^(-7)/(x*e + d)^(7/2)