3.2116 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.396223, 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 \[ -\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) \sqrt{d+e x}}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) (d+e x)^{3/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{5 e^7 (a+b x) (d+e x)^{5/2}}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2}}{7 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)}{5 e^7 (a+b x)}+\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}{e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^3}{e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 57.0448, size = 311, normalized size = 0.85 \[ \frac{128 b^{3} \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 e^{4}} + \frac{256 b^{3} \left (3 a + 3 b x\right ) \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{5}} + \frac{1024 b^{3} \sqrt{d + e x} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{6}} + \frac{2048 b^{3} \sqrt{d + e x} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{7} \left (a + b x\right )} - \frac{16 b^{2} \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e^{3} \sqrt{d + e x}} - \frac{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(7/2),x)

[Out]

128*b**3*sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(7*e**4) + 256*b**3*(
3*a + 3*b*x)*sqrt(d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*e**5
) + 1024*b**3*sqrt(d + e*x)*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(35*
e**6) + 2048*b**3*sqrt(d + e*x)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/
(35*e**7*(a + b*x)) - 16*b**2*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(e**
3*sqrt(d + e*x)) - 8*b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(5*e**2*(d + e*x)**(3
/2)) - 2*(a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(5*e*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.360549, size = 309, normalized size = 0.84 \[ -\frac{2 \sqrt{(a+b x)^2} \left (7 a^6 e^6+14 a^5 b e^5 (2 d+5 e x)+35 a^4 b^2 e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )-140 a^3 b^3 e^3 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+35 a^2 b^4 e^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )-14 a b^5 e \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )+b^6 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (a+b x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.014, size = 393, normalized size = 1.1 \[ -{\frac{-10\,{x}^{6}{b}^{6}{e}^{6}-84\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-350\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+280\,{x}^{4}a{b}^{5}d{e}^{5}-80\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-1400\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+2800\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-2240\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+640\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+1050\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-8400\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+16800\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-13440\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+3840\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+140\,x{a}^{5}b{e}^{6}+1400\,x{a}^{4}{b}^{2}d{e}^{5}-11200\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+22400\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-17920\,xa{b}^{5}{d}^{4}{e}^{2}+5120\,x{b}^{6}{d}^{5}e+14\,{a}^{6}{e}^{6}+56\,{a}^{5}bd{e}^{5}+560\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-4480\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+8960\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-7168\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{35\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

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)^(7/2),x)

[Out]

-2/35/(e*x+d)^(5/2)*(-5*b^6*e^6*x^6-42*a*b^5*e^6*x^5+12*b^6*d*e^5*x^5-175*a^2*b^
4*e^6*x^4+140*a*b^5*d*e^5*x^4-40*b^6*d^2*e^4*x^4-700*a^3*b^3*e^6*x^3+1400*a^2*b^
4*d*e^5*x^3-1120*a*b^5*d^2*e^4*x^3+320*b^6*d^3*e^3*x^3+525*a^4*b^2*e^6*x^2-4200*
a^3*b^3*d*e^5*x^2+8400*a^2*b^4*d^2*e^4*x^2-6720*a*b^5*d^3*e^3*x^2+1920*b^6*d^4*e
^2*x^2+70*a^5*b*e^6*x+700*a^4*b^2*d*e^5*x-5600*a^3*b^3*d^2*e^4*x+11200*a^2*b^4*d
^3*e^3*x-8960*a*b^5*d^4*e^2*x+2560*b^6*d^5*e*x+7*a^6*e^6+28*a^5*b*d*e^5+280*a^4*
b^2*d^2*e^4-2240*a^3*b^3*d^3*e^3+4480*a^2*b^4*d^4*e^2-3584*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 = 0.746331, size = 873, normalized size = 2.37 \[ \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \,{\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} a}{15 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (15 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 8960 \, a b^{4} d^{5} e - 8960 \, a^{2} b^{3} d^{4} e^{2} + 3360 \, a^{3} b^{2} d^{3} e^{3} - 280 \, a^{4} b d^{2} e^{4} - 14 \, a^{5} d e^{5} - 3 \,{\left (12 \, b^{5} d e^{5} - 35 \, a b^{4} e^{6}\right )} x^{5} + 10 \,{\left (12 \, b^{5} d^{2} e^{4} - 35 \, a b^{4} d e^{5} + 35 \, a^{2} b^{3} e^{6}\right )} x^{4} - 10 \,{\left (96 \, b^{5} d^{3} e^{3} - 280 \, a b^{4} d^{2} e^{4} + 280 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} - 15 \,{\left (384 \, b^{5} d^{4} e^{2} - 1120 \, a b^{4} d^{3} e^{3} + 1120 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 35 \, a^{4} b e^{6}\right )} x^{2} - 5 \,{\left (1536 \, b^{5} d^{5} e - 4480 \, a b^{4} d^{4} e^{2} + 4480 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 140 \, a^{4} b d e^{5} + 7 \, a^{5} e^{6}\right )} x\right )} b}{105 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt{e x + d}} \]

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)^(7/2),x, algorithm="maxima")

[Out]

2/15*(3*b^5*e^5*x^5 + 256*b^5*d^5 - 640*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 - 80*a
^3*b^2*d^2*e^3 - 10*a^4*b*d*e^4 - 3*a^5*e^5 - 5*(2*b^5*d*e^4 - 5*a*b^4*e^5)*x^4
+ 10*(8*b^5*d^2*e^3 - 20*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 + 30*(16*b^5*d^3*e^2
- 40*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 - 5*a^3*b^2*e^5)*x^2 + 5*(128*b^5*d^4*e -
320*a*b^4*d^3*e^2 + 240*a^2*b^3*d^2*e^3 - 40*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*a/(
(e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(e*x + d)) + 2/105*(15*b^5*e^6*x^6 - 3072*b^
5*d^6 + 8960*a*b^4*d^5*e - 8960*a^2*b^3*d^4*e^2 + 3360*a^3*b^2*d^3*e^3 - 280*a^4
*b*d^2*e^4 - 14*a^5*d*e^5 - 3*(12*b^5*d*e^5 - 35*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2
*e^4 - 35*a*b^4*d*e^5 + 35*a^2*b^3*e^6)*x^4 - 10*(96*b^5*d^3*e^3 - 280*a*b^4*d^2
*e^4 + 280*a^2*b^3*d*e^5 - 105*a^3*b^2*e^6)*x^3 - 15*(384*b^5*d^4*e^2 - 1120*a*b
^4*d^3*e^3 + 1120*a^2*b^3*d^2*e^4 - 420*a^3*b^2*d*e^5 + 35*a^4*b*e^6)*x^2 - 5*(1
536*b^5*d^5*e - 4480*a*b^4*d^4*e^2 + 4480*a^2*b^3*d^3*e^3 - 1680*a^3*b^2*d^2*e^4
+ 140*a^4*b*d*e^5 + 7*a^5*e^6)*x)*b/((e^9*x^2 + 2*d*e^8*x + d^2*e^7)*sqrt(e*x +
d))

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Fricas [A]  time = 0.282528, size = 509, normalized size = 1.38 \[ \frac{2 \,{\left (5 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 3584 \, a b^{5} d^{5} e - 4480 \, a^{2} b^{4} d^{4} e^{2} + 2240 \, a^{3} b^{3} d^{3} e^{3} - 280 \, a^{4} b^{2} d^{2} e^{4} - 28 \, a^{5} b d e^{5} - 7 \, a^{6} e^{6} - 6 \,{\left (2 \, b^{6} d e^{5} - 7 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (8 \, b^{6} d^{2} e^{4} - 28 \, a b^{5} d e^{5} + 35 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (16 \, b^{6} d^{3} e^{3} - 56 \, a b^{5} d^{2} e^{4} + 70 \, a^{2} b^{4} d e^{5} - 35 \, a^{3} b^{3} e^{6}\right )} x^{3} - 15 \,{\left (128 \, b^{6} d^{4} e^{2} - 448 \, a b^{5} d^{3} e^{3} + 560 \, a^{2} b^{4} d^{2} e^{4} - 280 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} - 10 \,{\left (256 \, b^{6} d^{5} e - 896 \, a b^{5} d^{4} e^{2} + 1120 \, a^{2} b^{4} d^{3} e^{3} - 560 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 7 \, a^{5} b e^{6}\right )} x\right )}}{35 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt{e x + d}} \]

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)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*b^6*e^6*x^6 - 1024*b^6*d^6 + 3584*a*b^5*d^5*e - 4480*a^2*b^4*d^4*e^2 + 2
240*a^3*b^3*d^3*e^3 - 280*a^4*b^2*d^2*e^4 - 28*a^5*b*d*e^5 - 7*a^6*e^6 - 6*(2*b^
6*d*e^5 - 7*a*b^5*e^6)*x^5 + 5*(8*b^6*d^2*e^4 - 28*a*b^5*d*e^5 + 35*a^2*b^4*e^6)
*x^4 - 20*(16*b^6*d^3*e^3 - 56*a*b^5*d^2*e^4 + 70*a^2*b^4*d*e^5 - 35*a^3*b^3*e^6
)*x^3 - 15*(128*b^6*d^4*e^2 - 448*a*b^5*d^3*e^3 + 560*a^2*b^4*d^2*e^4 - 280*a^3*
b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 - 10*(256*b^6*d^5*e - 896*a*b^5*d^4*e^2 + 1120*a
^2*b^4*d^3*e^3 - 560*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 7*a^5*b*e^6)*x)/((e^9*
x^2 + 2*d*e^8*x + d^2*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)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.325721, size = 845, normalized size = 2.3 \[ \text{result too large to display} \]

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)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*b^6*e^42*sign(b*x + a) - 42*(x*e + d)^(5/2)*b^6*d*e^42*s
ign(b*x + a) + 175*(x*e + d)^(3/2)*b^6*d^2*e^42*sign(b*x + a) - 700*sqrt(x*e + d
)*b^6*d^3*e^42*sign(b*x + a) + 42*(x*e + d)^(5/2)*a*b^5*e^43*sign(b*x + a) - 350
*(x*e + d)^(3/2)*a*b^5*d*e^43*sign(b*x + a) + 2100*sqrt(x*e + d)*a*b^5*d^2*e^43*
sign(b*x + a) + 175*(x*e + d)^(3/2)*a^2*b^4*e^44*sign(b*x + a) - 2100*sqrt(x*e +
d)*a^2*b^4*d*e^44*sign(b*x + a) + 700*sqrt(x*e + d)*a^3*b^3*e^45*sign(b*x + a))
*e^(-49) - 2/5*(75*(x*e + d)^2*b^6*d^4*sign(b*x + a) - 10*(x*e + d)*b^6*d^5*sign
(b*x + a) + b^6*d^6*sign(b*x + a) - 300*(x*e + d)^2*a*b^5*d^3*e*sign(b*x + a) +
50*(x*e + d)*a*b^5*d^4*e*sign(b*x + a) - 6*a*b^5*d^5*e*sign(b*x + a) + 450*(x*e
+ d)^2*a^2*b^4*d^2*e^2*sign(b*x + a) - 100*(x*e + d)*a^2*b^4*d^3*e^2*sign(b*x +
a) + 15*a^2*b^4*d^4*e^2*sign(b*x + a) - 300*(x*e + d)^2*a^3*b^3*d*e^3*sign(b*x +
a) + 100*(x*e + d)*a^3*b^3*d^2*e^3*sign(b*x + a) - 20*a^3*b^3*d^3*e^3*sign(b*x
+ a) + 75*(x*e + d)^2*a^4*b^2*e^4*sign(b*x + a) - 50*(x*e + d)*a^4*b^2*d*e^4*sig
n(b*x + a) + 15*a^4*b^2*d^2*e^4*sign(b*x + a) + 10*(x*e + d)*a^5*b*e^5*sign(b*x
+ a) - 6*a^5*b*d*e^5*sign(b*x + a) + a^6*e^6*sign(b*x + a))*e^(-7)/(x*e + d)^(5/
2)