3.1999 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=345 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) (d+e x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^7 (a+b x)}+\frac{15 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x)}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{2 e^7 (a+b x)}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)^2}{e^7 (a+b x)}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}{e^7 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.658257, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) (d+e x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 \log (d+e x)}{e^7 (a+b x)}+\frac{15 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x)}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)}-\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{2 e^7 (a+b x)}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)^2}{e^7 (a+b x)}-\frac{10 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}{e^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 62.3163, size = 265, normalized size = 0.77 \[ \frac{6 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 e^{2}} + \frac{3 b \left (5 a + 5 b x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{10 e^{3}} + \frac{2 b \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e^{4}} + \frac{b \left (3 a + 3 b x\right ) \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{5}} + \frac{6 b \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{6}} + \frac{6 b \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{7} \left (a + b x\right )} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{e \left (d + e x\right )} \]

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

[Out]

6*b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(5*e**2) + 3*b*(5*a + 5*b*x)*(a*e - b*d)
*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(10*e**3) + 2*b*(a*e - b*d)**2*(a**2 + 2*a*
b*x + b**2*x**2)**(3/2)/e**4 + b*(3*a + 3*b*x)*(a*e - b*d)**3*sqrt(a**2 + 2*a*b*
x + b**2*x**2)/e**5 + 6*b*(a*e - b*d)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/e**6 +
6*b*(a*e - b*d)**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(d + e*x)/(e**7*(a + b*x
)) - (a + b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(e*(d + e*x))

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Mathematica [A]  time = 0.466581, size = 320, normalized size = 0.93 \[ \frac{\sqrt{(a+b x)^2} \left (-10 a^6 e^6+60 a^5 b d e^5+150 a^4 b^2 e^4 \left (-d^2+d e x+e^2 x^2\right )+100 a^3 b^3 e^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+50 a^2 b^4 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b^5 e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-60 b (d+e x) (b d-a e)^5 \log (d+e x)+b^6 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )\right )}{10 e^7 (a+b x) (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x)^2]*(60*a^5*b*d*e^5 - 10*a^6*e^6 + 150*a^4*b^2*e^4*(-d^2 + d*e*x
+ e^2*x^2) + 100*a^3*b^3*e^3*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 50*a^
2*b^4*e^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b^5
*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*
x^5) + b^6*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x
^4 - 3*d*e^5*x^5 + 2*e^6*x^6) - 60*b*(b*d - a*e)^5*(d + e*x)*Log[d + e*x]))/(10*
e^7*(a + b*x)*(d + e*x))

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Maple [B]  time = 0.027, size = 601, normalized size = 1.7 \[{\frac{-100\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+600\,\ln \left ( ex+d \right ){a}^{3}{b}^{3}{d}^{3}{e}^{3}-600\,\ln \left ( ex+d \right ){a}^{2}{b}^{4}{d}^{4}{e}^{2}+300\,\ln \left ( ex+d \right ) a{b}^{5}{d}^{5}e-300\,\ln \left ( ex+d \right ){a}^{4}{b}^{2}{d}^{2}{e}^{4}+300\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-150\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-400\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+50\,x{b}^{6}{d}^{5}e+5\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-10\,{a}^{6}{e}^{6}-10\,{b}^{6}{d}^{6}+60\,{d}^{5}a{b}^{5}e+200\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+2\,{x}^{6}{b}^{6}{e}^{6}-300\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-25\,{x}^{4}a{b}^{5}d{e}^{5}-300\,\ln \left ( ex+d \right ) x{a}^{4}{b}^{2}d{e}^{5}+600\,\ln \left ( ex+d \right ) x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-600\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+300\,\ln \left ( ex+d \right ) xa{b}^{5}{d}^{4}{e}^{2}-150\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+450\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-240\,xa{b}^{5}{d}^{4}{e}^{2}+50\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+60\,\ln \left ( ex+d \right ){a}^{5}bd{e}^{5}+150\,x{a}^{4}{b}^{2}d{e}^{5}+60\,\ln \left ( ex+d \right ) x{a}^{5}b{e}^{6}-60\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e-150\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+60\,{a}^{5}bd{e}^{5}+15\,{x}^{5}a{b}^{5}{e}^{6}-3\,{x}^{5}{b}^{6}d{e}^{5}-10\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+100\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+50\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+150\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+30\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}}{10\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) } \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^2,x)

[Out]

1/10*((b*x+a)^2)^(5/2)*(-100*x^3*a^2*b^4*d*e^5+600*ln(e*x+d)*a^3*b^3*d^3*e^3-600
*ln(e*x+d)*a^2*b^4*d^4*e^2+300*ln(e*x+d)*a*b^5*d^5*e-300*ln(e*x+d)*a^4*b^2*d^2*e
^4+300*x^2*a^2*b^4*d^2*e^4-150*x^2*a*b^5*d^3*e^3-400*x*a^3*b^3*d^2*e^4+50*x*b^6*
d^5*e+5*x^4*b^6*d^2*e^4-10*a^6*e^6-10*b^6*d^6+60*d^5*a*b^5*e+200*a^3*b^3*d^3*e^3
-60*ln(e*x+d)*b^6*d^6+2*x^6*b^6*e^6-300*x^2*a^3*b^3*d*e^5-25*x^4*a*b^5*d*e^5-300
*ln(e*x+d)*x*a^4*b^2*d*e^5+600*ln(e*x+d)*x*a^3*b^3*d^2*e^4-600*ln(e*x+d)*x*a^2*b
^4*d^3*e^3+300*ln(e*x+d)*x*a*b^5*d^4*e^2-150*d^4*e^2*a^2*b^4+450*x*a^2*b^4*d^3*e
^3-240*x*a*b^5*d^4*e^2+50*x^3*a*b^5*d^2*e^4+60*ln(e*x+d)*a^5*b*d*e^5+150*x*a^4*b
^2*d*e^5+60*ln(e*x+d)*x*a^5*b*e^6-60*ln(e*x+d)*x*b^6*d^5*e-150*b^2*a^4*d^2*e^4+6
0*a^5*b*d*e^5+15*x^5*a*b^5*e^6-3*x^5*b^6*d*e^5-10*x^3*b^6*d^3*e^3+100*x^3*a^3*b^
3*e^6+50*x^4*a^2*b^4*e^6+150*x^2*a^4*b^2*e^6+30*x^2*b^6*d^4*e^2)/(b*x+a)^5/e^7/(
e*x+d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28484, size = 670, normalized size = 1.94 \[ \frac{2 \, b^{6} e^{6} x^{6} - 10 \, b^{6} d^{6} + 60 \, a b^{5} d^{5} e - 150 \, a^{2} b^{4} d^{4} e^{2} + 200 \, a^{3} b^{3} d^{3} e^{3} - 150 \, a^{4} b^{2} d^{2} e^{4} + 60 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} - 3 \,{\left (b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - 5 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} - 10 \,{\left (b^{6} d^{3} e^{3} - 5 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} - 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \,{\left (b^{6} d^{4} e^{2} - 5 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} - 10 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \,{\left (5 \, b^{6} d^{5} e - 24 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} - 40 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5}\right )} x - 60 \,{\left (b^{6} d^{6} - 5 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} - a^{5} b d e^{5} +{\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (e x + d\right )}{10 \,{\left (e^{8} x + d e^{7}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/10*(2*b^6*e^6*x^6 - 10*b^6*d^6 + 60*a*b^5*d^5*e - 150*a^2*b^4*d^4*e^2 + 200*a^
3*b^3*d^3*e^3 - 150*a^4*b^2*d^2*e^4 + 60*a^5*b*d*e^5 - 10*a^6*e^6 - 3*(b^6*d*e^5
- 5*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 5*a*b^5*d*e^5 + 10*a^2*b^4*e^6)*x^4 - 10*
(b^6*d^3*e^3 - 5*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 - 10*a^3*b^3*e^6)*x^3 + 30*(b^
6*d^4*e^2 - 5*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 - 10*a^3*b^3*d*e^5 + 5*a^4*b^2*
e^6)*x^2 + 10*(5*b^6*d^5*e - 24*a*b^5*d^4*e^2 + 45*a^2*b^4*d^3*e^3 - 40*a^3*b^3*
d^2*e^4 + 15*a^4*b^2*d*e^5)*x - 60*(b^6*d^6 - 5*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2
- 10*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 - a^5*b*d*e^5 + (b^6*d^5*e - 5*a*b^5*d
^4*e^2 + 10*a^2*b^4*d^3*e^3 - 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 - a^5*b*e^6)*
x)*log(e*x + d))/(e^8*x + d*e^7)

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.290445, size = 701, normalized size = 2.03 \[ -6 \,{\left (b^{6} d^{5}{\rm sign}\left (b x + a\right ) - 5 \, a b^{5} d^{4} e{\rm sign}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{3} e^{2}{\rm sign}\left (b x + a\right ) - 10 \, a^{3} b^{3} d^{2} e^{3}{\rm sign}\left (b x + a\right ) + 5 \, a^{4} b^{2} d e^{4}{\rm sign}\left (b x + a\right ) - a^{5} b e^{5}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{10} \,{\left (2 \, b^{6} x^{5} e^{8}{\rm sign}\left (b x + a\right ) - 5 \, b^{6} d x^{4} e^{7}{\rm sign}\left (b x + a\right ) + 10 \, b^{6} d^{2} x^{3} e^{6}{\rm sign}\left (b x + a\right ) - 20 \, b^{6} d^{3} x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 50 \, b^{6} d^{4} x e^{4}{\rm sign}\left (b x + a\right ) + 15 \, a b^{5} x^{4} e^{8}{\rm sign}\left (b x + a\right ) - 40 \, a b^{5} d x^{3} e^{7}{\rm sign}\left (b x + a\right ) + 90 \, a b^{5} d^{2} x^{2} e^{6}{\rm sign}\left (b x + a\right ) - 240 \, a b^{5} d^{3} x e^{5}{\rm sign}\left (b x + a\right ) + 50 \, a^{2} b^{4} x^{3} e^{8}{\rm sign}\left (b x + a\right ) - 150 \, a^{2} b^{4} d x^{2} e^{7}{\rm sign}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} x e^{6}{\rm sign}\left (b x + a\right ) + 100 \, a^{3} b^{3} x^{2} e^{8}{\rm sign}\left (b x + a\right ) - 400 \, a^{3} b^{3} d x e^{7}{\rm sign}\left (b x + a\right ) + 150 \, a^{4} b^{2} x e^{8}{\rm sign}\left (b x + a\right )\right )} e^{\left (-10\right )} - \frac{{\left (b^{6} d^{6}{\rm sign}\left (b x + a\right ) - 6 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 15 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 15 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) - 6 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) + a^{6} e^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}}{x e + d} \]

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)^2,x, algorithm="giac")

[Out]

-6*(b^6*d^5*sign(b*x + a) - 5*a*b^5*d^4*e*sign(b*x + a) + 10*a^2*b^4*d^3*e^2*sig
n(b*x + a) - 10*a^3*b^3*d^2*e^3*sign(b*x + a) + 5*a^4*b^2*d*e^4*sign(b*x + a) -
a^5*b*e^5*sign(b*x + a))*e^(-7)*ln(abs(x*e + d)) + 1/10*(2*b^6*x^5*e^8*sign(b*x
+ a) - 5*b^6*d*x^4*e^7*sign(b*x + a) + 10*b^6*d^2*x^3*e^6*sign(b*x + a) - 20*b^6
*d^3*x^2*e^5*sign(b*x + a) + 50*b^6*d^4*x*e^4*sign(b*x + a) + 15*a*b^5*x^4*e^8*s
ign(b*x + a) - 40*a*b^5*d*x^3*e^7*sign(b*x + a) + 90*a*b^5*d^2*x^2*e^6*sign(b*x
+ a) - 240*a*b^5*d^3*x*e^5*sign(b*x + a) + 50*a^2*b^4*x^3*e^8*sign(b*x + a) - 15
0*a^2*b^4*d*x^2*e^7*sign(b*x + a) + 450*a^2*b^4*d^2*x*e^6*sign(b*x + a) + 100*a^
3*b^3*x^2*e^8*sign(b*x + a) - 400*a^3*b^3*d*x*e^7*sign(b*x + a) + 150*a^4*b^2*x*
e^8*sign(b*x + a))*e^(-10) - (b^6*d^6*sign(b*x + a) - 6*a*b^5*d^5*e*sign(b*x + a
) + 15*a^2*b^4*d^4*e^2*sign(b*x + a) - 20*a^3*b^3*d^3*e^3*sign(b*x + a) + 15*a^4
*b^2*d^2*e^4*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)