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3.367 \(\int (e x)^m (a+b x) (a c-b c x)^4 \, dx\)

Optimal. Leaf size=145 \[ \frac{a^5 c^4 (e x)^{m+1}}{e (m+1)}-\frac{3 a^4 b c^4 (e x)^{m+2}}{e^2 (m+2)}+\frac{2 a^3 b^2 c^4 (e x)^{m+3}}{e^3 (m+3)}+\frac{2 a^2 b^3 c^4 (e x)^{m+4}}{e^4 (m+4)}-\frac{3 a b^4 c^4 (e x)^{m+5}}{e^5 (m+5)}+\frac{b^5 c^4 (e x)^{m+6}}{e^6 (m+6)} \]

[Out]

(a^5*c^4*(e*x)^(1 + m))/(e*(1 + m)) - (3*a^4*b*c^4*(e*x)^(2 + m))/(e^2*(2 + m))
+ (2*a^3*b^2*c^4*(e*x)^(3 + m))/(e^3*(3 + m)) + (2*a^2*b^3*c^4*(e*x)^(4 + m))/(e
^4*(4 + m)) - (3*a*b^4*c^4*(e*x)^(5 + m))/(e^5*(5 + m)) + (b^5*c^4*(e*x)^(6 + m)
)/(e^6*(6 + m))

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Rubi [A]  time = 0.229892, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{a^5 c^4 (e x)^{m+1}}{e (m+1)}-\frac{3 a^4 b c^4 (e x)^{m+2}}{e^2 (m+2)}+\frac{2 a^3 b^2 c^4 (e x)^{m+3}}{e^3 (m+3)}+\frac{2 a^2 b^3 c^4 (e x)^{m+4}}{e^4 (m+4)}-\frac{3 a b^4 c^4 (e x)^{m+5}}{e^5 (m+5)}+\frac{b^5 c^4 (e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(a^5*c^4*(e*x)^(1 + m))/(e*(1 + m)) - (3*a^4*b*c^4*(e*x)^(2 + m))/(e^2*(2 + m))
+ (2*a^3*b^2*c^4*(e*x)^(3 + m))/(e^3*(3 + m)) + (2*a^2*b^3*c^4*(e*x)^(4 + m))/(e
^4*(4 + m)) - (3*a*b^4*c^4*(e*x)^(5 + m))/(e^5*(5 + m)) + (b^5*c^4*(e*x)^(6 + m)
)/(e^6*(6 + m))

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Rubi in Sympy [A]  time = 52.452, size = 136, normalized size = 0.94 \[ \frac{a^{5} c^{4} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} - \frac{3 a^{4} b c^{4} \left (e x\right )^{m + 2}}{e^{2} \left (m + 2\right )} + \frac{2 a^{3} b^{2} c^{4} \left (e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{2 a^{2} b^{3} c^{4} \left (e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} - \frac{3 a b^{4} c^{4} \left (e x\right )^{m + 5}}{e^{5} \left (m + 5\right )} + \frac{b^{5} c^{4} \left (e x\right )^{m + 6}}{e^{6} \left (m + 6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

a**5*c**4*(e*x)**(m + 1)/(e*(m + 1)) - 3*a**4*b*c**4*(e*x)**(m + 2)/(e**2*(m + 2
)) + 2*a**3*b**2*c**4*(e*x)**(m + 3)/(e**3*(m + 3)) + 2*a**2*b**3*c**4*(e*x)**(m
 + 4)/(e**4*(m + 4)) - 3*a*b**4*c**4*(e*x)**(m + 5)/(e**5*(m + 5)) + b**5*c**4*(
e*x)**(m + 6)/(e**6*(m + 6))

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Mathematica [A]  time = 0.191307, size = 228, normalized size = 1.57 \[ \frac{c^4 x (e x)^m \left (a^5 \left (m^5+20 m^4+155 m^3+580 m^2+1044 m+720\right )-3 a^4 b \left (m^5+19 m^4+137 m^3+461 m^2+702 m+360\right ) x+2 a^3 b^2 \left (m^5+18 m^4+121 m^3+372 m^2+508 m+240\right ) x^2+2 a^2 b^3 \left (m^5+17 m^4+107 m^3+307 m^2+396 m+180\right ) x^3-3 a b^4 \left (m^5+16 m^4+95 m^3+260 m^2+324 m+144\right ) x^4+b^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )}{(m+1) (m+2) (m+3) (m+4) (m+5) (m+6)} \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^m*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(c^4*x*(e*x)^m*(a^5*(720 + 1044*m + 580*m^2 + 155*m^3 + 20*m^4 + m^5) - 3*a^4*b*
(360 + 702*m + 461*m^2 + 137*m^3 + 19*m^4 + m^5)*x + 2*a^3*b^2*(240 + 508*m + 37
2*m^2 + 121*m^3 + 18*m^4 + m^5)*x^2 + 2*a^2*b^3*(180 + 396*m + 307*m^2 + 107*m^3
 + 17*m^4 + m^5)*x^3 - 3*a*b^4*(144 + 324*m + 260*m^2 + 95*m^3 + 16*m^4 + m^5)*x
^4 + b^5*(120 + 274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5))/((1 + m)*(2 + m)*
(3 + m)*(4 + m)*(5 + m)*(6 + m))

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Maple [B]  time = 0.01, size = 424, normalized size = 2.9 \[{\frac{{c}^{4} \left ( ex \right ) ^{m} \left ({b}^{5}{m}^{5}{x}^{5}-3\,a{b}^{4}{m}^{5}{x}^{4}+15\,{b}^{5}{m}^{4}{x}^{5}+2\,{a}^{2}{b}^{3}{m}^{5}{x}^{3}-48\,a{b}^{4}{m}^{4}{x}^{4}+85\,{b}^{5}{m}^{3}{x}^{5}+2\,{a}^{3}{b}^{2}{m}^{5}{x}^{2}+34\,{a}^{2}{b}^{3}{m}^{4}{x}^{3}-285\,a{b}^{4}{m}^{3}{x}^{4}+225\,{b}^{5}{m}^{2}{x}^{5}-3\,{a}^{4}b{m}^{5}x+36\,{a}^{3}{b}^{2}{m}^{4}{x}^{2}+214\,{a}^{2}{b}^{3}{m}^{3}{x}^{3}-780\,a{b}^{4}{m}^{2}{x}^{4}+274\,{b}^{5}m{x}^{5}+{a}^{5}{m}^{5}-57\,{a}^{4}b{m}^{4}x+242\,{a}^{3}{b}^{2}{m}^{3}{x}^{2}+614\,{a}^{2}{b}^{3}{m}^{2}{x}^{3}-972\,a{b}^{4}m{x}^{4}+120\,{b}^{5}{x}^{5}+20\,{a}^{5}{m}^{4}-411\,{a}^{4}b{m}^{3}x+744\,{a}^{3}{b}^{2}{m}^{2}{x}^{2}+792\,{a}^{2}{b}^{3}m{x}^{3}-432\,a{b}^{4}{x}^{4}+155\,{a}^{5}{m}^{3}-1383\,{a}^{4}b{m}^{2}x+1016\,{a}^{3}{b}^{2}m{x}^{2}+360\,{a}^{2}{b}^{3}{x}^{3}+580\,{a}^{5}{m}^{2}-2106\,{a}^{4}bmx+480\,{a}^{3}{b}^{2}{x}^{2}+1044\,{a}^{5}m-1080\,{a}^{4}bx+720\,{a}^{5} \right ) x}{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(b*x+a)*(-b*c*x+a*c)^4,x)

[Out]

c^4*(e*x)^m*(b^5*m^5*x^5-3*a*b^4*m^5*x^4+15*b^5*m^4*x^5+2*a^2*b^3*m^5*x^3-48*a*b
^4*m^4*x^4+85*b^5*m^3*x^5+2*a^3*b^2*m^5*x^2+34*a^2*b^3*m^4*x^3-285*a*b^4*m^3*x^4
+225*b^5*m^2*x^5-3*a^4*b*m^5*x+36*a^3*b^2*m^4*x^2+214*a^2*b^3*m^3*x^3-780*a*b^4*
m^2*x^4+274*b^5*m*x^5+a^5*m^5-57*a^4*b*m^4*x+242*a^3*b^2*m^3*x^2+614*a^2*b^3*m^2
*x^3-972*a*b^4*m*x^4+120*b^5*x^5+20*a^5*m^4-411*a^4*b*m^3*x+744*a^3*b^2*m^2*x^2+
792*a^2*b^3*m*x^3-432*a*b^4*x^4+155*a^5*m^3-1383*a^4*b*m^2*x+1016*a^3*b^2*m*x^2+
360*a^2*b^3*x^3+580*a^5*m^2-2106*a^4*b*m*x+480*a^3*b^2*x^2+1044*a^5*m-1080*a^4*b
*x+720*a^5)*x/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)

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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*c*x - a*c)^4*(b*x + a)*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223694, size = 644, normalized size = 4.44 \[ \frac{{\left ({\left (b^{5} c^{4} m^{5} + 15 \, b^{5} c^{4} m^{4} + 85 \, b^{5} c^{4} m^{3} + 225 \, b^{5} c^{4} m^{2} + 274 \, b^{5} c^{4} m + 120 \, b^{5} c^{4}\right )} x^{6} - 3 \,{\left (a b^{4} c^{4} m^{5} + 16 \, a b^{4} c^{4} m^{4} + 95 \, a b^{4} c^{4} m^{3} + 260 \, a b^{4} c^{4} m^{2} + 324 \, a b^{4} c^{4} m + 144 \, a b^{4} c^{4}\right )} x^{5} + 2 \,{\left (a^{2} b^{3} c^{4} m^{5} + 17 \, a^{2} b^{3} c^{4} m^{4} + 107 \, a^{2} b^{3} c^{4} m^{3} + 307 \, a^{2} b^{3} c^{4} m^{2} + 396 \, a^{2} b^{3} c^{4} m + 180 \, a^{2} b^{3} c^{4}\right )} x^{4} + 2 \,{\left (a^{3} b^{2} c^{4} m^{5} + 18 \, a^{3} b^{2} c^{4} m^{4} + 121 \, a^{3} b^{2} c^{4} m^{3} + 372 \, a^{3} b^{2} c^{4} m^{2} + 508 \, a^{3} b^{2} c^{4} m + 240 \, a^{3} b^{2} c^{4}\right )} x^{3} - 3 \,{\left (a^{4} b c^{4} m^{5} + 19 \, a^{4} b c^{4} m^{4} + 137 \, a^{4} b c^{4} m^{3} + 461 \, a^{4} b c^{4} m^{2} + 702 \, a^{4} b c^{4} m + 360 \, a^{4} b c^{4}\right )} x^{2} +{\left (a^{5} c^{4} m^{5} + 20 \, a^{5} c^{4} m^{4} + 155 \, a^{5} c^{4} m^{3} + 580 \, a^{5} c^{4} m^{2} + 1044 \, a^{5} c^{4} m + 720 \, a^{5} c^{4}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)*(e*x)^m,x, algorithm="fricas")

[Out]

((b^5*c^4*m^5 + 15*b^5*c^4*m^4 + 85*b^5*c^4*m^3 + 225*b^5*c^4*m^2 + 274*b^5*c^4*
m + 120*b^5*c^4)*x^6 - 3*(a*b^4*c^4*m^5 + 16*a*b^4*c^4*m^4 + 95*a*b^4*c^4*m^3 +
260*a*b^4*c^4*m^2 + 324*a*b^4*c^4*m + 144*a*b^4*c^4)*x^5 + 2*(a^2*b^3*c^4*m^5 +
17*a^2*b^3*c^4*m^4 + 107*a^2*b^3*c^4*m^3 + 307*a^2*b^3*c^4*m^2 + 396*a^2*b^3*c^4
*m + 180*a^2*b^3*c^4)*x^4 + 2*(a^3*b^2*c^4*m^5 + 18*a^3*b^2*c^4*m^4 + 121*a^3*b^
2*c^4*m^3 + 372*a^3*b^2*c^4*m^2 + 508*a^3*b^2*c^4*m + 240*a^3*b^2*c^4)*x^3 - 3*(
a^4*b*c^4*m^5 + 19*a^4*b*c^4*m^4 + 137*a^4*b*c^4*m^3 + 461*a^4*b*c^4*m^2 + 702*a
^4*b*c^4*m + 360*a^4*b*c^4)*x^2 + (a^5*c^4*m^5 + 20*a^5*c^4*m^4 + 155*a^5*c^4*m^
3 + 580*a^5*c^4*m^2 + 1044*a^5*c^4*m + 720*a^5*c^4)*x)*(e*x)^m/(m^6 + 21*m^5 + 1
75*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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Sympy [A]  time = 5.72525, size = 2338, normalized size = 16.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

Piecewise(((-a**5*c**4/(5*x**5) + 3*a**4*b*c**4/(4*x**4) - 2*a**3*b**2*c**4/(3*x
**3) - a**2*b**3*c**4/x**2 + 3*a*b**4*c**4/x + b**5*c**4*log(x))/e**6, Eq(m, -6)
), ((-a**5*c**4/(4*x**4) + a**4*b*c**4/x**3 - a**3*b**2*c**4/x**2 - 2*a**2*b**3*
c**4/x - 3*a*b**4*c**4*log(x) + b**5*c**4*x)/e**5, Eq(m, -5)), ((-a**5*c**4/(3*x
**3) + 3*a**4*b*c**4/(2*x**2) - 2*a**3*b**2*c**4/x + 2*a**2*b**3*c**4*log(x) - 3
*a*b**4*c**4*x + b**5*c**4*x**2/2)/e**4, Eq(m, -4)), ((-a**5*c**4/(2*x**2) + 3*a
**4*b*c**4/x + 2*a**3*b**2*c**4*log(x) + 2*a**2*b**3*c**4*x - 3*a*b**4*c**4*x**2
/2 + b**5*c**4*x**3/3)/e**3, Eq(m, -3)), ((-a**5*c**4/x - 3*a**4*b*c**4*log(x) +
 2*a**3*b**2*c**4*x + a**2*b**3*c**4*x**2 - a*b**4*c**4*x**3 + b**5*c**4*x**4/4)
/e**2, Eq(m, -2)), ((a**5*c**4*log(x) - 3*a**4*b*c**4*x + a**3*b**2*c**4*x**2 +
2*a**2*b**3*c**4*x**3/3 - 3*a*b**4*c**4*x**4/4 + b**5*c**4*x**5/5)/e, Eq(m, -1))
, (a**5*c**4*e**m*m**5*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 20*a**5*c**4*e**m*m**4*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735
*m**3 + 1624*m**2 + 1764*m + 720) + 155*a**5*c**4*e**m*m**3*x*x**m/(m**6 + 21*m*
*5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*a**5*c**4*e**m*m**2*x
*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1044*a
**5*c**4*e**m*m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*
m + 720) + 720*a**5*c**4*e**m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 162
4*m**2 + 1764*m + 720) - 3*a**4*b*c**4*e**m*m**5*x**2*x**m/(m**6 + 21*m**5 + 175
*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 57*a**4*b*c**4*e**m*m**4*x**2*x**
m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 411*a**4*b
*c**4*e**m*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 17
64*m + 720) - 1383*a**4*b*c**4*e**m*m**2*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) - 2106*a**4*b*c**4*e**m*m*x**2*x**m/(m**6 +
 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 1080*a**4*b*c**4*e*
*m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) +
 2*a**3*b**2*c**4*e**m*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 16
24*m**2 + 1764*m + 720) + 36*a**3*b**2*c**4*e**m*m**4*x**3*x**m/(m**6 + 21*m**5
+ 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 242*a**3*b**2*c**4*e**m*m**3
*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 7
44*a**3*b**2*c**4*e**m*m**2*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 16
24*m**2 + 1764*m + 720) + 1016*a**3*b**2*c**4*e**m*m*x**3*x**m/(m**6 + 21*m**5 +
 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 480*a**3*b**2*c**4*e**m*x**3*
x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*a**2*
b**3*c**4*e**m*m**5*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 34*a**2*b**3*c**4*e**m*m**4*x**4*x**m/(m**6 + 21*m**5 + 175*m*
*4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 214*a**2*b**3*c**4*e**m*m**3*x**4*x*
*m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 614*a**2*
b**3*c**4*e**m*m**2*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 792*a**2*b**3*c**4*e**m*m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4
 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 360*a**2*b**3*c**4*e**m*x**4*x**m/(m**
6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 3*a*b**4*c**4*e*
*m*m**5*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 7
20) - 48*a*b**4*c**4*e**m*m**4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 +
 1624*m**2 + 1764*m + 720) - 285*a*b**4*c**4*e**m*m**3*x**5*x**m/(m**6 + 21*m**5
 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 780*a*b**4*c**4*e**m*m**2*x
**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 972
*a*b**4*c**4*e**m*m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) - 432*a*b**4*c**4*e**m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 73
5*m**3 + 1624*m**2 + 1764*m + 720) + b**5*c**4*e**m*m**5*x**6*x**m/(m**6 + 21*m*
*5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 15*b**5*c**4*e**m*m**4*x*
*6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 85*b
**5*c**4*e**m*m**3*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 +
 1764*m + 720) + 225*b**5*c**4*e**m*m**2*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) + 274*b**5*c**4*e**m*m*x**6*x**m/(m**6 + 21
*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 120*b**5*c**4*e**m*x**
6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232284, size = 1069, normalized size = 7.37 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*c*x - a*c)^4*(b*x + a)*(e*x)^m,x, algorithm="giac")

[Out]

(b^5*c^4*m^5*x^6*e^(m*ln(x) + m) - 3*a*b^4*c^4*m^5*x^5*e^(m*ln(x) + m) + 15*b^5*
c^4*m^4*x^6*e^(m*ln(x) + m) + 2*a^2*b^3*c^4*m^5*x^4*e^(m*ln(x) + m) - 48*a*b^4*c
^4*m^4*x^5*e^(m*ln(x) + m) + 85*b^5*c^4*m^3*x^6*e^(m*ln(x) + m) + 2*a^3*b^2*c^4*
m^5*x^3*e^(m*ln(x) + m) + 34*a^2*b^3*c^4*m^4*x^4*e^(m*ln(x) + m) - 285*a*b^4*c^4
*m^3*x^5*e^(m*ln(x) + m) + 225*b^5*c^4*m^2*x^6*e^(m*ln(x) + m) - 3*a^4*b*c^4*m^5
*x^2*e^(m*ln(x) + m) + 36*a^3*b^2*c^4*m^4*x^3*e^(m*ln(x) + m) + 214*a^2*b^3*c^4*
m^3*x^4*e^(m*ln(x) + m) - 780*a*b^4*c^4*m^2*x^5*e^(m*ln(x) + m) + 274*b^5*c^4*m*
x^6*e^(m*ln(x) + m) + a^5*c^4*m^5*x*e^(m*ln(x) + m) - 57*a^4*b*c^4*m^4*x^2*e^(m*
ln(x) + m) + 242*a^3*b^2*c^4*m^3*x^3*e^(m*ln(x) + m) + 614*a^2*b^3*c^4*m^2*x^4*e
^(m*ln(x) + m) - 972*a*b^4*c^4*m*x^5*e^(m*ln(x) + m) + 120*b^5*c^4*x^6*e^(m*ln(x
) + m) + 20*a^5*c^4*m^4*x*e^(m*ln(x) + m) - 411*a^4*b*c^4*m^3*x^2*e^(m*ln(x) + m
) + 744*a^3*b^2*c^4*m^2*x^3*e^(m*ln(x) + m) + 792*a^2*b^3*c^4*m*x^4*e^(m*ln(x) +
 m) - 432*a*b^4*c^4*x^5*e^(m*ln(x) + m) + 155*a^5*c^4*m^3*x*e^(m*ln(x) + m) - 13
83*a^4*b*c^4*m^2*x^2*e^(m*ln(x) + m) + 1016*a^3*b^2*c^4*m*x^3*e^(m*ln(x) + m) +
360*a^2*b^3*c^4*x^4*e^(m*ln(x) + m) + 580*a^5*c^4*m^2*x*e^(m*ln(x) + m) - 2106*a
^4*b*c^4*m*x^2*e^(m*ln(x) + m) + 480*a^3*b^2*c^4*x^3*e^(m*ln(x) + m) + 1044*a^5*
c^4*m*x*e^(m*ln(x) + m) - 1080*a^4*b*c^4*x^2*e^(m*ln(x) + m) + 720*a^5*c^4*x*e^(
m*ln(x) + m))/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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