3.487 \(\int (e x)^m (A+B x) (a+c x^2)^2 \, dx\)
Optimal. Leaf size=121 \[ \frac{a^2 A (e x)^{m+1}}{e (m+1)}+\frac{a^2 B (e x)^{m+2}}{e^2 (m+2)}+\frac{2 a A c (e x)^{m+3}}{e^3 (m+3)}+\frac{2 a B c (e x)^{m+4}}{e^4 (m+4)}+\frac{A c^2 (e x)^{m+5}}{e^5 (m+5)}+\frac{B c^2 (e x)^{m+6}}{e^6 (m+6)} \]
[Out]
(a^2*A*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*B*(e*x)^(2 + m))/(e^2*(2 + m)) + (2*a*A*c*(e*x)^(3 + m))/(e^3*(3 + m)
) + (2*a*B*c*(e*x)^(4 + m))/(e^4*(4 + m)) + (A*c^2*(e*x)^(5 + m))/(e^5*(5 + m)) + (B*c^2*(e*x)^(6 + m))/(e^6*(
6 + m))
________________________________________________________________________________________
Rubi [A] time = 0.0620844, antiderivative size = 121, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used =
{766} \[ \frac{a^2 A (e x)^{m+1}}{e (m+1)}+\frac{a^2 B (e x)^{m+2}}{e^2 (m+2)}+\frac{2 a A c (e x)^{m+3}}{e^3 (m+3)}+\frac{2 a B c (e x)^{m+4}}{e^4 (m+4)}+\frac{A c^2 (e x)^{m+5}}{e^5 (m+5)}+\frac{B c^2 (e x)^{m+6}}{e^6 (m+6)} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m*(A + B*x)*(a + c*x^2)^2,x]
[Out]
(a^2*A*(e*x)^(1 + m))/(e*(1 + m)) + (a^2*B*(e*x)^(2 + m))/(e^2*(2 + m)) + (2*a*A*c*(e*x)^(3 + m))/(e^3*(3 + m)
) + (2*a*B*c*(e*x)^(4 + m))/(e^4*(4 + m)) + (A*c^2*(e*x)^(5 + m))/(e^5*(5 + m)) + (B*c^2*(e*x)^(6 + m))/(e^6*(
6 + m))
Rule 766
Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]
Rubi steps
\begin{align*} \int (e x)^m (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A (e x)^m+\frac{a^2 B (e x)^{1+m}}{e}+\frac{2 a A c (e x)^{2+m}}{e^2}+\frac{2 a B c (e x)^{3+m}}{e^3}+\frac{A c^2 (e x)^{4+m}}{e^4}+\frac{B c^2 (e x)^{5+m}}{e^5}\right ) \, dx\\ &=\frac{a^2 A (e x)^{1+m}}{e (1+m)}+\frac{a^2 B (e x)^{2+m}}{e^2 (2+m)}+\frac{2 a A c (e x)^{3+m}}{e^3 (3+m)}+\frac{2 a B c (e x)^{4+m}}{e^4 (4+m)}+\frac{A c^2 (e x)^{5+m}}{e^5 (5+m)}+\frac{B c^2 (e x)^{6+m}}{e^6 (6+m)}\\ \end{align*}
Mathematica [A] time = 0.0802563, size = 74, normalized size = 0.61 \[ x (e x)^m \left (a^2 \left (\frac{A}{m+1}+\frac{B x}{m+2}\right )+2 a c x^2 \left (\frac{A}{m+3}+\frac{B x}{m+4}\right )+c^2 x^4 \left (\frac{A}{m+5}+\frac{B x}{m+6}\right )\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m*(A + B*x)*(a + c*x^2)^2,x]
[Out]
x*(e*x)^m*(a^2*(A/(1 + m) + (B*x)/(2 + m)) + 2*a*c*x^2*(A/(3 + m) + (B*x)/(4 + m)) + c^2*x^4*(A/(5 + m) + (B*x
)/(6 + m)))
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Maple [B] time = 0.005, size = 395, normalized size = 3.3 \begin{align*}{\frac{ \left ( B{c}^{2}{m}^{5}{x}^{5}+A{c}^{2}{m}^{5}{x}^{4}+15\,B{c}^{2}{m}^{4}{x}^{5}+16\,A{c}^{2}{m}^{4}{x}^{4}+2\,Bac{m}^{5}{x}^{3}+85\,B{c}^{2}{m}^{3}{x}^{5}+2\,Aac{m}^{5}{x}^{2}+95\,A{c}^{2}{m}^{3}{x}^{4}+34\,Bac{m}^{4}{x}^{3}+225\,B{c}^{2}{m}^{2}{x}^{5}+36\,Aac{m}^{4}{x}^{2}+260\,A{c}^{2}{m}^{2}{x}^{4}+B{a}^{2}{m}^{5}x+214\,Bac{m}^{3}{x}^{3}+274\,B{c}^{2}m{x}^{5}+A{a}^{2}{m}^{5}+242\,Aac{m}^{3}{x}^{2}+324\,A{c}^{2}m{x}^{4}+19\,B{a}^{2}{m}^{4}x+614\,Bac{m}^{2}{x}^{3}+120\,B{c}^{2}{x}^{5}+20\,A{a}^{2}{m}^{4}+744\,Aac{m}^{2}{x}^{2}+144\,A{c}^{2}{x}^{4}+137\,B{a}^{2}{m}^{3}x+792\,Bacm{x}^{3}+155\,A{a}^{2}{m}^{3}+1016\,Aacm{x}^{2}+461\,B{a}^{2}{m}^{2}x+360\,aBc{x}^{3}+580\,A{a}^{2}{m}^{2}+480\,aAc{x}^{2}+702\,B{a}^{2}mx+1044\,A{a}^{2}m+360\,{a}^{2}Bx+720\,A{a}^{2} \right ) x \left ( ex \right ) ^{m}}{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*(B*x+A)*(c*x^2+a)^2,x)
[Out]
x*(B*c^2*m^5*x^5+A*c^2*m^5*x^4+15*B*c^2*m^4*x^5+16*A*c^2*m^4*x^4+2*B*a*c*m^5*x^3+85*B*c^2*m^3*x^5+2*A*a*c*m^5*
x^2+95*A*c^2*m^3*x^4+34*B*a*c*m^4*x^3+225*B*c^2*m^2*x^5+36*A*a*c*m^4*x^2+260*A*c^2*m^2*x^4+B*a^2*m^5*x+214*B*a
*c*m^3*x^3+274*B*c^2*m*x^5+A*a^2*m^5+242*A*a*c*m^3*x^2+324*A*c^2*m*x^4+19*B*a^2*m^4*x+614*B*a*c*m^2*x^3+120*B*
c^2*x^5+20*A*a^2*m^4+744*A*a*c*m^2*x^2+144*A*c^2*x^4+137*B*a^2*m^3*x+792*B*a*c*m*x^3+155*A*a^2*m^3+1016*A*a*c*
m*x^2+461*B*a^2*m^2*x+360*B*a*c*x^3+580*A*a^2*m^2+480*A*a*c*x^2+702*B*a^2*m*x+1044*A*a^2*m+360*B*a^2*x+720*A*a
^2)*(e*x)^m/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x+A)*(c*x^2+a)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.71465, size = 833, normalized size = 6.88 \begin{align*} \frac{{\left ({\left (B c^{2} m^{5} + 15 \, B c^{2} m^{4} + 85 \, B c^{2} m^{3} + 225 \, B c^{2} m^{2} + 274 \, B c^{2} m + 120 \, B c^{2}\right )} x^{6} +{\left (A c^{2} m^{5} + 16 \, A c^{2} m^{4} + 95 \, A c^{2} m^{3} + 260 \, A c^{2} m^{2} + 324 \, A c^{2} m + 144 \, A c^{2}\right )} x^{5} + 2 \,{\left (B a c m^{5} + 17 \, B a c m^{4} + 107 \, B a c m^{3} + 307 \, B a c m^{2} + 396 \, B a c m + 180 \, B a c\right )} x^{4} + 2 \,{\left (A a c m^{5} + 18 \, A a c m^{4} + 121 \, A a c m^{3} + 372 \, A a c m^{2} + 508 \, A a c m + 240 \, A a c\right )} x^{3} +{\left (B a^{2} m^{5} + 19 \, B a^{2} m^{4} + 137 \, B a^{2} m^{3} + 461 \, B a^{2} m^{2} + 702 \, B a^{2} m + 360 \, B a^{2}\right )} x^{2} +{\left (A a^{2} m^{5} + 20 \, A a^{2} m^{4} + 155 \, A a^{2} m^{3} + 580 \, A a^{2} m^{2} + 1044 \, A a^{2} m + 720 \, A a^{2}\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x+A)*(c*x^2+a)^2,x, algorithm="fricas")
[Out]
((B*c^2*m^5 + 15*B*c^2*m^4 + 85*B*c^2*m^3 + 225*B*c^2*m^2 + 274*B*c^2*m + 120*B*c^2)*x^6 + (A*c^2*m^5 + 16*A*c
^2*m^4 + 95*A*c^2*m^3 + 260*A*c^2*m^2 + 324*A*c^2*m + 144*A*c^2)*x^5 + 2*(B*a*c*m^5 + 17*B*a*c*m^4 + 107*B*a*c
*m^3 + 307*B*a*c*m^2 + 396*B*a*c*m + 180*B*a*c)*x^4 + 2*(A*a*c*m^5 + 18*A*a*c*m^4 + 121*A*a*c*m^3 + 372*A*a*c*
m^2 + 508*A*a*c*m + 240*A*a*c)*x^3 + (B*a^2*m^5 + 19*B*a^2*m^4 + 137*B*a^2*m^3 + 461*B*a^2*m^2 + 702*B*a^2*m +
360*B*a^2)*x^2 + (A*a^2*m^5 + 20*A*a^2*m^4 + 155*A*a^2*m^3 + 580*A*a^2*m^2 + 1044*A*a^2*m + 720*A*a^2)*x)*(e*
x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)
________________________________________________________________________________________
Sympy [A] time = 1.79343, size = 2076, normalized size = 17.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m*(B*x+A)*(c*x**2+a)**2,x)
[Out]
Piecewise(((-A*a**2/(5*x**5) - 2*A*a*c/(3*x**3) - A*c**2/x - B*a**2/(4*x**4) - B*a*c/x**2 + B*c**2*log(x))/e**
6, Eq(m, -6)), ((-A*a**2/(4*x**4) - A*a*c/x**2 + A*c**2*log(x) - B*a**2/(3*x**3) - 2*B*a*c/x + B*c**2*x)/e**5,
Eq(m, -5)), ((-A*a**2/(3*x**3) - 2*A*a*c/x + A*c**2*x - B*a**2/(2*x**2) + 2*B*a*c*log(x) + B*c**2*x**2/2)/e**
4, Eq(m, -4)), ((-A*a**2/(2*x**2) + 2*A*a*c*log(x) + A*c**2*x**2/2 - B*a**2/x + 2*B*a*c*x + B*c**2*x**3/3)/e**
3, Eq(m, -3)), ((-A*a**2/x + 2*A*a*c*x + A*c**2*x**3/3 + B*a**2*log(x) + B*a*c*x**2 + B*c**2*x**4/4)/e**2, Eq(
m, -2)), ((A*a**2*log(x) + A*a*c*x**2 + A*c**2*x**4/4 + B*a**2*x + 2*B*a*c*x**3/3 + B*c**2*x**5/5)/e, Eq(m, -1
)), (A*a**2*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*a**2*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*a**2*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*a**2*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*a**2*e**m*m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 7
35*m**3 + 1624*m**2 + 1764*m + 720) + 720*A*a**2*e**m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2
+ 1764*m + 720) + 2*A*a*c*e**m*m**5*x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 72
0) + 36*A*a*c*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*a*
c*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) + 744*A*a*c*e**m*m**2*
x**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1016*A*a*c*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*a*c*e**m*x**3*x**m/(m**6 + 21*m**5 + 175
*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + A*c**2*e**m*m**5*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**
3 + 1624*m**2 + 1764*m + 720) + 16*A*c**2*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) + 95*A*c**2*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) + 260*A*c**2*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) + 324
*A*c**2*e**m*m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 144*A*c**2*e**m*x
**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + B*a**2*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) + 19*B*a**2*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) + 137*B*a**2*e**m*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) + 461*B*a**2*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) + 702*B*a**2*e**m*m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 17
64*m + 720) + 360*B*a**2*e**m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*
B*a*c*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*B*a*c*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*B*a*c*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*B*a*c*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*B*a*c*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*B*a*c*e**m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 162
4*m**2 + 1764*m + 720) + B*c**2*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*c**2*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*c**2*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*c**2*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*c**2*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*c**2*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 [B] time = 1.27514, size = 791, normalized size = 6.54 \begin{align*} \frac{B c^{2} m^{5} x^{6} x^{m} e^{m} + A c^{2} m^{5} x^{5} x^{m} e^{m} + 15 \, B c^{2} m^{4} x^{6} x^{m} e^{m} + 2 \, B a c m^{5} x^{4} x^{m} e^{m} + 16 \, A c^{2} m^{4} x^{5} x^{m} e^{m} + 85 \, B c^{2} m^{3} x^{6} x^{m} e^{m} + 2 \, A a c m^{5} x^{3} x^{m} e^{m} + 34 \, B a c m^{4} x^{4} x^{m} e^{m} + 95 \, A c^{2} m^{3} x^{5} x^{m} e^{m} + 225 \, B c^{2} m^{2} x^{6} x^{m} e^{m} + B a^{2} m^{5} x^{2} x^{m} e^{m} + 36 \, A a c m^{4} x^{3} x^{m} e^{m} + 214 \, B a c m^{3} x^{4} x^{m} e^{m} + 260 \, A c^{2} m^{2} x^{5} x^{m} e^{m} + 274 \, B c^{2} m x^{6} x^{m} e^{m} + A a^{2} m^{5} x x^{m} e^{m} + 19 \, B a^{2} m^{4} x^{2} x^{m} e^{m} + 242 \, A a c m^{3} x^{3} x^{m} e^{m} + 614 \, B a c m^{2} x^{4} x^{m} e^{m} + 324 \, A c^{2} m x^{5} x^{m} e^{m} + 120 \, B c^{2} x^{6} x^{m} e^{m} + 20 \, A a^{2} m^{4} x x^{m} e^{m} + 137 \, B a^{2} m^{3} x^{2} x^{m} e^{m} + 744 \, A a c m^{2} x^{3} x^{m} e^{m} + 792 \, B a c m x^{4} x^{m} e^{m} + 144 \, A c^{2} x^{5} x^{m} e^{m} + 155 \, A a^{2} m^{3} x x^{m} e^{m} + 461 \, B a^{2} m^{2} x^{2} x^{m} e^{m} + 1016 \, A a c m x^{3} x^{m} e^{m} + 360 \, B a c x^{4} x^{m} e^{m} + 580 \, A a^{2} m^{2} x x^{m} e^{m} + 702 \, B a^{2} m x^{2} x^{m} e^{m} + 480 \, A a c x^{3} x^{m} e^{m} + 1044 \, A a^{2} m x x^{m} e^{m} + 360 \, B a^{2} x^{2} x^{m} e^{m} + 720 \, A a^{2} x x^{m} e^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*(B*x+A)*(c*x^2+a)^2,x, algorithm="giac")
[Out]
(B*c^2*m^5*x^6*x^m*e^m + A*c^2*m^5*x^5*x^m*e^m + 15*B*c^2*m^4*x^6*x^m*e^m + 2*B*a*c*m^5*x^4*x^m*e^m + 16*A*c^2
*m^4*x^5*x^m*e^m + 85*B*c^2*m^3*x^6*x^m*e^m + 2*A*a*c*m^5*x^3*x^m*e^m + 34*B*a*c*m^4*x^4*x^m*e^m + 95*A*c^2*m^
3*x^5*x^m*e^m + 225*B*c^2*m^2*x^6*x^m*e^m + B*a^2*m^5*x^2*x^m*e^m + 36*A*a*c*m^4*x^3*x^m*e^m + 214*B*a*c*m^3*x
^4*x^m*e^m + 260*A*c^2*m^2*x^5*x^m*e^m + 274*B*c^2*m*x^6*x^m*e^m + A*a^2*m^5*x*x^m*e^m + 19*B*a^2*m^4*x^2*x^m*
e^m + 242*A*a*c*m^3*x^3*x^m*e^m + 614*B*a*c*m^2*x^4*x^m*e^m + 324*A*c^2*m*x^5*x^m*e^m + 120*B*c^2*x^6*x^m*e^m
+ 20*A*a^2*m^4*x*x^m*e^m + 137*B*a^2*m^3*x^2*x^m*e^m + 744*A*a*c*m^2*x^3*x^m*e^m + 792*B*a*c*m*x^4*x^m*e^m + 1
44*A*c^2*x^5*x^m*e^m + 155*A*a^2*m^3*x*x^m*e^m + 461*B*a^2*m^2*x^2*x^m*e^m + 1016*A*a*c*m*x^3*x^m*e^m + 360*B*
a*c*x^4*x^m*e^m + 580*A*a^2*m^2*x*x^m*e^m + 702*B*a^2*m*x^2*x^m*e^m + 480*A*a*c*x^3*x^m*e^m + 1044*A*a^2*m*x*x
^m*e^m + 360*B*a^2*x^2*x^m*e^m + 720*A*a^2*x*x^m*e^m)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m +
720)