Integrand size = 20, antiderivative size = 121 \[ \int (e x)^m (A+B x) \left (a+c x^2\right )^2 \, dx=\frac {a^2 A (e x)^{1+m}}{e (1+m)}+\frac {a^2 B (e x)^{2+m}}{e^2 (2+m)}+\frac {2 a A c (e x)^{3+m}}{e^3 (3+m)}+\frac {2 a B c (e x)^{4+m}}{e^4 (4+m)}+\frac {A c^2 (e x)^{5+m}}{e^5 (5+m)}+\frac {B c^2 (e x)^{6+m}}{e^6 (6+m)} \]
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)
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.61 \[ \int (e x)^m (A+B x) \left (a+c x^2\right )^2 \, dx=x (e x)^m \left (a^2 \left (\frac {A}{1+m}+\frac {B x}{2+m}\right )+2 a c x^2 \left (\frac {A}{3+m}+\frac {B x}{4+m}\right )+c^2 x^4 \left (\frac {A}{5+m}+\frac {B x}{6+m}\right )\right ) \]
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)))
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a+c x^2\right )^2 (A+B x) (e x)^m \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (a^2 A (e x)^m+\frac {a^2 B (e x)^{m+1}}{e}+\frac {2 a A c (e x)^{m+2}}{e^2}+\frac {2 a B c (e x)^{m+3}}{e^3}+\frac {A c^2 (e x)^{m+4}}{e^4}+\frac {B c^2 (e x)^{m+5}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 A (e x)^{m+1}}{e (m+1)}+\frac {a^2 B (e x)^{m+2}}{e^2 (m+2)}+\frac {2 a A c (e x)^{m+3}}{e^3 (m+3)}+\frac {2 a B c (e x)^{m+4}}{e^4 (m+4)}+\frac {A c^2 (e x)^{m+5}}{e^5 (m+5)}+\frac {B c^2 (e x)^{m+6}}{e^6 (m+6)}\) |
(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))
3.5.87.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {A \,a^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {A \,c^{2} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {B \,a^{2} x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}+\frac {B \,c^{2} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {2 A a c \,x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {2 B a c \,x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}\) | \(120\) |
| gosper | \(\frac {x \left (B \,c^{2} m^{5} x^{5}+A \,c^{2} m^{5} x^{4}+15 B \,c^{2} m^{4} x^{5}+16 A \,c^{2} m^{4} x^{4}+2 B a c \,m^{5} x^{3}+85 B \,c^{2} m^{3} x^{5}+2 A a c \,m^{5} x^{2}+95 A \,c^{2} m^{3} x^{4}+34 B a c \,m^{4} x^{3}+225 B \,c^{2} m^{2} x^{5}+36 A a c \,m^{4} x^{2}+260 A \,c^{2} m^{2} x^{4}+B \,a^{2} m^{5} x +214 B a c \,m^{3} x^{3}+274 m \,x^{5} B \,c^{2}+A \,a^{2} m^{5}+242 A a c \,m^{3} x^{2}+324 m \,x^{4} A \,c^{2}+19 B \,a^{2} m^{4} x +614 B a c \,m^{2} x^{3}+120 B \,c^{2} x^{5}+20 A \,a^{2} m^{4}+744 A a c \,m^{2} x^{2}+144 A \,c^{2} x^{4}+137 B \,a^{2} m^{3} x +792 a B c \,x^{3} m +155 A \,a^{2} m^{3}+1016 a A c \,x^{2} m +461 B \,a^{2} m^{2} x +360 a B c \,x^{3}+580 A \,a^{2} m^{2}+480 a A c \,x^{2}+702 a^{2} B x m +1044 A \,a^{2} m +360 a^{2} B x +720 A \,a^{2}\right ) \left (e x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(395\) |
| risch | \(\frac {x \left (B \,c^{2} m^{5} x^{5}+A \,c^{2} m^{5} x^{4}+15 B \,c^{2} m^{4} x^{5}+16 A \,c^{2} m^{4} x^{4}+2 B a c \,m^{5} x^{3}+85 B \,c^{2} m^{3} x^{5}+2 A a c \,m^{5} x^{2}+95 A \,c^{2} m^{3} x^{4}+34 B a c \,m^{4} x^{3}+225 B \,c^{2} m^{2} x^{5}+36 A a c \,m^{4} x^{2}+260 A \,c^{2} m^{2} x^{4}+B \,a^{2} m^{5} x +214 B a c \,m^{3} x^{3}+274 m \,x^{5} B \,c^{2}+A \,a^{2} m^{5}+242 A a c \,m^{3} x^{2}+324 m \,x^{4} A \,c^{2}+19 B \,a^{2} m^{4} x +614 B a c \,m^{2} x^{3}+120 B \,c^{2} x^{5}+20 A \,a^{2} m^{4}+744 A a c \,m^{2} x^{2}+144 A \,c^{2} x^{4}+137 B \,a^{2} m^{3} x +792 a B c \,x^{3} m +155 A \,a^{2} m^{3}+1016 a A c \,x^{2} m +461 B \,a^{2} m^{2} x +360 a B c \,x^{3}+580 A \,a^{2} m^{2}+480 a A c \,x^{2}+702 a^{2} B x m +1044 A \,a^{2} m +360 a^{2} B x +720 A \,a^{2}\right ) \left (e x \right )^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(395\) |
| parallelrisch | \(\frac {120 B \,x^{6} \left (e x \right )^{m} c^{2}+144 A \,x^{5} \left (e x \right )^{m} c^{2}+360 B \,x^{2} \left (e x \right )^{m} a^{2}+720 A x \left (e x \right )^{m} a^{2}+36 A \,x^{3} \left (e x \right )^{m} a c \,m^{4}+214 B \,x^{4} \left (e x \right )^{m} a c \,m^{3}+242 A \,x^{3} \left (e x \right )^{m} a c \,m^{3}+614 B \,x^{4} \left (e x \right )^{m} a c \,m^{2}+744 A \,x^{3} \left (e x \right )^{m} a c \,m^{2}+792 B \,x^{4} \left (e x \right )^{m} a c m +1016 A \,x^{3} \left (e x \right )^{m} a c m +2 B \,x^{4} \left (e x \right )^{m} a c \,m^{5}+2 A \,x^{3} \left (e x \right )^{m} a c \,m^{5}+34 B \,x^{4} \left (e x \right )^{m} a c \,m^{4}+B \,x^{6} \left (e x \right )^{m} c^{2} m^{5}+A \,x^{5} \left (e x \right )^{m} c^{2} m^{5}+15 B \,x^{6} \left (e x \right )^{m} c^{2} m^{4}+16 A \,x^{5} \left (e x \right )^{m} c^{2} m^{4}+85 B \,x^{6} \left (e x \right )^{m} c^{2} m^{3}+95 A \,x^{5} \left (e x \right )^{m} c^{2} m^{3}+225 B \,x^{6} \left (e x \right )^{m} c^{2} m^{2}+260 A \,x^{5} \left (e x \right )^{m} c^{2} m^{2}+274 B \,x^{6} \left (e x \right )^{m} c^{2} m +B \,x^{2} \left (e x \right )^{m} a^{2} m^{5}+324 A \,x^{5} \left (e x \right )^{m} c^{2} m +A x \left (e x \right )^{m} a^{2} m^{5}+19 B \,x^{2} \left (e x \right )^{m} a^{2} m^{4}+20 A x \left (e x \right )^{m} a^{2} m^{4}+137 B \,x^{2} \left (e x \right )^{m} a^{2} m^{3}+155 A x \left (e x \right )^{m} a^{2} m^{3}+360 B \,x^{4} \left (e x \right )^{m} a c +461 B \,x^{2} \left (e x \right )^{m} a^{2} m^{2}+480 A \,x^{3} \left (e x \right )^{m} a c +580 A x \left (e x \right )^{m} a^{2} m^{2}+702 B \,x^{2} \left (e x \right )^{m} a^{2} m +1044 A x \left (e x \right )^{m} a^{2} m}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(587\) |
A*a^2/(1+m)*x*exp(m*ln(e*x))+A*c^2/(5+m)*x^5*exp(m*ln(e*x))+B*a^2/(2+m)*x^ 2*exp(m*ln(e*x))+B*c^2/(6+m)*x^6*exp(m*ln(e*x))+2*A*a*c/(3+m)*x^3*exp(m*ln (e*x))+2*B*a*c/(4+m)*x^4*exp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (121) = 242\).
Time = 0.34 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.83 \[ \int (e x)^m (A+B x) \left (a+c x^2\right )^2 \, dx=\frac {{\left ({\left (B c^{2} m^{5} + 15 \, B c^{2} m^{4} + 85 \, B c^{2} m^{3} + 225 \, B c^{2} m^{2} + 274 \, B c^{2} m + 120 \, B c^{2}\right )} x^{6} + {\left (A c^{2} m^{5} + 16 \, A c^{2} m^{4} + 95 \, A c^{2} m^{3} + 260 \, A c^{2} m^{2} + 324 \, A c^{2} m + 144 \, A c^{2}\right )} x^{5} + 2 \, {\left (B a c m^{5} + 17 \, B a c m^{4} + 107 \, B a c m^{3} + 307 \, B a c m^{2} + 396 \, B a c m + 180 \, B a c\right )} x^{4} + 2 \, {\left (A a c m^{5} + 18 \, A a c m^{4} + 121 \, A a c m^{3} + 372 \, A a c m^{2} + 508 \, A a c m + 240 \, A a c\right )} x^{3} + {\left (B a^{2} m^{5} + 19 \, B a^{2} m^{4} + 137 \, B a^{2} m^{3} + 461 \, B a^{2} m^{2} + 702 \, B a^{2} m + 360 \, B a^{2}\right )} x^{2} + {\left (A a^{2} m^{5} + 20 \, A a^{2} m^{4} + 155 \, A a^{2} m^{3} + 580 \, A a^{2} m^{2} + 1044 \, A a^{2} m + 720 \, A a^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
((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 + 1 624*m^2 + 1764*m + 720)
Leaf count of result is larger than twice the leaf count of optimal. 2014 vs. \(2 (112) = 224\).
Time = 0.50 (sec) , antiderivative size = 2014, normalized size of antiderivative = 16.64 \[ \int (e x)^m (A+B x) \left (a+c x^2\right )^2 \, dx=\text {Too large to display} \]
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, E q(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, E q(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*m**5*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 20*A*a**2*m**4*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735 *m**3 + 1624*m**2 + 1764*m + 720) + 155*A*a**2*m**3*x*(e*x)**m/(m**6 + 21* m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*A*a**2*m**2*x *(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720 ) + 1044*A*a**2*m*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624* m**2 + 1764*m + 720) + 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) + 2*A*a*c*m**5*x**3*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 36*A*a*c*m**4* x**3*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 242*A*a*c*m**3*x**3*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m...
Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.96 \[ \int (e x)^m (A+B x) \left (a+c x^2\right )^2 \, dx=\frac {B c^{2} e^{m} x^{6} x^{m}}{m + 6} + \frac {A c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, B a c e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, A a c e^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{2} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} A a^{2}}{e {\left (m + 1\right )}} \]
B*c^2*e^m*x^6*x^m/(m + 6) + A*c^2*e^m*x^5*x^m/(m + 5) + 2*B*a*c*e^m*x^4*x^ m/(m + 4) + 2*A*a*c*e^m*x^3*x^m/(m + 3) + B*a^2*e^m*x^2*x^m/(m + 2) + (e*x )^(m + 1)*A*a^2/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (121) = 242\).
Time = 0.28 (sec) , antiderivative size = 586, normalized size of antiderivative = 4.84 \[ \int (e x)^m (A+B x) \left (a+c x^2\right )^2 \, dx=\frac {\left (e x\right )^{m} B c^{2} m^{5} x^{6} + \left (e x\right )^{m} A c^{2} m^{5} x^{5} + 15 \, \left (e x\right )^{m} B c^{2} m^{4} x^{6} + 2 \, \left (e x\right )^{m} B a c m^{5} x^{4} + 16 \, \left (e x\right )^{m} A c^{2} m^{4} x^{5} + 85 \, \left (e x\right )^{m} B c^{2} m^{3} x^{6} + 2 \, \left (e x\right )^{m} A a c m^{5} x^{3} + 34 \, \left (e x\right )^{m} B a c m^{4} x^{4} + 95 \, \left (e x\right )^{m} A c^{2} m^{3} x^{5} + 225 \, \left (e x\right )^{m} B c^{2} m^{2} x^{6} + \left (e x\right )^{m} B a^{2} m^{5} x^{2} + 36 \, \left (e x\right )^{m} A a c m^{4} x^{3} + 214 \, \left (e x\right )^{m} B a c m^{3} x^{4} + 260 \, \left (e x\right )^{m} A c^{2} m^{2} x^{5} + 274 \, \left (e x\right )^{m} B c^{2} m x^{6} + \left (e x\right )^{m} A a^{2} m^{5} x + 19 \, \left (e x\right )^{m} B a^{2} m^{4} x^{2} + 242 \, \left (e x\right )^{m} A a c m^{3} x^{3} + 614 \, \left (e x\right )^{m} B a c m^{2} x^{4} + 324 \, \left (e x\right )^{m} A c^{2} m x^{5} + 120 \, \left (e x\right )^{m} B c^{2} x^{6} + 20 \, \left (e x\right )^{m} A a^{2} m^{4} x + 137 \, \left (e x\right )^{m} B a^{2} m^{3} x^{2} + 744 \, \left (e x\right )^{m} A a c m^{2} x^{3} + 792 \, \left (e x\right )^{m} B a c m x^{4} + 144 \, \left (e x\right )^{m} A c^{2} x^{5} + 155 \, \left (e x\right )^{m} A a^{2} m^{3} x + 461 \, \left (e x\right )^{m} B a^{2} m^{2} x^{2} + 1016 \, \left (e x\right )^{m} A a c m x^{3} + 360 \, \left (e x\right )^{m} B a c x^{4} + 580 \, \left (e x\right )^{m} A a^{2} m^{2} x + 702 \, \left (e x\right )^{m} B a^{2} m x^{2} + 480 \, \left (e x\right )^{m} A a c x^{3} + 1044 \, \left (e x\right )^{m} A a^{2} m x + 360 \, \left (e x\right )^{m} B a^{2} x^{2} + 720 \, \left (e x\right )^{m} A a^{2} x}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
((e*x)^m*B*c^2*m^5*x^6 + (e*x)^m*A*c^2*m^5*x^5 + 15*(e*x)^m*B*c^2*m^4*x^6 + 2*(e*x)^m*B*a*c*m^5*x^4 + 16*(e*x)^m*A*c^2*m^4*x^5 + 85*(e*x)^m*B*c^2*m^ 3*x^6 + 2*(e*x)^m*A*a*c*m^5*x^3 + 34*(e*x)^m*B*a*c*m^4*x^4 + 95*(e*x)^m*A* c^2*m^3*x^5 + 225*(e*x)^m*B*c^2*m^2*x^6 + (e*x)^m*B*a^2*m^5*x^2 + 36*(e*x) ^m*A*a*c*m^4*x^3 + 214*(e*x)^m*B*a*c*m^3*x^4 + 260*(e*x)^m*A*c^2*m^2*x^5 + 274*(e*x)^m*B*c^2*m*x^6 + (e*x)^m*A*a^2*m^5*x + 19*(e*x)^m*B*a^2*m^4*x^2 + 242*(e*x)^m*A*a*c*m^3*x^3 + 614*(e*x)^m*B*a*c*m^2*x^4 + 324*(e*x)^m*A*c^ 2*m*x^5 + 120*(e*x)^m*B*c^2*x^6 + 20*(e*x)^m*A*a^2*m^4*x + 137*(e*x)^m*B*a ^2*m^3*x^2 + 744*(e*x)^m*A*a*c*m^2*x^3 + 792*(e*x)^m*B*a*c*m*x^4 + 144*(e* x)^m*A*c^2*x^5 + 155*(e*x)^m*A*a^2*m^3*x + 461*(e*x)^m*B*a^2*m^2*x^2 + 101 6*(e*x)^m*A*a*c*m*x^3 + 360*(e*x)^m*B*a*c*x^4 + 580*(e*x)^m*A*a^2*m^2*x + 702*(e*x)^m*B*a^2*m*x^2 + 480*(e*x)^m*A*a*c*x^3 + 1044*(e*x)^m*A*a^2*m*x + 360*(e*x)^m*B*a^2*x^2 + 720*(e*x)^m*A*a^2*x)/(m^6 + 21*m^5 + 175*m^4 + 73 5*m^3 + 1624*m^2 + 1764*m + 720)
Time = 10.14 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.07 \[ \int (e x)^m (A+B x) \left (a+c x^2\right )^2 \, dx={\left (e\,x\right )}^m\,\left (\frac {A\,a^2\,x\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {B\,a^2\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {A\,c^2\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {B\,c^2\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,A\,a\,c\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,B\,a\,c\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \]
(e*x)^m*((A*a^2*x*(1044*m + 580*m^2 + 155*m^3 + 20*m^4 + m^5 + 720))/(1764 *m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (B*a^2*x^2*(702* m + 461*m^2 + 137*m^3 + 19*m^4 + m^5 + 360))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (A*c^2*x^5*(324*m + 260*m^2 + 95*m^3 + 1 6*m^4 + m^5 + 144))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (B*c^2*x^6*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764 *m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (2*A*a*c*x^3*(50 8*m + 372*m^2 + 121*m^3 + 18*m^4 + m^5 + 240))/(1764*m + 1624*m^2 + 735*m^ 3 + 175*m^4 + 21*m^5 + m^6 + 720) + (2*B*a*c*x^4*(396*m + 307*m^2 + 107*m^ 3 + 17*m^4 + m^5 + 180))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))