Integrand size = 21, antiderivative size = 116 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\frac {4 (e x)^{1+m}}{e (1+m)}+\frac {4 a (e x)^{2+m}}{e^2 (2+m)}-\frac {8 a^2 (e x)^{3+m}}{e^3 (3+m)}-\frac {8 a^3 (e x)^{4+m}}{e^4 (4+m)}+\frac {4 a^4 (e x)^{5+m}}{e^5 (5+m)}+\frac {4 a^5 (e x)^{6+m}}{e^6 (6+m)} \]
4*(e*x)^(1+m)/e/(1+m)+4*a*(e*x)^(2+m)/e^2/(2+m)-8*a^2*(e*x)^(3+m)/e^3/(3+m )-8*a^3*(e*x)^(4+m)/e^4/(4+m)+4*a^4*(e*x)^(5+m)/e^5/(5+m)+4*a^5*(e*x)^(6+m )/e^6/(6+m)
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.62 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=4 x (e x)^m \left (\frac {1}{1+m}+\frac {a x}{2+m}-\frac {2 a^2 x^2}{3+m}-\frac {2 a^3 x^3}{4+m}+\frac {a^4 x^4}{5+m}+\frac {a^5 x^5}{6+m}\right ) \]
4*x*(e*x)^m*((1 + m)^(-1) + (a*x)/(2 + m) - (2*a^2*x^2)/(3 + m) - (2*a^3*x ^3)/(4 + m) + (a^4*x^4)/(5 + m) + (a^5*x^5)/(6 + m))
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {99, 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.
| \(\displaystyle \int (2-2 a x)^2 (a x+1)^3 (e x)^m \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {4 a^5 (e x)^{m+5}}{e^5}+\frac {4 a^4 (e x)^{m+4}}{e^4}-\frac {8 a^3 (e x)^{m+3}}{e^3}-\frac {8 a^2 (e x)^{m+2}}{e^2}+\frac {4 a (e x)^{m+1}}{e}+4 (e x)^m\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 a^5 (e x)^{m+6}}{e^6 (m+6)}+\frac {4 a^4 (e x)^{m+5}}{e^5 (m+5)}-\frac {8 a^3 (e x)^{m+4}}{e^4 (m+4)}-\frac {8 a^2 (e x)^{m+3}}{e^3 (m+3)}+\frac {4 a (e x)^{m+2}}{e^2 (m+2)}+\frac {4 (e x)^{m+1}}{e (m+1)}\) |
(4*(e*x)^(1 + m))/(e*(1 + m)) + (4*a*(e*x)^(2 + m))/(e^2*(2 + m)) - (8*a^2 *(e*x)^(3 + m))/(e^3*(3 + m)) - (8*a^3*(e*x)^(4 + m))/(e^4*(4 + m)) + (4*a ^4*(e*x)^(5 + m))/(e^5*(5 + m)) + (4*a^5*(e*x)^(6 + m))/(e^6*(6 + m))
3.1.57.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.41 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {4 x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {4 a \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}-\frac {8 a^{2} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}-\frac {8 a^{3} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {4 a^{4} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {4 a^{5} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}\) | \(115\) |
| gosper | \(\frac {4 \left (e x \right )^{m} \left (a^{5} m^{5} x^{5}+15 a^{5} m^{4} x^{5}+85 a^{5} m^{3} x^{5}+a^{4} m^{5} x^{4}+225 a^{5} m^{2} x^{5}+16 a^{4} m^{4} x^{4}+274 a^{5} x^{5} m +95 a^{4} m^{3} x^{4}-2 a^{3} m^{5} x^{3}+120 a^{5} x^{5}+260 a^{4} m^{2} x^{4}-34 a^{3} m^{4} x^{3}+324 a^{4} x^{4} m -214 a^{3} m^{3} x^{3}-2 a^{2} m^{5} x^{2}+144 a^{4} x^{4}-614 a^{3} m^{2} x^{3}-36 a^{2} m^{4} x^{2}-792 a^{3} x^{3} m -242 a^{2} m^{3} x^{2}+a \,m^{5} x -360 a^{3} x^{3}-744 a^{2} m^{2} x^{2}+19 a \,m^{4} x -1016 a^{2} m \,x^{2}+137 a \,m^{3} x +m^{5}-480 a^{2} x^{2}+461 a \,m^{2} x +20 m^{4}+702 a x m +155 m^{3}+360 a x +580 m^{2}+1044 m +720\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(340\) |
| risch | \(\frac {4 \left (e x \right )^{m} \left (a^{5} m^{5} x^{5}+15 a^{5} m^{4} x^{5}+85 a^{5} m^{3} x^{5}+a^{4} m^{5} x^{4}+225 a^{5} m^{2} x^{5}+16 a^{4} m^{4} x^{4}+274 a^{5} x^{5} m +95 a^{4} m^{3} x^{4}-2 a^{3} m^{5} x^{3}+120 a^{5} x^{5}+260 a^{4} m^{2} x^{4}-34 a^{3} m^{4} x^{3}+324 a^{4} x^{4} m -214 a^{3} m^{3} x^{3}-2 a^{2} m^{5} x^{2}+144 a^{4} x^{4}-614 a^{3} m^{2} x^{3}-36 a^{2} m^{4} x^{2}-792 a^{3} x^{3} m -242 a^{2} m^{3} x^{2}+a \,m^{5} x -360 a^{3} x^{3}-744 a^{2} m^{2} x^{2}+19 a \,m^{4} x -1016 a^{2} m \,x^{2}+137 a \,m^{3} x +m^{5}-480 a^{2} x^{2}+461 a \,m^{2} x +20 m^{4}+702 a x m +155 m^{3}+360 a x +580 m^{2}+1044 m +720\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(340\) |
| parallelrisch | \(\frac {-144 x^{3} \left (e x \right )^{m} a^{2} m^{4}-3168 x^{4} \left (e x \right )^{m} a^{3} m -968 x^{3} \left (e x \right )^{m} a^{2} m^{3}+4 x^{2} \left (e x \right )^{m} a \,m^{5}-2976 x^{3} \left (e x \right )^{m} a^{2} m^{2}+76 x^{2} \left (e x \right )^{m} a \,m^{4}-4064 x^{3} \left (e x \right )^{m} a^{2} m +548 x^{2} \left (e x \right )^{m} a \,m^{3}+1844 x^{2} \left (e x \right )^{m} a \,m^{2}+2808 x^{2} \left (e x \right )^{m} a m +340 x^{6} \left (e x \right )^{m} a^{5} m^{3}+4 x^{5} \left (e x \right )^{m} a^{4} m^{5}+900 x^{6} \left (e x \right )^{m} a^{5} m^{2}+64 x^{5} \left (e x \right )^{m} a^{4} m^{4}+1096 x^{6} \left (e x \right )^{m} a^{5} m +380 x^{5} \left (e x \right )^{m} a^{4} m^{3}-8 x^{4} \left (e x \right )^{m} a^{3} m^{5}+1040 x^{5} \left (e x \right )^{m} a^{4} m^{2}-136 x^{4} \left (e x \right )^{m} a^{3} m^{4}+1296 x^{5} \left (e x \right )^{m} a^{4} m -856 x^{4} \left (e x \right )^{m} a^{3} m^{3}-8 x^{3} \left (e x \right )^{m} a^{2} m^{5}-2456 x^{4} \left (e x \right )^{m} a^{3} m^{2}+4 x^{6} \left (e x \right )^{m} a^{5} m^{5}+60 x^{6} \left (e x \right )^{m} a^{5} m^{4}+480 x^{6} \left (e x \right )^{m} a^{5}+576 x^{5} \left (e x \right )^{m} a^{4}-1440 x^{4} \left (e x \right )^{m} a^{3}+4 x \left (e x \right )^{m} m^{5}-1920 x^{3} \left (e x \right )^{m} a^{2}+80 x \left (e x \right )^{m} m^{4}+620 x \left (e x \right )^{m} m^{3}+1440 x^{2} \left (e x \right )^{m} a +2320 x \left (e x \right )^{m} m^{2}+4176 x \left (e x \right )^{m} m +2880 \left (e x \right )^{m} x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(537\) |
4/(1+m)*x*exp(m*ln(e*x))+4*a/(2+m)*x^2*exp(m*ln(e*x))-8*a^2/(3+m)*x^3*exp( m*ln(e*x))-8*a^3/(4+m)*x^4*exp(m*ln(e*x))+4*a^4/(5+m)*x^5*exp(m*ln(e*x))+4 *a^5/(6+m)*x^6*exp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (116) = 232\).
Time = 0.23 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.48 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\frac {4 \, {\left ({\left (a^{5} m^{5} + 15 \, a^{5} m^{4} + 85 \, a^{5} m^{3} + 225 \, a^{5} m^{2} + 274 \, a^{5} m + 120 \, a^{5}\right )} x^{6} + {\left (a^{4} m^{5} + 16 \, a^{4} m^{4} + 95 \, a^{4} m^{3} + 260 \, a^{4} m^{2} + 324 \, a^{4} m + 144 \, a^{4}\right )} x^{5} - 2 \, {\left (a^{3} m^{5} + 17 \, a^{3} m^{4} + 107 \, a^{3} m^{3} + 307 \, a^{3} m^{2} + 396 \, a^{3} m + 180 \, a^{3}\right )} x^{4} - 2 \, {\left (a^{2} m^{5} + 18 \, a^{2} m^{4} + 121 \, a^{2} m^{3} + 372 \, a^{2} m^{2} + 508 \, a^{2} m + 240 \, a^{2}\right )} x^{3} + {\left (a m^{5} + 19 \, a m^{4} + 137 \, a m^{3} + 461 \, a m^{2} + 702 \, a m + 360 \, a\right )} x^{2} + {\left (m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
4*((a^5*m^5 + 15*a^5*m^4 + 85*a^5*m^3 + 225*a^5*m^2 + 274*a^5*m + 120*a^5) *x^6 + (a^4*m^5 + 16*a^4*m^4 + 95*a^4*m^3 + 260*a^4*m^2 + 324*a^4*m + 144* a^4)*x^5 - 2*(a^3*m^5 + 17*a^3*m^4 + 107*a^3*m^3 + 307*a^3*m^2 + 396*a^3*m + 180*a^3)*x^4 - 2*(a^2*m^5 + 18*a^2*m^4 + 121*a^2*m^3 + 372*a^2*m^2 + 50 8*a^2*m + 240*a^2)*x^3 + (a*m^5 + 19*a*m^4 + 137*a*m^3 + 461*a*m^2 + 702*a *m + 360*a)*x^2 + (m^5 + 20*m^4 + 155*m^3 + 580*m^2 + 1044*m + 720)*x)*(e* x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)
Leaf count of result is larger than twice the leaf count of optimal. 1867 vs. \(2 (104) = 208\).
Time = 0.45 (sec) , antiderivative size = 1867, normalized size of antiderivative = 16.09 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\text {Too large to display} \]
Piecewise(((4*a**5*log(x) - 4*a**4/x + 4*a**3/x**2 + 8*a**2/(3*x**3) - a/x **4 - 4/(5*x**5))/e**6, Eq(m, -6)), ((4*a**5*x + 4*a**4*log(x) + 8*a**3/x + 4*a**2/x**2 - 4*a/(3*x**3) - 1/x**4)/e**5, Eq(m, -5)), ((2*a**5*x**2 + 4 *a**4*x - 8*a**3*log(x) + 8*a**2/x - 2*a/x**2 - 4/(3*x**3))/e**4, Eq(m, -4 )), ((4*a**5*x**3/3 + 2*a**4*x**2 - 8*a**3*x - 8*a**2*log(x) - 4*a/x - 2/x **2)/e**3, Eq(m, -3)), ((a**5*x**4 + 4*a**4*x**3/3 - 4*a**3*x**2 - 8*a**2* x + 4*a*log(x) - 4/x)/e**2, Eq(m, -2)), ((4*a**5*x**5/5 + a**4*x**4 - 8*a* *3*x**3/3 - 4*a**2*x**2 + 4*a*x + 4*log(x))/e, Eq(m, -1)), (4*a**5*m**5*x* *6*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 7 20) + 60*a**5*m**4*x**6*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1 624*m**2 + 1764*m + 720) + 340*a**5*m**3*x**6*(e*x)**m/(m**6 + 21*m**5 + 1 75*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 900*a**5*m**2*x**6*(e*x)* *m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 109 6*a**5*m*x**6*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 480*a**5*x**6*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m **3 + 1624*m**2 + 1764*m + 720) + 4*a**4*m**5*x**5*(e*x)**m/(m**6 + 21*m** 5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 64*a**4*m**4*x**5*(e *x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 380*a**4*m**3*x**5*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624* m**2 + 1764*m + 720) + 1040*a**4*m**2*x**5*(e*x)**m/(m**6 + 21*m**5 + 1...
Time = 0.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\frac {4 \, a^{5} e^{m} x^{6} x^{m}}{m + 6} + \frac {4 \, a^{4} e^{m} x^{5} x^{m}}{m + 5} - \frac {8 \, a^{3} e^{m} x^{4} x^{m}}{m + 4} - \frac {8 \, a^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {4 \, a e^{m} x^{2} x^{m}}{m + 2} + \frac {4 \, \left (e x\right )^{m + 1}}{e {\left (m + 1\right )}} \]
4*a^5*e^m*x^6*x^m/(m + 6) + 4*a^4*e^m*x^5*x^m/(m + 5) - 8*a^3*e^m*x^4*x^m/ (m + 4) - 8*a^2*e^m*x^3*x^m/(m + 3) + 4*a*e^m*x^2*x^m/(m + 2) + 4*(e*x)^(m + 1)/(e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (116) = 232\).
Time = 0.27 (sec) , antiderivative size = 533, normalized size of antiderivative = 4.59 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx=\frac {4 \, {\left (\left (e x\right )^{m} a^{5} m^{5} x^{6} + 15 \, \left (e x\right )^{m} a^{5} m^{4} x^{6} + \left (e x\right )^{m} a^{4} m^{5} x^{5} + 85 \, \left (e x\right )^{m} a^{5} m^{3} x^{6} + 16 \, \left (e x\right )^{m} a^{4} m^{4} x^{5} + 225 \, \left (e x\right )^{m} a^{5} m^{2} x^{6} - 2 \, \left (e x\right )^{m} a^{3} m^{5} x^{4} + 95 \, \left (e x\right )^{m} a^{4} m^{3} x^{5} + 274 \, \left (e x\right )^{m} a^{5} m x^{6} - 34 \, \left (e x\right )^{m} a^{3} m^{4} x^{4} + 260 \, \left (e x\right )^{m} a^{4} m^{2} x^{5} + 120 \, \left (e x\right )^{m} a^{5} x^{6} - 2 \, \left (e x\right )^{m} a^{2} m^{5} x^{3} - 214 \, \left (e x\right )^{m} a^{3} m^{3} x^{4} + 324 \, \left (e x\right )^{m} a^{4} m x^{5} - 36 \, \left (e x\right )^{m} a^{2} m^{4} x^{3} - 614 \, \left (e x\right )^{m} a^{3} m^{2} x^{4} + 144 \, \left (e x\right )^{m} a^{4} x^{5} + \left (e x\right )^{m} a m^{5} x^{2} - 242 \, \left (e x\right )^{m} a^{2} m^{3} x^{3} - 792 \, \left (e x\right )^{m} a^{3} m x^{4} + 19 \, \left (e x\right )^{m} a m^{4} x^{2} - 744 \, \left (e x\right )^{m} a^{2} m^{2} x^{3} - 360 \, \left (e x\right )^{m} a^{3} x^{4} + \left (e x\right )^{m} m^{5} x + 137 \, \left (e x\right )^{m} a m^{3} x^{2} - 1016 \, \left (e x\right )^{m} a^{2} m x^{3} + 20 \, \left (e x\right )^{m} m^{4} x + 461 \, \left (e x\right )^{m} a m^{2} x^{2} - 480 \, \left (e x\right )^{m} a^{2} x^{3} + 155 \, \left (e x\right )^{m} m^{3} x + 702 \, \left (e x\right )^{m} a m x^{2} + 580 \, \left (e x\right )^{m} m^{2} x + 360 \, \left (e x\right )^{m} a x^{2} + 1044 \, \left (e x\right )^{m} m x + 720 \, \left (e x\right )^{m} x\right )}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]
4*((e*x)^m*a^5*m^5*x^6 + 15*(e*x)^m*a^5*m^4*x^6 + (e*x)^m*a^4*m^5*x^5 + 85 *(e*x)^m*a^5*m^3*x^6 + 16*(e*x)^m*a^4*m^4*x^5 + 225*(e*x)^m*a^5*m^2*x^6 - 2*(e*x)^m*a^3*m^5*x^4 + 95*(e*x)^m*a^4*m^3*x^5 + 274*(e*x)^m*a^5*m*x^6 - 3 4*(e*x)^m*a^3*m^4*x^4 + 260*(e*x)^m*a^4*m^2*x^5 + 120*(e*x)^m*a^5*x^6 - 2* (e*x)^m*a^2*m^5*x^3 - 214*(e*x)^m*a^3*m^3*x^4 + 324*(e*x)^m*a^4*m*x^5 - 36 *(e*x)^m*a^2*m^4*x^3 - 614*(e*x)^m*a^3*m^2*x^4 + 144*(e*x)^m*a^4*x^5 + (e* x)^m*a*m^5*x^2 - 242*(e*x)^m*a^2*m^3*x^3 - 792*(e*x)^m*a^3*m*x^4 + 19*(e*x )^m*a*m^4*x^2 - 744*(e*x)^m*a^2*m^2*x^3 - 360*(e*x)^m*a^3*x^4 + (e*x)^m*m^ 5*x + 137*(e*x)^m*a*m^3*x^2 - 1016*(e*x)^m*a^2*m*x^3 + 20*(e*x)^m*m^4*x + 461*(e*x)^m*a*m^2*x^2 - 480*(e*x)^m*a^2*x^3 + 155*(e*x)^m*m^3*x + 702*(e*x )^m*a*m*x^2 + 580*(e*x)^m*m^2*x + 360*(e*x)^m*a*x^2 + 1044*(e*x)^m*m*x + 7 20*(e*x)^m*x)/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)
Time = 0.70 (sec) , antiderivative size = 367, normalized size of antiderivative = 3.16 \[ \int (e x)^m (2-2 a x)^2 (1+a x)^3 \, dx={\left (e\,x\right )}^m\,\left (\frac {x\,\left (4\,m^5+80\,m^4+620\,m^3+2320\,m^2+4176\,m+2880\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {4\,a\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {4\,a^5\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {4\,a^4\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {8\,a^3\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {8\,a^2\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \]
(e*x)^m*((x*(4176*m + 2320*m^2 + 620*m^3 + 80*m^4 + 4*m^5 + 2880))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (4*a*x^2*(702*m + 461*m^2 + 137*m^3 + 19*m^4 + m^5 + 360))/(1764*m + 1624*m^2 + 735*m^3 + 17 5*m^4 + 21*m^5 + m^6 + 720) + (4*a^5*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 + 72 0) + (4*a^4*x^5*(324*m + 260*m^2 + 95*m^3 + 16*m^4 + m^5 + 144))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) - (8*a^3*x^4*(396*m + 307*m^2 + 107*m^3 + 17*m^4 + m^5 + 180))/(1764*m + 1624*m^2 + 735*m^3 + 17 5*m^4 + 21*m^5 + m^6 + 720) - (8*a^2*x^3*(508*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 + 7 20))