Integrand size = 22, antiderivative size = 182 \[ \int (d x)^m \left (a+b x^n+c x^{2 n}\right )^3 \, dx=\frac {3 a^2 b x^{1+n} (d x)^m}{1+m+n}+\frac {3 a \left (b^2+a c\right ) x^{1+2 n} (d x)^m}{1+m+2 n}+\frac {b \left (b^2+6 a c\right ) x^{1+3 n} (d x)^m}{1+m+3 n}+\frac {3 c \left (b^2+a c\right ) x^{1+4 n} (d x)^m}{1+m+4 n}+\frac {3 b c^2 x^{1+5 n} (d x)^m}{1+m+5 n}+\frac {c^3 x^{1+6 n} (d x)^m}{1+m+6 n}+\frac {a^3 (d x)^{1+m}}{d (1+m)} \]
3*a^2*b*x^(1+n)*(d*x)^m/(1+m+n)+3*a*(a*c+b^2)*x^(1+2*n)*(d*x)^m/(1+m+2*n)+ b*(6*a*c+b^2)*x^(1+3*n)*(d*x)^m/(1+m+3*n)+3*c*(a*c+b^2)*x^(1+4*n)*(d*x)^m/ (1+m+4*n)+3*b*c^2*x^(1+5*n)*(d*x)^m/(1+m+5*n)+c^3*x^(1+6*n)*(d*x)^m/(1+m+6 *n)+a^3*(d*x)^(1+m)/d/(1+m)
Time = 0.56 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.75 \[ \int (d x)^m \left (a+b x^n+c x^{2 n}\right )^3 \, dx=x (d x)^m \left (\frac {a^3}{1+m}+\frac {3 a^2 b x^n}{1+m+n}+\frac {3 a \left (b^2+a c\right ) x^{2 n}}{1+m+2 n}+\frac {b \left (b^2+6 a c\right ) x^{3 n}}{1+m+3 n}+\frac {3 c \left (b^2+a c\right ) x^{4 n}}{1+m+4 n}+\frac {3 b c^2 x^{5 n}}{1+m+5 n}+\frac {c^3 x^{6 n}}{1+m+6 n}\right ) \]
x*(d*x)^m*(a^3/(1 + m) + (3*a^2*b*x^n)/(1 + m + n) + (3*a*(b^2 + a*c)*x^(2 *n))/(1 + m + 2*n) + (b*(b^2 + 6*a*c)*x^(3*n))/(1 + m + 3*n) + (3*c*(b^2 + a*c)*x^(4*n))/(1 + m + 4*n) + (3*b*c^2*x^(5*n))/(1 + m + 5*n) + (c^3*x^(6 *n))/(1 + m + 6*n))
Time = 0.38 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1691, 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.
\(\displaystyle \int (d x)^m \left (a+b x^n+c x^{2 n}\right )^3 \, dx\) |
\(\Big \downarrow \) 1691 |
\(\displaystyle \int \left (a^3 (d x)^m+3 a^2 b x^n (d x)^m+3 a b^2 x^{2 n} \left (\frac {a c}{b^2}+1\right ) (d x)^m+3 b^2 c x^{4 n} \left (\frac {a c}{b^2}+1\right ) (d x)^m+b^3 x^{3 n} \left (\frac {6 a c}{b^2}+1\right ) (d x)^m+3 b c^2 x^{5 n} (d x)^m+c^3 x^{6 n} (d x)^m\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 (d x)^{m+1}}{d (m+1)}+\frac {3 a^2 b x^{n+1} (d x)^m}{m+n+1}+\frac {3 a x^{2 n+1} \left (a c+b^2\right ) (d x)^m}{m+2 n+1}+\frac {b x^{3 n+1} \left (6 a c+b^2\right ) (d x)^m}{m+3 n+1}+\frac {3 c x^{4 n+1} \left (a c+b^2\right ) (d x)^m}{m+4 n+1}+\frac {3 b c^2 x^{5 n+1} (d x)^m}{m+5 n+1}+\frac {c^3 x^{6 n+1} (d x)^m}{m+6 n+1}\) |
(3*a^2*b*x^(1 + n)*(d*x)^m)/(1 + m + n) + (3*a*(b^2 + a*c)*x^(1 + 2*n)*(d* x)^m)/(1 + m + 2*n) + (b*(b^2 + 6*a*c)*x^(1 + 3*n)*(d*x)^m)/(1 + m + 3*n) + (3*c*(b^2 + a*c)*x^(1 + 4*n)*(d*x)^m)/(1 + m + 4*n) + (3*b*c^2*x^(1 + 5* n)*(d*x)^m)/(1 + m + 5*n) + (c^3*x^(1 + 6*n)*(d*x)^m)/(1 + m + 6*n) + (a^3 *(d*x)^(1 + m))/(d*(1 + m))
3.6.96.3.1 Defintions of rubi rules used
Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[n2, 2*n] && IGtQ[p, 0] && !IntegerQ [Simplify[(m + 1)/n]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.56 (sec) , antiderivative size = 3765, normalized size of antiderivative = 20.69
method | result | size |
risch | \(\text {Expression too large to display}\) | \(3765\) |
parallelrisch | \(\text {Expression too large to display}\) | \(5804\) |
x*(3*a*c^2*m^6*(x^n)^4+1284*b^2*c*m*n^2*(x^n)^4+240*b*c^2*m*n*(x^n)^5+300* a^2*b*m^4*n*x^n+1860*a^2*b*m^3*n^2*x^n+5220*a^2*b*m^2*n^3*x^n+6264*a^2*b*m *n^4*x^n+570*a^2*c*m^3*n*(x^n)^2+2466*a^2*c*m^2*n^2*(x^n)^2+4149*a^2*c*m*n ^3*(x^n)^2+570*a*b^2*m^3*n*(x^n)^2+2466*a*b^2*m^2*n^2*(x^n)^2+4149*a*b^2*m *n^3*(x^n)^2+120*a*b*c*m^3*(x^n)^3+2904*a*b*c*m*n^2*(x^n)^3+540*a*b*c*m*n* (x^n)^3+6*(x^n)^3*c*a*b+3*b^2*c*m^6*(x^n)^4+a^3+3*(x^n)^4*c^2*a+3*(x^n)^4* c*b^2+3*(x^n)^2*c*a^2+3*(x^n)^5*c^2*b+15*c^3*m^5*n*(x^n)^6+1383*a^2*c*m^3* n^3*(x^n)^2+2106*a^2*c*m^2*n^4*(x^n)^2+1080*a^2*c*m*n^5*(x^n)^2+57*a*b^2*m ^5*n*(x^n)^2+411*a*b^2*m^4*n^2*(x^n)^2+1383*a*b^2*m^3*n^3*(x^n)^2+2106*a*b ^2*m^2*n^4*(x^n)^2+1080*a*b^2*m*n^5*(x^n)^2+36*a*b*c*m^5*(x^n)^3+1440*a*b* c*n^5*(x^n)^3+2106*a^2*c*n^4*(x^n)^2+45*a*b^2*m^4*(x^n)^2+2106*a*b^2*n^4*( x^n)^2+45*a*c^2*m^2*(x^n)^4+321*a*c^2*n^2*(x^n)^4+540*a*c^2*m*n^5*(x^n)^4+ 51*b^2*c*m^5*n*(x^n)^4+321*b^2*c*m^4*n^2*(x^n)^4+921*b^2*c*m^3*n^3*(x^n)^4 +1188*b^2*c*m^2*n^4*(x^n)^4+180*b^3*m^3*n*(x^n)^3+726*b^3*m^2*n^2*(x^n)^3+ 1116*b^3*m*n^3*(x^n)^3+60*b^2*c*m^3*(x^n)^4+921*b^2*c*n^3*(x^n)^4+6*b^3*m^ 5*(x^n)^3+240*b^3*n^5*(x^n)^3+15*c^3*m^2*(x^n)^6+85*c^3*n^2*(x^n)^6+15*b^3 *m^4*(x^n)^3+508*b^3*n^4*(x^n)^3+675*c^3*m*n^3*(x^n)^6+18*a*c^2*m^5*(x^n)^ 4+540*a*c^2*n^5*(x^n)^4+6096*a*b*c*m*n^4*(x^n)^3+1080*a*b*c*m^3*n*(x^n)^3+ 4356*a*b*c*m^2*n^2*(x^n)^3+85*c^3*m^4*n^2*(x^n)^6+2763*b^2*c*m^2*n^3*(x^n) ^4+2376*b^2*c*m*n^4*(x^n)^4+480*b*c^2*m^3*n*(x^n)^5+1710*b*c^2*m^2*n^2*...
Leaf count of result is larger than twice the leaf count of optimal. 2303 vs. \(2 (182) = 364\).
Time = 0.33 (sec) , antiderivative size = 2303, normalized size of antiderivative = 12.65 \[ \int (d x)^m \left (a+b x^n+c x^{2 n}\right )^3 \, dx=\text {Too large to display} \]
((c^3*m^6 + 6*c^3*m^5 + 15*c^3*m^4 + 20*c^3*m^3 + 120*(c^3*m + c^3)*n^5 + 15*c^3*m^2 + 274*(c^3*m^2 + 2*c^3*m + c^3)*n^4 + 6*c^3*m + 225*(c^3*m^3 + 3*c^3*m^2 + 3*c^3*m + c^3)*n^3 + c^3 + 85*(c^3*m^4 + 4*c^3*m^3 + 6*c^3*m^2 + 4*c^3*m + c^3)*n^2 + 15*(c^3*m^5 + 5*c^3*m^4 + 10*c^3*m^3 + 10*c^3*m^2 + 5*c^3*m + c^3)*n)*x*x^(6*n)*e^(m*log(d) + m*log(x)) + 3*(b*c^2*m^6 + 6*b *c^2*m^5 + 15*b*c^2*m^4 + 20*b*c^2*m^3 + 144*(b*c^2*m + b*c^2)*n^5 + 15*b* c^2*m^2 + 324*(b*c^2*m^2 + 2*b*c^2*m + b*c^2)*n^4 + 6*b*c^2*m + 260*(b*c^2 *m^3 + 3*b*c^2*m^2 + 3*b*c^2*m + b*c^2)*n^3 + b*c^2 + 95*(b*c^2*m^4 + 4*b* c^2*m^3 + 6*b*c^2*m^2 + 4*b*c^2*m + b*c^2)*n^2 + 16*(b*c^2*m^5 + 5*b*c^2*m ^4 + 10*b*c^2*m^3 + 10*b*c^2*m^2 + 5*b*c^2*m + b*c^2)*n)*x*x^(5*n)*e^(m*lo g(d) + m*log(x)) + 3*((b^2*c + a*c^2)*m^6 + 6*(b^2*c + a*c^2)*m^5 + 180*(b ^2*c + a*c^2 + (b^2*c + a*c^2)*m)*n^5 + 15*(b^2*c + a*c^2)*m^4 + 396*(b^2* c + a*c^2 + (b^2*c + a*c^2)*m^2 + 2*(b^2*c + a*c^2)*m)*n^4 + 20*(b^2*c + a *c^2)*m^3 + 307*((b^2*c + a*c^2)*m^3 + b^2*c + a*c^2 + 3*(b^2*c + a*c^2)*m ^2 + 3*(b^2*c + a*c^2)*m)*n^3 + b^2*c + a*c^2 + 15*(b^2*c + a*c^2)*m^2 + 1 07*((b^2*c + a*c^2)*m^4 + 4*(b^2*c + a*c^2)*m^3 + b^2*c + a*c^2 + 6*(b^2*c + a*c^2)*m^2 + 4*(b^2*c + a*c^2)*m)*n^2 + 6*(b^2*c + a*c^2)*m + 17*((b^2* c + a*c^2)*m^5 + 5*(b^2*c + a*c^2)*m^4 + 10*(b^2*c + a*c^2)*m^3 + b^2*c + a*c^2 + 10*(b^2*c + a*c^2)*m^2 + 5*(b^2*c + a*c^2)*m)*n)*x*x^(4*n)*e^(m*lo g(d) + m*log(x)) + ((b^3 + 6*a*b*c)*m^6 + 6*(b^3 + 6*a*b*c)*m^5 + 240*(...
Leaf count of result is larger than twice the leaf count of optimal. 68491 vs. \(2 (170) = 340\).
Time = 65.90 (sec) , antiderivative size = 68491, normalized size of antiderivative = 376.32 \[ \int (d x)^m \left (a+b x^n+c x^{2 n}\right )^3 \, dx=\text {Too large to display} \]
Piecewise(((a + b + c)**3*log(x)/d, Eq(m, -1) & Eq(n, 0)), ((a**3*log(x) + 3*a**2*b*x**n/n + 3*a**2*c*x**(2*n)/(2*n) + 3*a*b**2*x**(2*n)/(2*n) + 2*a *b*c*x**(3*n)/n + 3*a*c**2*x**(4*n)/(4*n) + b**3*x**(3*n)/(3*n) + 3*b**2*c *x**(4*n)/(4*n) + 3*b*c**2*x**(5*n)/(5*n) + c**3*x**(6*n)/(6*n))/d, Eq(m, -1)), (a**3*Piecewise((0**(-6*n - 1)*x, Eq(d, 0)), (Piecewise((-1/(6*n*(d* x)**(6*n)), Ne(n, 0)), (log(d*x), True))/d, True)) + 3*a**2*b*Piecewise((- x*x**n*(d*x)**(-6*n - 1)/(5*n), Ne(n, 0)), (x*x**n*(d*x)**(-6*n - 1)*log(x ), True)) + 3*a**2*c*Piecewise((-x*x**(2*n)*(d*x)**(-6*n - 1)/(4*n), Ne(n, 0)), (x*x**(2*n)*(d*x)**(-6*n - 1)*log(x), True)) + 3*a*b**2*Piecewise((- x*x**(2*n)*(d*x)**(-6*n - 1)/(4*n), Ne(n, 0)), (x*x**(2*n)*(d*x)**(-6*n - 1)*log(x), True)) + 6*a*b*c*Piecewise((-x*x**(3*n)*(d*x)**(-6*n - 1)/(3*n) , Ne(n, 0)), (x*x**(3*n)*(d*x)**(-6*n - 1)*log(x), True)) + 3*a*c**2*Piece wise((-x*x**(4*n)*(d*x)**(-6*n - 1)/(2*n), Ne(n, 0)), (x*x**(4*n)*(d*x)**( -6*n - 1)*log(x), True)) + b**3*Piecewise((-x*x**(3*n)*(d*x)**(-6*n - 1)/( 3*n), Ne(n, 0)), (x*x**(3*n)*(d*x)**(-6*n - 1)*log(x), True)) + 3*b**2*c*P iecewise((-x*x**(4*n)*(d*x)**(-6*n - 1)/(2*n), Ne(n, 0)), (x*x**(4*n)*(d*x )**(-6*n - 1)*log(x), True)) + 3*b*c**2*Piecewise((-x*x**(5*n)*(d*x)**(-6* n - 1)/n, Ne(n, 0)), (x*x**(5*n)*(d*x)**(-6*n - 1)*log(x), True)) + c**3*x *x**(6*n)*(d*x)**(-6*n - 1)*log(x), Eq(m, -6*n - 1)), (a**3*Piecewise((0** (-5*n - 1)*x, Eq(d, 0)), (Piecewise((-1/(5*n*(d*x)**(5*n)), Ne(n, 0)), ...
Time = 0.26 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.50 \[ \int (d x)^m \left (a+b x^n+c x^{2 n}\right )^3 \, dx=\frac {c^{3} d^{m} x e^{\left (m \log \left (x\right ) + 6 \, n \log \left (x\right )\right )}}{m + 6 \, n + 1} + \frac {3 \, b c^{2} d^{m} x e^{\left (m \log \left (x\right ) + 5 \, n \log \left (x\right )\right )}}{m + 5 \, n + 1} + \frac {3 \, b^{2} c d^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {3 \, a c^{2} d^{m} x e^{\left (m \log \left (x\right ) + 4 \, n \log \left (x\right )\right )}}{m + 4 \, n + 1} + \frac {b^{3} d^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {6 \, a b c d^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )}}{m + 3 \, n + 1} + \frac {3 \, a b^{2} d^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {3 \, a^{2} c d^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )}}{m + 2 \, n + 1} + \frac {3 \, a^{2} b d^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m + n + 1} + \frac {\left (d x\right )^{m + 1} a^{3}}{d {\left (m + 1\right )}} \]
c^3*d^m*x*e^(m*log(x) + 6*n*log(x))/(m + 6*n + 1) + 3*b*c^2*d^m*x*e^(m*log (x) + 5*n*log(x))/(m + 5*n + 1) + 3*b^2*c*d^m*x*e^(m*log(x) + 4*n*log(x))/ (m + 4*n + 1) + 3*a*c^2*d^m*x*e^(m*log(x) + 4*n*log(x))/(m + 4*n + 1) + b^ 3*d^m*x*e^(m*log(x) + 3*n*log(x))/(m + 3*n + 1) + 6*a*b*c*d^m*x*e^(m*log(x ) + 3*n*log(x))/(m + 3*n + 1) + 3*a*b^2*d^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*a^2*c*d^m*x*e^(m*log(x) + 2*n*log(x))/(m + 2*n + 1) + 3*a^ 2*b*d^m*x*e^(m*log(x) + n*log(x))/(m + n + 1) + (d*x)^(m + 1)*a^3/(d*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 25656 vs. \(2 (182) = 364\).
Time = 0.49 (sec) , antiderivative size = 25656, normalized size of antiderivative = 140.97 \[ \int (d x)^m \left (a+b x^n+c x^{2 n}\right )^3 \, dx=\text {Too large to display} \]
(c^3*m^6*x*x^(6*n)*e^(m*log(d) + m*log(x)) + 15*c^3*m^5*n*x*x^(6*n)*e^(m*l og(d) + m*log(x)) + 85*c^3*m^4*n^2*x*x^(6*n)*e^(m*log(d) + m*log(x)) + 225 *c^3*m^3*n^3*x*x^(6*n)*e^(m*log(d) + m*log(x)) + 274*c^3*m^2*n^4*x*x^(6*n) *e^(m*log(d) + m*log(x)) + 120*c^3*m*n^5*x*x^(6*n)*e^(m*log(d) + m*log(x)) + 3*b*c^2*m^6*x*x^(5*n)*e^(m*log(d) + m*log(x)) + c^3*m^6*x*x^(5*n)*e^(m* log(d) + m*log(x)) + 48*b*c^2*m^5*n*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 15 *c^3*m^5*n*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 285*b*c^2*m^4*n^2*x*x^(5*n) *e^(m*log(d) + m*log(x)) + 85*c^3*m^4*n^2*x*x^(5*n)*e^(m*log(d) + m*log(x) ) + 780*b*c^2*m^3*n^3*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 225*c^3*m^3*n^3* x*x^(5*n)*e^(m*log(d) + m*log(x)) + 972*b*c^2*m^2*n^4*x*x^(5*n)*e^(m*log(d ) + m*log(x)) + 274*c^3*m^2*n^4*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 432*b* c^2*m*n^5*x*x^(5*n)*e^(m*log(d) + m*log(x)) + 120*c^3*m*n^5*x*x^(5*n)*e^(m *log(d) + m*log(x)) + 3*b^2*c*m^6*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 3*a* c^2*m^6*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 3*b*c^2*m^6*x*x^(4*n)*e^(m*log (d) + m*log(x)) + c^3*m^6*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 51*b^2*c*m^5 *n*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 51*a*c^2*m^5*n*x*x^(4*n)*e^(m*log(d ) + m*log(x)) + 48*b*c^2*m^5*n*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 15*c^3* m^5*n*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 321*b^2*c*m^4*n^2*x*x^(4*n)*e^(m *log(d) + m*log(x)) + 321*a*c^2*m^4*n^2*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 285*b*c^2*m^4*n^2*x*x^(4*n)*e^(m*log(d) + m*log(x)) + 85*c^3*m^4*n^2*...
Time = 9.43 (sec) , antiderivative size = 1734, normalized size of antiderivative = 9.53 \[ \int (d x)^m \left (a+b x^n+c x^{2 n}\right )^3 \, dx=\text {Too large to display} \]
(a^3*x*(d*x)^m)/(m + 1) + (c^3*x*x^(6*n)*(d*x)^m*(5*m + 15*n + 60*m*n + 25 5*m*n^2 + 90*m^2*n + 450*m*n^3 + 60*m^3*n + 274*m*n^4 + 15*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 85*n^2 + 225*n^3 + 274*n^4 + 120*n^5 + 255*m^2*n^ 2 + 225*m^2*n^3 + 85*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210 *m^2*n + 2205*m*n^3 + 210*m^3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21 *m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m^5 + m^6 + 175*n^2 + 735*n^3 + 1624 *n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^2 + 2205*m^2*n^3 + 700*m^3*n^2 + 16 24*m^2*n^4 + 735*m^3*n^3 + 175*m^4*n^2 + 1) + (3*a*x*x^(2*n)*(d*x)^m*(a*c + b^2)*(5*m + 19*n + 76*m*n + 411*m*n^2 + 114*m^2*n + 922*m*n^3 + 76*m^3*n + 702*m*n^4 + 19*m^4*n + 10*m^2 + 10*m^3 + 5*m^4 + m^5 + 137*n^2 + 461*n^ 3 + 702*n^4 + 360*n^5 + 411*m^2*n^2 + 461*m^2*n^3 + 137*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2*n + 2205*m*n^3 + 210*m^3*n + 3248* m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*m^5*n + 15*m^2 + 20*m^3 + 15*m^4 + 6*m ^5 + m^6 + 175*n^2 + 735*n^3 + 1624*n^4 + 1764*n^5 + 720*n^6 + 1050*m^2*n^ 2 + 2205*m^2*n^3 + 700*m^3*n^2 + 1624*m^2*n^4 + 735*m^3*n^3 + 175*m^4*n^2 + 1) + (b*x*x^(3*n)*(d*x)^m*(6*a*c + b^2)*(5*m + 18*n + 72*m*n + 363*m*n^2 + 108*m^2*n + 744*m*n^3 + 72*m^3*n + 508*m*n^4 + 18*m^4*n + 10*m^2 + 10*m ^3 + 5*m^4 + m^5 + 121*n^2 + 372*n^3 + 508*n^4 + 240*n^5 + 363*m^2*n^2 + 3 72*m^2*n^3 + 121*m^3*n^2 + 1))/(6*m + 21*n + 105*m*n + 700*m*n^2 + 210*m^2 *n + 2205*m*n^3 + 210*m^3*n + 3248*m*n^4 + 105*m^4*n + 1764*m*n^5 + 21*...