Integrand size = 22, antiderivative size = 91 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^2 \, dx=\frac {a^2 c (e x)^{3+m}}{e^3 (3+m)}+\frac {a (2 b c+a d) (e x)^{4+m}}{e^4 (4+m)}+\frac {b (b c+2 a d) (e x)^{5+m}}{e^5 (5+m)}+\frac {b^2 d (e x)^{6+m}}{e^6 (6+m)} \] Output:
a^2*c*(e*x)^(3+m)/e^3/(3+m)+a*(a*d+2*b*c)*(e*x)^(4+m)/e^4/(4+m)+b*(2*a*d+b *c)*(e*x)^(5+m)/e^5/(5+m)+b^2*d*(e*x)^(6+m)/e^6/(6+m)
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.81 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^2 \, dx=\frac {x^3 (e x)^m \left (d (a+b x)^3+(-a d (3+m)+b c (6+m)) \left (\frac {a^2}{3+m}+\frac {2 a b x}{4+m}+\frac {b^2 x^2}{5+m}\right )\right )}{b (6+m)} \] Input:
Integrate[(e*x)^m*(c + d*x)*(a*x + b*x^2)^2,x]
Output:
(x^3*(e*x)^m*(d*(a + b*x)^3 + (-(a*d*(3 + m)) + b*c*(6 + m))*(a^2/(3 + m) + (2*a*b*x)/(4 + m) + (b^2*x^2)/(5 + m))))/(b*(6 + m))
Time = 0.41 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {9, 85, 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.
| \(\displaystyle \int \left (a x+b x^2\right )^2 (c+d x) (e x)^m \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \frac {\int (e x)^{m+2} (a+b x)^2 (c+d x)dx}{e^2}\) |
| \(\Big \downarrow \) 85 |
| \(\displaystyle \frac {\int \left (a^2 c (e x)^{m+2}+\frac {a (2 b c+a d) (e x)^{m+3}}{e}+\frac {b (b c+2 a d) (e x)^{m+4}}{e^2}+\frac {b^2 d (e x)^{m+5}}{e^3}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^2 c (e x)^{m+3}}{e (m+3)}+\frac {b (e x)^{m+5} (2 a d+b c)}{e^3 (m+5)}+\frac {a (e x)^{m+4} (a d+2 b c)}{e^2 (m+4)}+\frac {b^2 d (e x)^{m+6}}{e^4 (m+6)}}{e^2}\) |
Input:
Int[(e*x)^m*(c + d*x)*(a*x + b*x^2)^2,x]
Output:
((a^2*c*(e*x)^(3 + m))/(e*(3 + m)) + (a*(2*b*c + a*d)*(e*x)^(4 + m))/(e^2* (4 + m)) + (b*(b*c + 2*a*d)*(e*x)^(5 + m))/(e^3*(5 + m)) + (b^2*d*(e*x)^(6 + m))/(e^4*(6 + m)))/e^2
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Time = 0.99 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01
| method | result | size |
| norman | \(\frac {a \left (a d +2 b c \right ) x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {a^{2} c \,x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {b \left (2 a d +b c \right ) x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {b^{2} d \,x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}\) | \(92\) |
| gosper | \(\frac {\left (e x \right )^{m} \left (b^{2} d \,m^{3} x^{3}+2 a b d \,m^{3} x^{2}+b^{2} c \,m^{3} x^{2}+12 b^{2} d \,m^{2} x^{3}+a^{2} d \,m^{3} x +2 a b c \,m^{3} x +26 a b d \,m^{2} x^{2}+13 b^{2} c \,m^{2} x^{2}+47 m \,x^{3} b^{2} d +a^{2} c \,m^{3}+14 a^{2} d \,m^{2} x +28 a b c \,m^{2} x +108 a b d m \,x^{2}+54 b^{2} c m \,x^{2}+60 b^{2} d \,x^{3}+15 a^{2} c \,m^{2}+63 a^{2} d m x +126 a b c m x +144 a b d \,x^{2}+72 b^{2} c \,x^{2}+74 a^{2} c m +90 a^{2} d x +180 a b c x +120 a^{2} c \right ) x^{3}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right )}\) | \(249\) |
| risch | \(\frac {\left (e x \right )^{m} \left (b^{2} d \,m^{3} x^{3}+2 a b d \,m^{3} x^{2}+b^{2} c \,m^{3} x^{2}+12 b^{2} d \,m^{2} x^{3}+a^{2} d \,m^{3} x +2 a b c \,m^{3} x +26 a b d \,m^{2} x^{2}+13 b^{2} c \,m^{2} x^{2}+47 m \,x^{3} b^{2} d +a^{2} c \,m^{3}+14 a^{2} d \,m^{2} x +28 a b c \,m^{2} x +108 a b d m \,x^{2}+54 b^{2} c m \,x^{2}+60 b^{2} d \,x^{3}+15 a^{2} c \,m^{2}+63 a^{2} d m x +126 a b c m x +144 a b d \,x^{2}+72 b^{2} c \,x^{2}+74 a^{2} c m +90 a^{2} d x +180 a b c x +120 a^{2} c \right ) x^{3}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right )}\) | \(249\) |
| orering | \(\frac {\left (b^{2} d \,m^{3} x^{3}+2 a b d \,m^{3} x^{2}+b^{2} c \,m^{3} x^{2}+12 b^{2} d \,m^{2} x^{3}+a^{2} d \,m^{3} x +2 a b c \,m^{3} x +26 a b d \,m^{2} x^{2}+13 b^{2} c \,m^{2} x^{2}+47 m \,x^{3} b^{2} d +a^{2} c \,m^{3}+14 a^{2} d \,m^{2} x +28 a b c \,m^{2} x +108 a b d m \,x^{2}+54 b^{2} c m \,x^{2}+60 b^{2} d \,x^{3}+15 a^{2} c \,m^{2}+63 a^{2} d m x +126 a b c m x +144 a b d \,x^{2}+72 b^{2} c \,x^{2}+74 a^{2} c m +90 a^{2} d x +180 a b c x +120 a^{2} c \right ) x \left (e x \right )^{m} \left (b \,x^{2}+a x \right )^{2}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (b x +a \right )^{2}}\) | \(265\) |
| parallelrisch | \(\frac {60 x^{6} \left (e x \right )^{m} b^{2} d +2 x^{5} \left (e x \right )^{m} a b d \,m^{3}+26 x^{5} \left (e x \right )^{m} a b d \,m^{2}+2 x^{4} \left (e x \right )^{m} a b c \,m^{3}+108 x^{5} \left (e x \right )^{m} a b d m +28 x^{4} \left (e x \right )^{m} a b c \,m^{2}+126 x^{4} \left (e x \right )^{m} a b c m +x^{5} \left (e x \right )^{m} b^{2} c \,m^{3}+47 x^{6} \left (e x \right )^{m} b^{2} d m +13 x^{5} \left (e x \right )^{m} b^{2} c \,m^{2}+x^{4} \left (e x \right )^{m} a^{2} d \,m^{3}+63 x^{4} \left (e x \right )^{m} a^{2} d m +15 x^{3} \left (e x \right )^{m} a^{2} c \,m^{2}+180 x^{4} \left (e x \right )^{m} a b c +74 x^{3} \left (e x \right )^{m} a^{2} c m +72 x^{5} \left (e x \right )^{m} b^{2} c +90 x^{4} \left (e x \right )^{m} a^{2} d +120 x^{3} \left (e x \right )^{m} a^{2} c +54 x^{5} \left (e x \right )^{m} b^{2} c m +14 x^{4} \left (e x \right )^{m} a^{2} d \,m^{2}+x^{3} \left (e x \right )^{m} a^{2} c \,m^{3}+x^{6} \left (e x \right )^{m} b^{2} d \,m^{3}+12 x^{6} \left (e x \right )^{m} b^{2} d \,m^{2}+144 x^{5} \left (e x \right )^{m} a b d}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right )}\) | \(389\) |
Input:
int((e*x)^m*(d*x+c)*(b*x^2+a*x)^2,x,method=_RETURNVERBOSE)
Output:
a*(a*d+2*b*c)/(4+m)*x^4*exp(m*ln(e*x))+a^2*c/(3+m)*x^3*exp(m*ln(e*x))+b*(2 *a*d+b*c)/(5+m)*x^5*exp(m*ln(e*x))+b^2*d/(6+m)*x^6*exp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (91) = 182\).
Time = 0.09 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.41 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^2 \, dx=\frac {{\left ({\left (b^{2} d m^{3} + 12 \, b^{2} d m^{2} + 47 \, b^{2} d m + 60 \, b^{2} d\right )} x^{6} + {\left ({\left (b^{2} c + 2 \, a b d\right )} m^{3} + 72 \, b^{2} c + 144 \, a b d + 13 \, {\left (b^{2} c + 2 \, a b d\right )} m^{2} + 54 \, {\left (b^{2} c + 2 \, a b d\right )} m\right )} x^{5} + {\left ({\left (2 \, a b c + a^{2} d\right )} m^{3} + 180 \, a b c + 90 \, a^{2} d + 14 \, {\left (2 \, a b c + a^{2} d\right )} m^{2} + 63 \, {\left (2 \, a b c + a^{2} d\right )} m\right )} x^{4} + {\left (a^{2} c m^{3} + 15 \, a^{2} c m^{2} + 74 \, a^{2} c m + 120 \, a^{2} c\right )} x^{3}\right )} \left (e x\right )^{m}}{m^{4} + 18 \, m^{3} + 119 \, m^{2} + 342 \, m + 360} \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a*x)^2,x, algorithm="fricas")
Output:
((b^2*d*m^3 + 12*b^2*d*m^2 + 47*b^2*d*m + 60*b^2*d)*x^6 + ((b^2*c + 2*a*b* d)*m^3 + 72*b^2*c + 144*a*b*d + 13*(b^2*c + 2*a*b*d)*m^2 + 54*(b^2*c + 2*a *b*d)*m)*x^5 + ((2*a*b*c + a^2*d)*m^3 + 180*a*b*c + 90*a^2*d + 14*(2*a*b*c + a^2*d)*m^2 + 63*(2*a*b*c + a^2*d)*m)*x^4 + (a^2*c*m^3 + 15*a^2*c*m^2 + 74*a^2*c*m + 120*a^2*c)*x^3)*(e*x)^m/(m^4 + 18*m^3 + 119*m^2 + 342*m + 360 )
Leaf count of result is larger than twice the leaf count of optimal. 1081 vs. \(2 (83) = 166\).
Time = 0.41 (sec) , antiderivative size = 1081, normalized size of antiderivative = 11.88 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^2 \, dx =\text {Too large to display} \] Input:
integrate((e*x)**m*(d*x+c)*(b*x**2+a*x)**2,x)
Output:
Piecewise(((-a**2*c/(3*x**3) - a**2*d/(2*x**2) - a*b*c/x**2 - 2*a*b*d/x - b**2*c/x + b**2*d*log(x))/e**6, Eq(m, -6)), ((-a**2*c/(2*x**2) - a**2*d/x - 2*a*b*c/x + 2*a*b*d*log(x) + b**2*c*log(x) + b**2*d*x)/e**5, Eq(m, -5)), ((-a**2*c/x + a**2*d*log(x) + 2*a*b*c*log(x) + 2*a*b*d*x + b**2*c*x + b** 2*d*x**2/2)/e**4, Eq(m, -4)), ((a**2*c*log(x) + a**2*d*x + 2*a*b*c*x + a*b *d*x**2 + b**2*c*x**2/2 + b**2*d*x**3/3)/e**3, Eq(m, -3)), (a**2*c*m**3*x* *3*(e*x)**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 15*a**2*c*m**2*x** 3*(e*x)**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 74*a**2*c*m*x**3*(e *x)**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 120*a**2*c*x**3*(e*x)** m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + a**2*d*m**3*x**4*(e*x)**m/(m **4 + 18*m**3 + 119*m**2 + 342*m + 360) + 14*a**2*d*m**2*x**4*(e*x)**m/(m* *4 + 18*m**3 + 119*m**2 + 342*m + 360) + 63*a**2*d*m*x**4*(e*x)**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 90*a**2*d*x**4*(e*x)**m/(m**4 + 18*m* *3 + 119*m**2 + 342*m + 360) + 2*a*b*c*m**3*x**4*(e*x)**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 28*a*b*c*m**2*x**4*(e*x)**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 126*a*b*c*m*x**4*(e*x)**m/(m**4 + 18*m**3 + 119* m**2 + 342*m + 360) + 180*a*b*c*x**4*(e*x)**m/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360) + 2*a*b*d*m**3*x**5*(e*x)**m/(m**4 + 18*m**3 + 119*m**2 + 34 2*m + 360) + 26*a*b*d*m**2*x**5*(e*x)**m/(m**4 + 18*m**3 + 119*m**2 + 342* m + 360) + 108*a*b*d*m*x**5*(e*x)**m/(m**4 + 18*m**3 + 119*m**2 + 342*m...
Time = 0.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.26 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^2 \, dx=\frac {b^{2} d e^{m} x^{6} x^{m}}{m + 6} + \frac {b^{2} c e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b d e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b c e^{m} x^{4} x^{m}}{m + 4} + \frac {a^{2} d e^{m} x^{4} x^{m}}{m + 4} + \frac {a^{2} c e^{m} x^{3} x^{m}}{m + 3} \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a*x)^2,x, algorithm="maxima")
Output:
b^2*d*e^m*x^6*x^m/(m + 6) + b^2*c*e^m*x^5*x^m/(m + 5) + 2*a*b*d*e^m*x^5*x^ m/(m + 5) + 2*a*b*c*e^m*x^4*x^m/(m + 4) + a^2*d*e^m*x^4*x^m/(m + 4) + a^2* c*e^m*x^3*x^m/(m + 3)
Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (91) = 182\).
Time = 0.18 (sec) , antiderivative size = 388, normalized size of antiderivative = 4.26 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^2 \, dx=\frac {\left (e x\right )^{m} b^{2} d m^{3} x^{6} + \left (e x\right )^{m} b^{2} c m^{3} x^{5} + 2 \, \left (e x\right )^{m} a b d m^{3} x^{5} + 12 \, \left (e x\right )^{m} b^{2} d m^{2} x^{6} + 2 \, \left (e x\right )^{m} a b c m^{3} x^{4} + \left (e x\right )^{m} a^{2} d m^{3} x^{4} + 13 \, \left (e x\right )^{m} b^{2} c m^{2} x^{5} + 26 \, \left (e x\right )^{m} a b d m^{2} x^{5} + 47 \, \left (e x\right )^{m} b^{2} d m x^{6} + \left (e x\right )^{m} a^{2} c m^{3} x^{3} + 28 \, \left (e x\right )^{m} a b c m^{2} x^{4} + 14 \, \left (e x\right )^{m} a^{2} d m^{2} x^{4} + 54 \, \left (e x\right )^{m} b^{2} c m x^{5} + 108 \, \left (e x\right )^{m} a b d m x^{5} + 60 \, \left (e x\right )^{m} b^{2} d x^{6} + 15 \, \left (e x\right )^{m} a^{2} c m^{2} x^{3} + 126 \, \left (e x\right )^{m} a b c m x^{4} + 63 \, \left (e x\right )^{m} a^{2} d m x^{4} + 72 \, \left (e x\right )^{m} b^{2} c x^{5} + 144 \, \left (e x\right )^{m} a b d x^{5} + 74 \, \left (e x\right )^{m} a^{2} c m x^{3} + 180 \, \left (e x\right )^{m} a b c x^{4} + 90 \, \left (e x\right )^{m} a^{2} d x^{4} + 120 \, \left (e x\right )^{m} a^{2} c x^{3}}{m^{4} + 18 \, m^{3} + 119 \, m^{2} + 342 \, m + 360} \] Input:
integrate((e*x)^m*(d*x+c)*(b*x^2+a*x)^2,x, algorithm="giac")
Output:
((e*x)^m*b^2*d*m^3*x^6 + (e*x)^m*b^2*c*m^3*x^5 + 2*(e*x)^m*a*b*d*m^3*x^5 + 12*(e*x)^m*b^2*d*m^2*x^6 + 2*(e*x)^m*a*b*c*m^3*x^4 + (e*x)^m*a^2*d*m^3*x^ 4 + 13*(e*x)^m*b^2*c*m^2*x^5 + 26*(e*x)^m*a*b*d*m^2*x^5 + 47*(e*x)^m*b^2*d *m*x^6 + (e*x)^m*a^2*c*m^3*x^3 + 28*(e*x)^m*a*b*c*m^2*x^4 + 14*(e*x)^m*a^2 *d*m^2*x^4 + 54*(e*x)^m*b^2*c*m*x^5 + 108*(e*x)^m*a*b*d*m*x^5 + 60*(e*x)^m *b^2*d*x^6 + 15*(e*x)^m*a^2*c*m^2*x^3 + 126*(e*x)^m*a*b*c*m*x^4 + 63*(e*x) ^m*a^2*d*m*x^4 + 72*(e*x)^m*b^2*c*x^5 + 144*(e*x)^m*a*b*d*x^5 + 74*(e*x)^m *a^2*c*m*x^3 + 180*(e*x)^m*a*b*c*x^4 + 90*(e*x)^m*a^2*d*x^4 + 120*(e*x)^m* a^2*c*x^3)/(m^4 + 18*m^3 + 119*m^2 + 342*m + 360)
Time = 8.83 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.99 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^2 \, dx={\left (e\,x\right )}^m\,\left (\frac {a\,x^4\,\left (a\,d+2\,b\,c\right )\,\left (m^3+14\,m^2+63\,m+90\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}+\frac {b\,x^5\,\left (2\,a\,d+b\,c\right )\,\left (m^3+13\,m^2+54\,m+72\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}+\frac {a^2\,c\,x^3\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}+\frac {b^2\,d\,x^6\,\left (m^3+12\,m^2+47\,m+60\right )}{m^4+18\,m^3+119\,m^2+342\,m+360}\right ) \] Input:
int((a*x + b*x^2)^2*(e*x)^m*(c + d*x),x)
Output:
(e*x)^m*((a*x^4*(a*d + 2*b*c)*(63*m + 14*m^2 + m^3 + 90))/(342*m + 119*m^2 + 18*m^3 + m^4 + 360) + (b*x^5*(2*a*d + b*c)*(54*m + 13*m^2 + m^3 + 72))/ (342*m + 119*m^2 + 18*m^3 + m^4 + 360) + (a^2*c*x^3*(74*m + 15*m^2 + m^3 + 120))/(342*m + 119*m^2 + 18*m^3 + m^4 + 360) + (b^2*d*x^6*(47*m + 12*m^2 + m^3 + 60))/(342*m + 119*m^2 + 18*m^3 + m^4 + 360))
Time = 0.20 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.74 \[ \int (e x)^m (c+d x) \left (a x+b x^2\right )^2 \, dx=\frac {x^{m} e^{m} x^{3} \left (b^{2} d \,m^{3} x^{3}+2 a b d \,m^{3} x^{2}+b^{2} c \,m^{3} x^{2}+12 b^{2} d \,m^{2} x^{3}+a^{2} d \,m^{3} x +2 a b c \,m^{3} x +26 a b d \,m^{2} x^{2}+13 b^{2} c \,m^{2} x^{2}+47 b^{2} d m \,x^{3}+a^{2} c \,m^{3}+14 a^{2} d \,m^{2} x +28 a b c \,m^{2} x +108 a b d m \,x^{2}+54 b^{2} c m \,x^{2}+60 b^{2} d \,x^{3}+15 a^{2} c \,m^{2}+63 a^{2} d m x +126 a b c m x +144 a b d \,x^{2}+72 b^{2} c \,x^{2}+74 a^{2} c m +90 a^{2} d x +180 a b c x +120 a^{2} c \right )}{m^{4}+18 m^{3}+119 m^{2}+342 m +360} \] Input:
int((e*x)^m*(d*x+c)*(b*x^2+a*x)^2,x)
Output:
(x**m*e**m*x**3*(a**2*c*m**3 + 15*a**2*c*m**2 + 74*a**2*c*m + 120*a**2*c + a**2*d*m**3*x + 14*a**2*d*m**2*x + 63*a**2*d*m*x + 90*a**2*d*x + 2*a*b*c* m**3*x + 28*a*b*c*m**2*x + 126*a*b*c*m*x + 180*a*b*c*x + 2*a*b*d*m**3*x**2 + 26*a*b*d*m**2*x**2 + 108*a*b*d*m*x**2 + 144*a*b*d*x**2 + b**2*c*m**3*x* *2 + 13*b**2*c*m**2*x**2 + 54*b**2*c*m*x**2 + 72*b**2*c*x**2 + b**2*d*m**3 *x**3 + 12*b**2*d*m**2*x**3 + 47*b**2*d*m*x**3 + 60*b**2*d*x**3))/(m**4 + 18*m**3 + 119*m**2 + 342*m + 360)