\(\int \frac {x^5 (c+d x^2)^3}{a+b x^2} \, dx\) [612]

Optimal result

Integrand size = 22, antiderivative size = 138 \[ \int \frac {x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx=-\frac {a (b c-a d)^3 x^2}{2 b^5}+\frac {(b c-a d)^3 x^4}{4 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^6}{6 b^3}+\frac {d^2 (3 b c-a d) x^8}{8 b^2}+\frac {d^3 x^{10}}{10 b}+\frac {a^2 (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^6} \] Output:

-1/2*a*(-a*d+b*c)^3*x^2/b^5+1/4*(-a*d+b*c)^3*x^4/b^4+1/6*d*(a^2*d^2-3*a*b* 
c*d+3*b^2*c^2)*x^6/b^3+1/8*d^2*(-a*d+3*b*c)*x^8/b^2+1/10*d^3*x^10/b+1/2*a^ 
2*(-a*d+b*c)^3*ln(b*x^2+a)/b^6
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \frac {x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {60 a b (-b c+a d)^3 x^2+30 b^2 (b c-a d)^3 x^4+20 b^3 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^6+15 b^4 d^2 (3 b c-a d) x^8+12 b^5 d^3 x^{10}+60 a^2 (b c-a d)^3 \log \left (a+b x^2\right )}{120 b^6} \] Input:

Integrate[(x^5*(c + d*x^2)^3)/(a + b*x^2),x]
 

Output:

(60*a*b*(-(b*c) + a*d)^3*x^2 + 30*b^2*(b*c - a*d)^3*x^4 + 20*b^3*d*(3*b^2* 
c^2 - 3*a*b*c*d + a^2*d^2)*x^6 + 15*b^4*d^2*(3*b*c - a*d)*x^8 + 12*b^5*d^3 
*x^10 + 60*a^2*(b*c - a*d)^3*Log[a + b*x^2])/(120*b^6)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 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.

\(\displaystyle \int \frac {x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} \int \frac {x^4 \left (d x^2+c\right )^3}{b x^2+a}dx^2\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {1}{2} \int \left (\frac {d^3 x^8}{b}+\frac {d^2 (3 b c-a d) x^6}{b^2}+\frac {d \left (3 b^2 c^2-3 a b d c+a^2 d^2\right ) x^4}{b^3}+\frac {(b c-a d)^3 x^2}{b^4}+\frac {a (a d-b c)^3}{b^5}-\frac {a^2 (a d-b c)^3}{b^5 \left (b x^2+a\right )}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {a^2 (b c-a d)^3 \log \left (a+b x^2\right )}{b^6}+\frac {d x^6 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}-\frac {a x^2 (b c-a d)^3}{b^5}+\frac {x^4 (b c-a d)^3}{2 b^4}+\frac {d^2 x^8 (3 b c-a d)}{4 b^2}+\frac {d^3 x^{10}}{5 b}\right )\)

Input:

Int[(x^5*(c + d*x^2)^3)/(a + b*x^2),x]
 

Output:

(-((a*(b*c - a*d)^3*x^2)/b^5) + ((b*c - a*d)^3*x^4)/(2*b^4) + (d*(3*b^2*c^ 
2 - 3*a*b*c*d + a^2*d^2)*x^6)/(3*b^3) + (d^2*(3*b*c - a*d)*x^8)/(4*b^2) + 
(d^3*x^10)/(5*b) + (a^2*(b*c - a*d)^3*Log[a + b*x^2])/b^6)/2
 

Defintions of rubi rules used

rule 99

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]))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.48

method	result	size
norman	\(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{4}}{4 b^{4}}+\frac {d^{3} x^{10}}{10 b}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 b^{5}}-\frac {d^{2} \left (a d -3 b c \right ) x^{8}}{8 b^{2}}+\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{6}}{6 b^{3}}-\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{6}}\)	\(204\)
default	\(\frac {\frac {1}{5} d^{3} x^{10} b^{4}-\frac {1}{4} a \,b^{3} d^{3} x^{8}+\frac {3}{4} b^{4} c \,d^{2} x^{8}+\frac {1}{3} a^{2} b^{2} d^{3} x^{6}-a \,b^{3} c \,d^{2} x^{6}+b^{4} c^{2} d \,x^{6}-\frac {1}{2} x^{4} a^{3} b \,d^{3}+\frac {3}{2} x^{4} a^{2} b^{2} c \,d^{2}-\frac {3}{2} x^{4} a \,b^{3} c^{2} d +\frac {1}{2} b^{4} c^{3} x^{4}+x^{2} a^{4} d^{3}-3 x^{2} a^{3} b c \,d^{2}+3 x^{2} a^{2} b^{2} c^{2} d -x^{2} a \,b^{3} c^{3}}{2 b^{5}}-\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{6}}\)	\(234\)
parallelrisch	\(-\frac {-12 d^{3} x^{10} b^{5}+15 x^{8} a \,b^{4} d^{3}-45 x^{8} b^{5} c \,d^{2}-20 x^{6} a^{2} b^{3} d^{3}+60 x^{6} a \,b^{4} c \,d^{2}-60 x^{6} b^{5} c^{2} d +30 x^{4} a^{3} b^{2} d^{3}-90 x^{4} a^{2} b^{3} c \,d^{2}+90 x^{4} a \,b^{4} c^{2} d -30 x^{4} b^{5} c^{3}-60 x^{2} a^{4} b \,d^{3}+180 x^{2} a^{3} b^{2} c \,d^{2}-180 x^{2} a^{2} b^{3} c^{2} d +60 x^{2} a \,b^{4} c^{3}+60 \ln \left (b \,x^{2}+a \right ) a^{5} d^{3}-180 \ln \left (b \,x^{2}+a \right ) a^{4} b c \,d^{2}+180 \ln \left (b \,x^{2}+a \right ) a^{3} b^{2} c^{2} d -60 \ln \left (b \,x^{2}+a \right ) a^{2} b^{3} c^{3}}{120 b^{6}}\)	\(261\)
risch	\(\frac {d^{3} x^{10}}{10 b}-\frac {a \,d^{3} x^{8}}{8 b^{2}}+\frac {3 c \,d^{2} x^{8}}{8 b}+\frac {a^{2} d^{3} x^{6}}{6 b^{3}}-\frac {a c \,d^{2} x^{6}}{2 b^{2}}+\frac {c^{2} d \,x^{6}}{2 b}-\frac {x^{4} a^{3} d^{3}}{4 b^{4}}+\frac {3 x^{4} a^{2} c \,d^{2}}{4 b^{3}}-\frac {3 x^{4} a \,c^{2} d}{4 b^{2}}+\frac {c^{3} x^{4}}{4 b}+\frac {x^{2} a^{4} d^{3}}{2 b^{5}}-\frac {3 x^{2} a^{3} c \,d^{2}}{2 b^{4}}+\frac {3 x^{2} a^{2} c^{2} d}{2 b^{3}}-\frac {x^{2} a \,c^{3}}{2 b^{2}}-\frac {a^{5} \ln \left (b \,x^{2}+a \right ) d^{3}}{2 b^{6}}+\frac {3 a^{4} \ln \left (b \,x^{2}+a \right ) c \,d^{2}}{2 b^{5}}-\frac {3 a^{3} \ln \left (b \,x^{2}+a \right ) c^{2} d}{2 b^{4}}+\frac {a^{2} \ln \left (b \,x^{2}+a \right ) c^{3}}{2 b^{3}}\)	\(263\)

Input:

int(x^5*(d*x^2+c)^3/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-1/4/b^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*x^4+1/10*d^3*x^10/b 
+1/2*a*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^5*x^2-1/8/b^2*d^2*( 
a*d-3*b*c)*x^8+1/6*d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*x^6/b^3-1/2*a^2*(a^3*d^ 
3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^6*ln(b*x^2+a)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.59 \[ \int \frac {x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {12 \, b^{5} d^{3} x^{10} + 15 \, {\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{8} + 20 \, {\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{6} + 30 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{4} - 60 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{2} + 60 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x^{2} + a\right )}{120 \, b^{6}} \] Input:

integrate(x^5*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")
 

Output:

1/120*(12*b^5*d^3*x^10 + 15*(3*b^5*c*d^2 - a*b^4*d^3)*x^8 + 20*(3*b^5*c^2* 
d - 3*a*b^4*c*d^2 + a^2*b^3*d^3)*x^6 + 30*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2 
*b^3*c*d^2 - a^3*b^2*d^3)*x^4 - 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^ 
2*c*d^2 - a^4*b*d^3)*x^2 + 60*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d 
^2 - a^5*d^3)*log(b*x^2 + a))/b^6
 

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.46 \[ \int \frac {x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx=- \frac {a^{2} \left (a d - b c\right )^{3} \log {\left (a + b x^{2} \right )}}{2 b^{6}} + x^{8} \left (- \frac {a d^{3}}{8 b^{2}} + \frac {3 c d^{2}}{8 b}\right ) + x^{6} \left (\frac {a^{2} d^{3}}{6 b^{3}} - \frac {a c d^{2}}{2 b^{2}} + \frac {c^{2} d}{2 b}\right ) + x^{4} \left (- \frac {a^{3} d^{3}}{4 b^{4}} + \frac {3 a^{2} c d^{2}}{4 b^{3}} - \frac {3 a c^{2} d}{4 b^{2}} + \frac {c^{3}}{4 b}\right ) + x^{2} \left (\frac {a^{4} d^{3}}{2 b^{5}} - \frac {3 a^{3} c d^{2}}{2 b^{4}} + \frac {3 a^{2} c^{2} d}{2 b^{3}} - \frac {a c^{3}}{2 b^{2}}\right ) + \frac {d^{3} x^{10}}{10 b} \] Input:

integrate(x**5*(d*x**2+c)**3/(b*x**2+a),x)
 

Output:

-a**2*(a*d - b*c)**3*log(a + b*x**2)/(2*b**6) + x**8*(-a*d**3/(8*b**2) + 3 
*c*d**2/(8*b)) + x**6*(a**2*d**3/(6*b**3) - a*c*d**2/(2*b**2) + c**2*d/(2* 
b)) + x**4*(-a**3*d**3/(4*b**4) + 3*a**2*c*d**2/(4*b**3) - 3*a*c**2*d/(4*b 
**2) + c**3/(4*b)) + x**2*(a**4*d**3/(2*b**5) - 3*a**3*c*d**2/(2*b**4) + 3 
*a**2*c**2*d/(2*b**3) - a*c**3/(2*b**2)) + d**3*x**10/(10*b)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.59 \[ \int \frac {x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {12 \, b^{4} d^{3} x^{10} + 15 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{8} + 20 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{6} + 30 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} - 60 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2}}{120 \, b^{5}} + \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{6}} \] Input:

integrate(x^5*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")
 

Output:

1/120*(12*b^4*d^3*x^10 + 15*(3*b^4*c*d^2 - a*b^3*d^3)*x^8 + 20*(3*b^4*c^2* 
d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^6 + 30*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2 
*b^2*c*d^2 - a^3*b*d^3)*x^4 - 60*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c* 
d^2 - a^4*d^3)*x^2)/b^5 + 1/2*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d 
^2 - a^5*d^3)*log(b*x^2 + a)/b^6
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.72 \[ \int \frac {x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {12 \, b^{4} d^{3} x^{10} + 45 \, b^{4} c d^{2} x^{8} - 15 \, a b^{3} d^{3} x^{8} + 60 \, b^{4} c^{2} d x^{6} - 60 \, a b^{3} c d^{2} x^{6} + 20 \, a^{2} b^{2} d^{3} x^{6} + 30 \, b^{4} c^{3} x^{4} - 90 \, a b^{3} c^{2} d x^{4} + 90 \, a^{2} b^{2} c d^{2} x^{4} - 30 \, a^{3} b d^{3} x^{4} - 60 \, a b^{3} c^{3} x^{2} + 180 \, a^{2} b^{2} c^{2} d x^{2} - 180 \, a^{3} b c d^{2} x^{2} + 60 \, a^{4} d^{3} x^{2}}{120 \, b^{5}} + \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} \] Input:

integrate(x^5*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")
 

Output:

1/120*(12*b^4*d^3*x^10 + 45*b^4*c*d^2*x^8 - 15*a*b^3*d^3*x^8 + 60*b^4*c^2* 
d*x^6 - 60*a*b^3*c*d^2*x^6 + 20*a^2*b^2*d^3*x^6 + 30*b^4*c^3*x^4 - 90*a*b^ 
3*c^2*d*x^4 + 90*a^2*b^2*c*d^2*x^4 - 30*a^3*b*d^3*x^4 - 60*a*b^3*c^3*x^2 + 
 180*a^2*b^2*c^2*d*x^2 - 180*a^3*b*c*d^2*x^2 + 60*a^4*d^3*x^2)/b^5 + 1/2*( 
a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*log(abs(b*x^2 + a 
))/b^6
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.71 \[ \int \frac {x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx=x^4\,\left (\frac {c^3}{4\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{4\,b}\right )-x^8\,\left (\frac {a\,d^3}{8\,b^2}-\frac {3\,c\,d^2}{8\,b}\right )+x^6\,\left (\frac {c^2\,d}{2\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{6\,b}\right )-\frac {\ln \left (b\,x^2+a\right )\,\left (a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3\right )}{2\,b^6}+\frac {d^3\,x^{10}}{10\,b}-\frac {a\,x^2\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{2\,b} \] Input:

int((x^5*(c + d*x^2)^3)/(a + b*x^2),x)
 

Output:

x^4*(c^3/(4*b) - (a*((3*c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/(4* 
b)) - x^8*((a*d^3)/(8*b^2) - (3*c*d^2)/(8*b)) + x^6*((c^2*d)/(2*b) + (a*(( 
a*d^3)/b^2 - (3*c*d^2)/b))/(6*b)) - (log(a + b*x^2)*(a^5*d^3 - a^2*b^3*c^3 
 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2))/(2*b^6) + (d^3*x^10)/(10*b) - (a*x^2* 
(c^3/b - (a*((3*c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/b))/(2*b)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.88 \[ \int \frac {x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx=\frac {-60 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{5} d^{3}+180 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{4} b c \,d^{2}-180 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{3} b^{2} c^{2} d +60 \,\mathrm {log}\left (b \,x^{2}+a \right ) a^{2} b^{3} c^{3}+60 a^{4} b \,d^{3} x^{2}-180 a^{3} b^{2} c \,d^{2} x^{2}-30 a^{3} b^{2} d^{3} x^{4}+180 a^{2} b^{3} c^{2} d \,x^{2}+90 a^{2} b^{3} c \,d^{2} x^{4}+20 a^{2} b^{3} d^{3} x^{6}-60 a \,b^{4} c^{3} x^{2}-90 a \,b^{4} c^{2} d \,x^{4}-60 a \,b^{4} c \,d^{2} x^{6}-15 a \,b^{4} d^{3} x^{8}+30 b^{5} c^{3} x^{4}+60 b^{5} c^{2} d \,x^{6}+45 b^{5} c \,d^{2} x^{8}+12 b^{5} d^{3} x^{10}}{120 b^{6}} \] Input:

int(x^5*(d*x^2+c)^3/(b*x^2+a),x)
 

Output:

( - 60*log(a + b*x**2)*a**5*d**3 + 180*log(a + b*x**2)*a**4*b*c*d**2 - 180 
*log(a + b*x**2)*a**3*b**2*c**2*d + 60*log(a + b*x**2)*a**2*b**3*c**3 + 60 
*a**4*b*d**3*x**2 - 180*a**3*b**2*c*d**2*x**2 - 30*a**3*b**2*d**3*x**4 + 1 
80*a**2*b**3*c**2*d*x**2 + 90*a**2*b**3*c*d**2*x**4 + 20*a**2*b**3*d**3*x* 
*6 - 60*a*b**4*c**3*x**2 - 90*a*b**4*c**2*d*x**4 - 60*a*b**4*c*d**2*x**6 - 
 15*a*b**4*d**3*x**8 + 30*b**5*c**3*x**4 + 60*b**5*c**2*d*x**6 + 45*b**5*c 
*d**2*x**8 + 12*b**5*d**3*x**10)/(120*b**6)