\(\int \frac {(a+b x^3)^{2/3}}{x^{12} (c+d x^3)} \, dx\) [705]

Optimal result

Integrand size = 24, antiderivative size = 320 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{11 c x^{11}}-\frac {(2 b c-11 a d) \left (a+b x^3\right )^{2/3}}{88 a c^2 x^8}+\frac {\left (6 b^2 c^2+11 a b c d-44 a^2 d^2\right ) \left (a+b x^3\right )^{2/3}}{220 a^2 c^3 x^5}-\frac {\left (18 b^3 c^3+33 a b^2 c^2 d+88 a^2 b c d^2-220 a^3 d^3\right ) \left (a+b x^3\right )^{2/3}}{440 a^3 c^4 x^2}-\frac {d^3 (b c-a d)^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{14/3}}-\frac {d^3 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 c^{14/3}}+\frac {d^3 (b c-a d)^{2/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{14/3}} \] Output:

-1/11*(b*x^3+a)^(2/3)/c/x^11-1/88*(-11*a*d+2*b*c)*(b*x^3+a)^(2/3)/a/c^2/x^ 
8+1/220*(-44*a^2*d^2+11*a*b*c*d+6*b^2*c^2)*(b*x^3+a)^(2/3)/a^2/c^3/x^5-1/4 
40*(-220*a^3*d^3+88*a^2*b*c*d^2+33*a*b^2*c^2*d+18*b^3*c^3)*(b*x^3+a)^(2/3) 
/a^3/c^4/x^2-1/3*d^3*(-a*d+b*c)^(2/3)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c 
^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/c^(14/3)-1/6*d^3*(-a*d+b*c)^(2/3) 
*ln(d*x^3+c)/c^(14/3)+1/2*d^3*(-a*d+b*c)^(2/3)*ln((-a*d+b*c)^(1/3)*x/c^(1/ 
3)-(b*x^3+a)^(1/3))/c^(14/3)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.80 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx=\frac {\frac {3 c^{2/3} \left (a+b x^3\right )^{2/3} \left (-18 b^3 c^3 x^9+3 a b^2 c^2 x^6 \left (4 c-11 d x^3\right )-2 a^2 b c x^3 \left (5 c^2-11 c d x^3+44 d^2 x^6\right )+a^3 \left (-40 c^3+55 c^2 d x^3-88 c d^2 x^6+220 d^3 x^9\right )\right )}{a^3 x^{11}}+220 \sqrt {-6+6 i \sqrt {3}} d^3 (b c-a d)^{2/3} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )-220 i \left (-i+\sqrt {3}\right ) d^3 (b c-a d)^{2/3} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+110 \left (1+i \sqrt {3}\right ) d^3 (b c-a d)^{2/3} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{1320 c^{14/3}} \] Input:

Integrate[(a + b*x^3)^(2/3)/(x^12*(c + d*x^3)),x]
 

Output:

((3*c^(2/3)*(a + b*x^3)^(2/3)*(-18*b^3*c^3*x^9 + 3*a*b^2*c^2*x^6*(4*c - 11 
*d*x^3) - 2*a^2*b*c*x^3*(5*c^2 - 11*c*d*x^3 + 44*d^2*x^6) + a^3*(-40*c^3 + 
 55*c^2*d*x^3 - 88*c*d^2*x^6 + 220*d^3*x^9)))/(a^3*x^11) + 220*Sqrt[-6 + ( 
6*I)*Sqrt[3]]*d^3*(b*c - a*d)^(2/3)*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3 
]*(b*c - a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))] - (220 
*I)*(-I + Sqrt[3])*d^3*(b*c - a*d)^(2/3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + 
I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3)] + 110*(1 + I*Sqrt[3])*d^3*(b*c - a*d 
)^(2/3)*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3])*c^(1/3)*(b*c - a*d) 
^(1/3)*x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2/3)*(a + b*x^3)^(2/3)])/( 
1320*c^(14/3))
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {975, 1053, 27, 1053, 1053, 27, 901}

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.

\(\displaystyle \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx\)

\(\Big \downarrow \) 975

\(\displaystyle \frac {\int \frac {-9 b d x^3+2 b c-11 a d}{x^9 \sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{11 c}-\frac {\left (a+b x^3\right )^{2/3}}{11 c x^{11}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {-\frac {\int \frac {2 \left (3 b d (2 b c-11 a d) x^3+6 b^2 c^2-44 a^2 d^2+11 a b c d\right )}{x^6 \sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{8 a c}-\frac {\left (a+b x^3\right )^{2/3} (2 b c-11 a d)}{8 a c x^8}}{11 c}-\frac {\left (a+b x^3\right )^{2/3}}{11 c x^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {3 b d (2 b c-11 a d) x^3+6 b^2 c^2-44 a^2 d^2+11 a b c d}{x^6 \sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{4 a c}-\frac {\left (a+b x^3\right )^{2/3} (2 b c-11 a d)}{8 a c x^8}}{11 c}-\frac {\left (a+b x^3\right )^{2/3}}{11 c x^{11}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {-\frac {-\frac {\int \frac {18 b^3 c^3+33 a b^2 d c^2+88 a^2 b d^2 c-220 a^3 d^3+3 b d \left (6 b^2 c^2+11 a b d c-44 a^2 d^2\right ) x^3}{x^3 \sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{5 a c}-\frac {\left (a+b x^3\right )^{2/3} \left (\frac {6 b^2 c}{a}-\frac {44 a d^2}{c}+11 b d\right )}{5 x^5}}{4 a c}-\frac {\left (a+b x^3\right )^{2/3} (2 b c-11 a d)}{8 a c x^8}}{11 c}-\frac {\left (a+b x^3\right )^{2/3}}{11 c x^{11}}\)

\(\Big \downarrow \) 1053

\(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {440 a^3 d^3 (b c-a d)}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{2 a c}-\frac {\left (a+b x^3\right )^{2/3} \left (-220 a^3 d^3+88 a^2 b c d^2+33 a b^2 c^2 d+18 b^3 c^3\right )}{2 a c x^2}}{5 a c}-\frac {\left (a+b x^3\right )^{2/3} \left (\frac {6 b^2 c}{a}-\frac {44 a d^2}{c}+11 b d\right )}{5 x^5}}{4 a c}-\frac {\left (a+b x^3\right )^{2/3} (2 b c-11 a d)}{8 a c x^8}}{11 c}-\frac {\left (a+b x^3\right )^{2/3}}{11 c x^{11}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {-\frac {220 a^2 d^3 (b c-a d) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{c}-\frac {\left (a+b x^3\right )^{2/3} \left (-220 a^3 d^3+88 a^2 b c d^2+33 a b^2 c^2 d+18 b^3 c^3\right )}{2 a c x^2}}{5 a c}-\frac {\left (a+b x^3\right )^{2/3} \left (\frac {6 b^2 c}{a}-\frac {44 a d^2}{c}+11 b d\right )}{5 x^5}}{4 a c}-\frac {\left (a+b x^3\right )^{2/3} (2 b c-11 a d)}{8 a c x^8}}{11 c}-\frac {\left (a+b x^3\right )^{2/3}}{11 c x^{11}}\)

\(\Big \downarrow \) 901

\(\displaystyle \frac {-\frac {-\frac {-\frac {220 a^2 d^3 (b c-a d) \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{c}-\frac {\left (a+b x^3\right )^{2/3} \left (-220 a^3 d^3+88 a^2 b c d^2+33 a b^2 c^2 d+18 b^3 c^3\right )}{2 a c x^2}}{5 a c}-\frac {\left (a+b x^3\right )^{2/3} \left (\frac {6 b^2 c}{a}-\frac {44 a d^2}{c}+11 b d\right )}{5 x^5}}{4 a c}-\frac {\left (a+b x^3\right )^{2/3} (2 b c-11 a d)}{8 a c x^8}}{11 c}-\frac {\left (a+b x^3\right )^{2/3}}{11 c x^{11}}\)

Input:

Int[(a + b*x^3)^(2/3)/(x^12*(c + d*x^3)),x]
 

Output:

-1/11*(a + b*x^3)^(2/3)/(c*x^11) + (-1/8*((2*b*c - 11*a*d)*(a + b*x^3)^(2/ 
3))/(a*c*x^8) - (-1/5*(((6*b^2*c)/a + 11*b*d - (44*a*d^2)/c)*(a + b*x^3)^( 
2/3))/x^5 - (-1/2*((18*b^3*c^3 + 33*a*b^2*c^2*d + 88*a^2*b*c*d^2 - 220*a^3 
*d^3)*(a + b*x^3)^(2/3))/(a*c*x^2) - (220*a^2*d^3*(b*c - a*d)*(ArcTan[(1 + 
 (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c^ 
(2/3)*(b*c - a*d)^(1/3)) + Log[c + d*x^3]/(6*c^(2/3)*(b*c - a*d)^(1/3)) - 
Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]/(2*c^(2/3)*(b*c - a 
*d)^(1/3))))/c)/(5*a*c))/(4*a*c))/(11*c)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] 
 + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 975

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/ 
(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1))   Int[(e*x)^(m + n)*(a + b*x^n) 
^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m 
 + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi 
alQ[a, b, c, d, e, m, n, p, q, x]
 

rule 1053

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ 
))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b 
*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( 
m + 1))   Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) 
- e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 
) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 
0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 1.98 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(-\frac {6 c \left (\left (\frac {9}{20} b^{2} x^{6}-\frac {3}{4} a b \,x^{3}+a^{2}\right ) \left (b \,x^{3}+a \right ) c^{3}-\frac {11 a \left (-\frac {3 b \,x^{3}}{5}+a \right ) d \left (b \,x^{3}+a \right ) x^{3} c^{2}}{8}+\frac {11 \left (b \,x^{3}+a \right ) a^{2} c \,d^{2} x^{6}}{5}-\frac {11 a^{3} d^{3} x^{9}}{2}\right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}+11 a^{3} d^{3} x^{11} \left (a d -b c \right ) \left (-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right )}{66 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x^{11} c^{5} a^{3}}\)	\(307\)

Input:

int((b*x^3+a)^(2/3)/x^12/(d*x^3+c),x,method=_RETURNVERBOSE)
 

Output:

-1/66/((a*d-b*c)/c)^(1/3)*(6*c*((9/20*b^2*x^6-3/4*a*b*x^3+a^2)*(b*x^3+a)*c 
^3-11/8*a*(-3/5*b*x^3+a)*d*(b*x^3+a)*x^3*c^2+11/5*(b*x^3+a)*a^2*c*d^2*x^6- 
11/2*a^3*d^3*x^9)*(b*x^3+a)^(2/3)*((a*d-b*c)/c)^(1/3)+11*a^3*d^3*x^11*(a*d 
-b*c)*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(((a*d-b*c)/c)^(1/3)*x-2*(b*x^3+a)^(1 
/3))/((a*d-b*c)/c)^(1/3)/x)+ln((((a*d-b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3 
)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)-2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x 
^3+a)^(1/3))/x)))/x^11/c^5/a^3
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:

integrate((b*x^3+a)^(2/3)/x^12/(d*x^3+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx=\int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{x^{12} \left (c + d x^{3}\right )}\, dx \] Input:

integrate((b*x**3+a)**(2/3)/x**12/(d*x**3+c),x)
 

Output:

Integral((a + b*x**3)**(2/3)/(x**12*(c + d*x**3)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{12}} \,d x } \] Input:

integrate((b*x^3+a)^(2/3)/x^12/(d*x^3+c),x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^12), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )} x^{12}} \,d x } \] Input:

integrate((b*x^3+a)^(2/3)/x^12/(d*x^3+c),x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(2/3)/((d*x^3 + c)*x^12), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{2/3}}{x^{12}\,\left (d\,x^3+c\right )} \,d x \] Input:

int((a + b*x^3)^(2/3)/(x^12*(c + d*x^3)),x)
 

Output:

int((a + b*x^3)^(2/3)/(x^12*(c + d*x^3)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12} \left (c+d x^3\right )} \, dx=\frac {-40 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{3} c^{2}+55 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{3} c d \,x^{3}-88 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{3} d^{2} x^{6}-10 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} b \,c^{2} x^{3}+22 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} b c d \,x^{6}+132 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} b \,d^{2} x^{9}+12 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,b^{2} c^{2} x^{6}-33 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,b^{2} c d \,x^{9}-18 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{3} c^{2} x^{9}-440 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b d \,x^{9}+a d \,x^{6}+b c \,x^{6}+a c \,x^{3}}d x \right ) a^{4} d^{3} x^{11}+440 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b d \,x^{9}+a d \,x^{6}+b c \,x^{6}+a c \,x^{3}}d x \right ) a^{3} b c \,d^{2} x^{11}}{440 a^{3} c^{3} x^{11}} \] Input:

int((b*x^3+a)^(2/3)/x^12/(d*x^3+c),x)
 

Output:

( - 40*(a + b*x**3)**(2/3)*a**3*c**2 + 55*(a + b*x**3)**(2/3)*a**3*c*d*x** 
3 - 88*(a + b*x**3)**(2/3)*a**3*d**2*x**6 - 10*(a + b*x**3)**(2/3)*a**2*b* 
c**2*x**3 + 22*(a + b*x**3)**(2/3)*a**2*b*c*d*x**6 + 132*(a + b*x**3)**(2/ 
3)*a**2*b*d**2*x**9 + 12*(a + b*x**3)**(2/3)*a*b**2*c**2*x**6 - 33*(a + b* 
x**3)**(2/3)*a*b**2*c*d*x**9 - 18*(a + b*x**3)**(2/3)*b**3*c**2*x**9 - 440 
*int((a + b*x**3)**(2/3)/(a*c*x**3 + a*d*x**6 + b*c*x**6 + b*d*x**9),x)*a* 
*4*d**3*x**11 + 440*int((a + b*x**3)**(2/3)/(a*c*x**3 + a*d*x**6 + b*c*x** 
6 + b*d*x**9),x)*a**3*b*c*d**2*x**11)/(440*a**3*c**3*x**11)