\(\int (A+B x+C x^2) (4 c^3-27 c d^2 x^2-27 d^3 x^3)^p \, dx\) [41]

Optimal result

Integrand size = 36, antiderivative size = 246 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \, dx=-\frac {(c C p+3 B d (1+p)) (c-3 d x) (2 c+3 d x) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p}{27 d^3 (1+p) (2+3 p)}+\frac {C (c-3 d x)^2 (2 c+3 d x) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p}{81 d^3 (1+p)}-\frac {3^{-3+2 p} 4^{-p} \left (2 c^2 C-3 B c d+9 A d^2 (2+3 p)\right ) (c-3 d x) \left (1+\frac {3 d x}{2 c}\right )^{-2 p} \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \operatorname {Hypergeometric2F1}\left (-2 p,1+p,2+p,\frac {c-3 d x}{3 c}\right )}{d^3 (1+p) (2+3 p)} \] Output:

-1/27*(c*C*p+3*B*d*(p+1))*(-3*d*x+c)*(3*d*x+2*c)*(-27*d^3*x^3-27*c*d^2*x^2 
+4*c^3)^p/d^3/(p+1)/(2+3*p)+1/81*C*(-3*d*x+c)^2*(3*d*x+2*c)*(-27*d^3*x^3-2 
7*c*d^2*x^2+4*c^3)^p/d^3/(p+1)-3^(-3+2*p)*(2*C*c^2-3*B*c*d+9*A*d^2*(2+3*p) 
)*(-3*d*x+c)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^p*hypergeom([-2*p, p+1],[2+p 
],1/3*(-3*d*x+c)/c)/(4^p)/d^3/(p+1)/(2+3*p)/((1+3/2*d*x/c)^(2*p))
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.65 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \, dx=\frac {3^{-3+2 p} (c-3 d x) \left ((c-3 d x) (2 c+3 d x)^2\right )^p \left (2+\frac {3 d x}{c}\right )^{-2 p} \left (3 c (4 c C-3 B d) \operatorname {Hypergeometric2F1}\left (-1-2 p,1+p,2+p,\frac {1}{3}-\frac {d x}{c}\right )+\left (-4 c^2 C+6 B c d-9 A d^2\right ) \operatorname {Hypergeometric2F1}\left (-2 p,1+p,2+p,\frac {1}{3}-\frac {d x}{c}\right )-9 c^2 C \operatorname {Hypergeometric2F1}\left (-2 (1+p),1+p,2+p,\frac {1}{3}-\frac {d x}{c}\right )\right )}{d^3 (1+p)} \] Input:

Integrate[(A + B*x + C*x^2)*(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^p,x]
 

Output:

(3^(-3 + 2*p)*(c - 3*d*x)*((c - 3*d*x)*(2*c + 3*d*x)^2)^p*(3*c*(4*c*C - 3* 
B*d)*Hypergeometric2F1[-1 - 2*p, 1 + p, 2 + p, 1/3 - (d*x)/c] + (-4*c^2*C 
+ 6*B*c*d - 9*A*d^2)*Hypergeometric2F1[-2*p, 1 + p, 2 + p, 1/3 - (d*x)/c] 
- 9*c^2*C*Hypergeometric2F1[-2*(1 + p), 1 + p, 2 + p, 1/3 - (d*x)/c]))/(d^ 
3*(1 + p)*(2 + (3*d*x)/c)^(2*p))
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.54, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2526, 27, 2490, 2483, 27, 90, 80, 79}

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.

\(\displaystyle \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \, dx\)

\(\Big \downarrow \) 2526

\(\displaystyle -\frac {\int -27 d^2 (3 A d-(2 c C-3 B d) x) \left (4 c^3-27 d^2 x^2 c-27 d^3 x^3\right )^pdx}{81 d^3}-\frac {C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{p+1}}{81 d^3 (p+1)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (3 A d-(2 c C-3 B d) x) \left (4 c^3-27 d^2 x^2 c-27 d^3 x^3\right )^pdx}{3 d}-\frac {C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{p+1}}{81 d^3 (p+1)}\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {\int \left ((3 B d-2 c C) \left (\frac {c}{3 d}+x\right )-\frac {27 c d^2 (3 B d-2 c C)-243 A d^4}{81 d^3}\right ) \left (2 c^3+9 d \left (\frac {c}{3 d}+x\right ) c^2-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^pd\left (\frac {c}{3 d}+x\right )}{3 d}-\frac {C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{p+1}}{81 d^3 (p+1)}\)

\(\Big \downarrow \) 2483

\(\displaystyle \frac {\left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-p} \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-2 p} \left (2 c^3+9 c^2 d \left (\frac {c}{3 d}+x\right )-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^p \int -\frac {1}{3} \left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^p \left (6 c^3+18 d \left (\frac {c}{3 d}+x\right ) c^2\right )^{2 p} \left (-\frac {2 C c^2}{d}+3 B c-9 A d+3 (2 c C-3 B d) \left (\frac {c}{3 d}+x\right )\right )d\left (\frac {c}{3 d}+x\right )}{3 d}-\frac {C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{p+1}}{81 d^3 (p+1)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-p} \left (2 c^3+9 c^2 d \left (\frac {c}{3 d}+x\right )-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^p \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-2 p} \int \left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^p \left (6 c^3+18 d \left (\frac {c}{3 d}+x\right ) c^2\right )^{2 p} \left (-\frac {2 C c^2}{d}+3 B c-9 A d+3 (2 c C-3 B d) \left (\frac {c}{3 d}+x\right )\right )d\left (\frac {c}{3 d}+x\right )}{9 d}-\frac {C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{p+1}}{81 d^3 (p+1)}\)

\(\Big \downarrow \) 90

\(\displaystyle -\frac {\left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-p} \left (2 c^3+9 c^2 d \left (\frac {c}{3 d}+x\right )-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^p \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-2 p} \left (-\frac {\left (9 A d^2 (3 p+2)-3 B c d+2 c^2 C\right ) \int \left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^p \left (6 c^3+18 d \left (\frac {c}{3 d}+x\right ) c^2\right )^{2 p}d\left (\frac {c}{3 d}+x\right )}{d (3 p+2)}-\frac {(2 c C-3 B d) \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{2 p+1} \left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{p+1}}{54 c^4 d^2 (3 p+2)}\right )}{9 d}-\frac {C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{p+1}}{81 d^3 (p+1)}\)

\(\Big \downarrow \) 80

\(\displaystyle -\frac {\left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-p} \left (2 c^3+9 c^2 d \left (\frac {c}{3 d}+x\right )-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^p \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-2 p} \left (-\frac {9^p \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{2 p} \left (\frac {3 d \left (\frac {c}{3 d}+x\right )+c}{c}\right )^{-2 p} \left (9 A d^2 (3 p+2)-3 B c d+2 c^2 C\right ) \int \left (\frac {d \left (\frac {c}{3 d}+x\right )}{c}+\frac {1}{3}\right )^{2 p} \left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^pd\left (\frac {c}{3 d}+x\right )}{d (3 p+2)}-\frac {(2 c C-3 B d) \left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{p+1} \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{2 p+1}}{54 c^4 d^2 (3 p+2)}\right )}{9 d}-\frac {C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{p+1}}{81 d^3 (p+1)}\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {\left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-p} \left (2 c^3+9 c^2 d \left (\frac {c}{3 d}+x\right )-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^p \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{-2 p} \left (\frac {9^{p-1} \left (\frac {3 d \left (\frac {c}{3 d}+x\right )+c}{c}\right )^{-2 p} \left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{p+1} \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{2 p} \operatorname {Hypergeometric2F1}\left (-2 p,p+1,p+2,\frac {2 c-3 d \left (\frac {c}{3 d}+x\right )}{3 c}\right ) \left (9 A d^2 (3 p+2)-3 B c d+2 c^2 C\right )}{c^2 d^2 (p+1) (3 p+2)}-\frac {(2 c C-3 B d) \left (6 c^3-9 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{p+1} \left (6 c^3+18 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{2 p+1}}{54 c^4 d^2 (3 p+2)}\right )}{9 d}-\frac {C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{p+1}}{81 d^3 (p+1)}\)

Input:

Int[(A + B*x + C*x^2)*(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^p,x]
 

Output:

-1/81*(C*(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(1 + p))/(d^3*(1 + p)) - ((2* 
c^3 + 9*c^2*d*(c/(3*d) + x) - 27*d^3*(c/(3*d) + x)^3)^p*(-1/54*((2*c*C - 3 
*B*d)*(6*c^3 - 9*c^2*d*(c/(3*d) + x))^(1 + p)*(6*c^3 + 18*c^2*d*(c/(3*d) + 
 x))^(1 + 2*p))/(c^4*d^2*(2 + 3*p)) + (9^(-1 + p)*(2*c^2*C - 3*B*c*d + 9*A 
*d^2*(2 + 3*p))*(6*c^3 - 9*c^2*d*(c/(3*d) + x))^(1 + p)*(6*c^3 + 18*c^2*d* 
(c/(3*d) + x))^(2*p)*Hypergeometric2F1[-2*p, 1 + p, 2 + p, (2*c - 3*d*(c/( 
3*d) + x))/(3*c)])/(c^2*d^2*(1 + p)*(2 + 3*p)*((c + 3*d*(c/(3*d) + x))/c)^ 
(2*p))))/(9*d*(6*c^3 - 9*c^2*d*(c/(3*d) + x))^p*(6*c^3 + 18*c^2*d*(c/(3*d) 
 + x))^(2*p))
 

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 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 2483

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> Simp[(a + b*x + d*x^3)^p/((3*a - b*x)^p*(3*a + 2*b*x)^(2*p))   In 
t[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, 
e, f, m, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp 
[1/(n*Coeff[Qn, x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x 
, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

Maple [F]

\[\int \left (C \,x^{2}+B x +A \right ) \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{p}d x\]

Input:

int((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^p,x)
 

Output:

int((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^p,x)
 

Fricas [F]

\[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}\right )}^{p} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^p,x, algorithm="f 
ricas")
 

Output:

integral((C*x^2 + B*x + A)*(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)^p, x)
 

Sympy [F]

\[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \, dx=\int \left (- \left (- c + 3 d x\right ) \left (2 c + 3 d x\right )^{2}\right )^{p} \left (A + B x + C x^{2}\right )\, dx \] Input:

integrate((C*x**2+B*x+A)*(-27*d**3*x**3-27*c*d**2*x**2+4*c**3)**p,x)
 

Output:

Integral((-(-c + 3*d*x)*(2*c + 3*d*x)**2)**p*(A + B*x + C*x**2), x)
 

Maxima [F]

\[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}\right )}^{p} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^p,x, algorithm="m 
axima")
 

Output:

integrate((C*x^2 + B*x + A)*(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)^p, x)
 

Giac [F]

\[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}\right )}^{p} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^p,x, algorithm="g 
iac")
 

Output:

integrate((C*x^2 + B*x + A)*(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (4\,c^3-27\,c\,d^2\,x^2-27\,d^3\,x^3\right )}^p \,d x \] Input:

int((A + B*x + C*x^2)*(4*c^3 - 27*d^3*x^3 - 27*c*d^2*x^2)^p,x)
 

Output:

int((A + B*x + C*x^2)*(4*c^3 - 27*d^3*x^3 - 27*c*d^2*x^2)^p, x)
 

Reduce [F]

\[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^p \, dx=\text {too large to display} \] Input:

int((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^p,x)
 

Output:

(81*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*a*c*d**2*p**2 + 135*(4*c** 
3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*a*c*d**2*p + 54*(4*c**3 - 27*c*d**2* 
x**2 - 27*d**3*x**3)**p*a*c*d**2 + 243*(4*c**3 - 27*c*d**2*x**2 - 27*d**3* 
x**3)**p*a*d**3*p**2*x + 405*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*a 
*d**3*p*x + 162*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*a*d**3*x - 54* 
(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*b*c**2*d*p**2 - 81*(4*c**3 - 2 
7*c*d**2*x**2 - 27*d**3*x**3)**p*b*c**2*d*p - 27*(4*c**3 - 27*c*d**2*x**2 
- 27*d**3*x**3)**p*b*c**2*d + 81*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)* 
*p*b*c*d**2*p**2*x + 81*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*b*c*d* 
*2*p*x + 243*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*b*d**3*p**2*x**2 
+ 324*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*b*d**3*p*x**2 + 81*(4*c* 
*3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*b*d**3*x**2 + 18*(4*c**3 - 27*c*d** 
2*x**2 - 27*d**3*x**3)**p*c**4*p + 10*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x 
**3)**p*c**4 - 54*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*c**3*d*p**2* 
x - 54*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*c**3*d*p*x + 81*(4*c**3 
 - 27*c*d**2*x**2 - 27*d**3*x**3)**p*c**2*d**2*p**2*x**2 + 27*(4*c**3 - 27 
*c*d**2*x**2 - 27*d**3*x**3)**p*c**2*d**2*p*x**2 + 243*(4*c**3 - 27*c*d**2 
*x**2 - 27*d**3*x**3)**p*c*d**3*p**2*x**3 + 243*(4*c**3 - 27*c*d**2*x**2 - 
 27*d**3*x**3)**p*c*d**3*p*x**3 + 54*(4*c**3 - 27*c*d**2*x**2 - 27*d**3*x* 
*3)**p*c*d**3*x**3 + 13122*int((4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)...