\(\int (a^2+2 a c x^n+c^2 x^{2 n})^3 \, dx\) [106]

Optimal result

Integrand size = 22, antiderivative size = 126 \[ \int \left (a^2+2 a c x^n+c^2 x^{2 n}\right )^3 \, dx=a^6 x+\frac {6 a^5 c x^{1+n}}{1+n}+\frac {15 a^4 c^2 x^{1+2 n}}{1+2 n}+\frac {20 a^3 c^3 x^{1+3 n}}{1+3 n}+\frac {15 a^2 c^4 x^{1+4 n}}{1+4 n}+\frac {6 a c^5 x^{1+5 n}}{1+5 n}+\frac {c^6 x^{1+6 n}}{1+6 n} \] Output:

a^6*x+6*a^5*c*x^(1+n)/(1+n)+15*a^4*c^2*x^(1+2*n)/(1+2*n)+20*a^3*c^3*x^(1+3 
*n)/(1+3*n)+15*a^2*c^4*x^(1+4*n)/(1+4*n)+6*a*c^5*x^(1+5*n)/(1+5*n)+c^6*x^( 
1+6*n)/(1+6*n)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.90 \[ \int \left (a^2+2 a c x^n+c^2 x^{2 n}\right )^3 \, dx=x \left (a^6+\frac {6 a^5 c x^n}{1+n}+\frac {15 a^4 c^2 x^{2 n}}{1+2 n}+\frac {20 a^3 c^3 x^{3 n}}{1+3 n}+\frac {15 a^2 c^4 x^{4 n}}{1+4 n}+\frac {6 a c^5 x^{5 n}}{1+5 n}+\frac {c^6 x^{6 n}}{1+6 n}\right ) \] Input:

Integrate[(a^2 + 2*a*c*x^n + c^2*x^(2*n))^3,x]
 

Output:

x*(a^6 + (6*a^5*c*x^n)/(1 + n) + (15*a^4*c^2*x^(2*n))/(1 + 2*n) + (20*a^3* 
c^3*x^(3*n))/(1 + 3*n) + (15*a^2*c^4*x^(4*n))/(1 + 4*n) + (6*a*c^5*x^(5*n) 
)/(1 + 5*n) + (c^6*x^(6*n))/(1 + 6*n))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1379, 775, 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.

\(\displaystyle \int \left (a^2+2 a c x^n+c^2 x^{2 n}\right )^3 \, dx\)

\(\Big \downarrow \) 1379

\(\displaystyle \frac {\int \left (c^2 x^n+a c\right )^6dx}{c^6}\)

\(\Big \downarrow \) 775

\(\displaystyle \frac {\int \left (6 a^5 c^7 x^n+15 a^4 c^8 x^{2 n}+20 a^3 c^9 x^{3 n}+15 a^2 c^{10} x^{4 n}+6 a c^{11} x^{5 n}+c^{12} x^{6 n}+a^6 c^6\right )dx}{c^6}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^6 c^6 x+\frac {6 a^5 c^7 x^{n+1}}{n+1}+\frac {15 a^4 c^8 x^{2 n+1}}{2 n+1}+\frac {20 a^3 c^9 x^{3 n+1}}{3 n+1}+\frac {15 a^2 c^{10} x^{4 n+1}}{4 n+1}+\frac {6 a c^{11} x^{5 n+1}}{5 n+1}+\frac {c^{12} x^{6 n+1}}{6 n+1}}{c^6}\)

Input:

Int[(a^2 + 2*a*c*x^n + c^2*x^(2*n))^3,x]
 

Output:

(a^6*c^6*x + (6*a^5*c^7*x^(1 + n))/(1 + n) + (15*a^4*c^8*x^(1 + 2*n))/(1 + 
 2*n) + (20*a^3*c^9*x^(1 + 3*n))/(1 + 3*n) + (15*a^2*c^10*x^(1 + 4*n))/(1 
+ 4*n) + (6*a*c^11*x^(1 + 5*n))/(1 + 5*n) + (c^12*x^(1 + 6*n))/(1 + 6*n))/ 
c^6
 

Defintions of rubi rules used

rule 775

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b* 
x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && IGtQ[p, 0]
 

rule 1379

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/ 
c^p   Int[(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n 
2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && NeQ[p, 1]
 

rule 2009

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

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96

method	result	size
risch	\(a^{6} x +\frac {c^{6} x \,x^{6 n}}{1+6 n}+\frac {6 a \,c^{5} x \,x^{5 n}}{1+5 n}+\frac {15 a^{2} c^{4} x \,x^{4 n}}{1+4 n}+\frac {20 a^{3} c^{3} x \,x^{3 n}}{1+3 n}+\frac {15 a^{4} c^{2} x \,x^{2 n}}{1+2 n}+\frac {6 a^{5} c x \,x^{n}}{1+n}\)	\(121\)
norman	\(a^{6} x +\frac {c^{6} x \,{\mathrm e}^{6 n \ln \left (x \right )}}{1+6 n}+\frac {6 a \,c^{5} x \,{\mathrm e}^{5 n \ln \left (x \right )}}{1+5 n}+\frac {15 a^{2} c^{4} x \,{\mathrm e}^{4 n \ln \left (x \right )}}{1+4 n}+\frac {20 a^{3} c^{3} x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}+\frac {15 a^{4} c^{2} x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}+\frac {6 a^{5} c x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}\)	\(133\)
parallelrisch	\(\text {Expression too large to display}\)	\(1020\)
orering	\(\text {Expression too large to display}\)	\(2911\)

Input:

int((a^2+2*a*c*x^n+c^2*x^(2*n))^3,x,method=_RETURNVERBOSE)
 

Output:

a^6*x+c^6/(1+6*n)*x*(x^n)^6+6*a*c^5/(1+5*n)*x*(x^n)^5+15*a^2*c^4/(1+4*n)*x 
*(x^n)^4+20*a^3*c^3/(1+3*n)*x*(x^n)^3+15*a^4*c^2/(1+2*n)*x*(x^n)^2+6*a^5*c 
/(1+n)*x*x^n
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (126) = 252\).

Time = 0.09 (sec) , antiderivative size = 454, normalized size of antiderivative = 3.60 \[ \int \left (a^2+2 a c x^n+c^2 x^{2 n}\right )^3 \, dx=\frac {{\left (120 \, c^{6} n^{5} + 274 \, c^{6} n^{4} + 225 \, c^{6} n^{3} + 85 \, c^{6} n^{2} + 15 \, c^{6} n + c^{6}\right )} x x^{6 \, n} + 6 \, {\left (144 \, a c^{5} n^{5} + 324 \, a c^{5} n^{4} + 260 \, a c^{5} n^{3} + 95 \, a c^{5} n^{2} + 16 \, a c^{5} n + a c^{5}\right )} x x^{5 \, n} + 15 \, {\left (180 \, a^{2} c^{4} n^{5} + 396 \, a^{2} c^{4} n^{4} + 307 \, a^{2} c^{4} n^{3} + 107 \, a^{2} c^{4} n^{2} + 17 \, a^{2} c^{4} n + a^{2} c^{4}\right )} x x^{4 \, n} + 20 \, {\left (240 \, a^{3} c^{3} n^{5} + 508 \, a^{3} c^{3} n^{4} + 372 \, a^{3} c^{3} n^{3} + 121 \, a^{3} c^{3} n^{2} + 18 \, a^{3} c^{3} n + a^{3} c^{3}\right )} x x^{3 \, n} + 15 \, {\left (360 \, a^{4} c^{2} n^{5} + 702 \, a^{4} c^{2} n^{4} + 461 \, a^{4} c^{2} n^{3} + 137 \, a^{4} c^{2} n^{2} + 19 \, a^{4} c^{2} n + a^{4} c^{2}\right )} x x^{2 \, n} + 6 \, {\left (720 \, a^{5} c n^{5} + 1044 \, a^{5} c n^{4} + 580 \, a^{5} c n^{3} + 155 \, a^{5} c n^{2} + 20 \, a^{5} c n + a^{5} c\right )} x x^{n} + {\left (720 \, a^{6} n^{6} + 1764 \, a^{6} n^{5} + 1624 \, a^{6} n^{4} + 735 \, a^{6} n^{3} + 175 \, a^{6} n^{2} + 21 \, a^{6} n + a^{6}\right )} x}{720 \, n^{6} + 1764 \, n^{5} + 1624 \, n^{4} + 735 \, n^{3} + 175 \, n^{2} + 21 \, n + 1} \] Input:

integrate((a^2+2*a*c*x^n+c^2*x^(2*n))^3,x, algorithm="fricas")
 

Output:

((120*c^6*n^5 + 274*c^6*n^4 + 225*c^6*n^3 + 85*c^6*n^2 + 15*c^6*n + c^6)*x 
*x^(6*n) + 6*(144*a*c^5*n^5 + 324*a*c^5*n^4 + 260*a*c^5*n^3 + 95*a*c^5*n^2 
 + 16*a*c^5*n + a*c^5)*x*x^(5*n) + 15*(180*a^2*c^4*n^5 + 396*a^2*c^4*n^4 + 
 307*a^2*c^4*n^3 + 107*a^2*c^4*n^2 + 17*a^2*c^4*n + a^2*c^4)*x*x^(4*n) + 2 
0*(240*a^3*c^3*n^5 + 508*a^3*c^3*n^4 + 372*a^3*c^3*n^3 + 121*a^3*c^3*n^2 + 
 18*a^3*c^3*n + a^3*c^3)*x*x^(3*n) + 15*(360*a^4*c^2*n^5 + 702*a^4*c^2*n^4 
 + 461*a^4*c^2*n^3 + 137*a^4*c^2*n^2 + 19*a^4*c^2*n + a^4*c^2)*x*x^(2*n) + 
 6*(720*a^5*c*n^5 + 1044*a^5*c*n^4 + 580*a^5*c*n^3 + 155*a^5*c*n^2 + 20*a^ 
5*c*n + a^5*c)*x*x^n + (720*a^6*n^6 + 1764*a^6*n^5 + 1624*a^6*n^4 + 735*a^ 
6*n^3 + 175*a^6*n^2 + 21*a^6*n + a^6)*x)/(720*n^6 + 1764*n^5 + 1624*n^4 + 
735*n^3 + 175*n^2 + 21*n + 1)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2445 vs. \(2 (114) = 228\).

Time = 6.24 (sec) , antiderivative size = 2445, normalized size of antiderivative = 19.40 \[ \int \left (a^2+2 a c x^n+c^2 x^{2 n}\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((a**2+2*a*c*x**n+c**2*x**(2*n))**3,x)
 

Output:

Piecewise((a**6*x + 6*a**5*c*log(x) - 15*a**4*c**2/x - 10*a**3*c**3/x**2 - 
 5*a**2*c**4/x**3 - 3*a*c**5/(2*x**4) - c**6/(5*x**5), Eq(n, -1)), (a**6*x 
 + 12*a**5*c*sqrt(x) + 15*a**4*c**2*log(x) - 40*a**3*c**3/sqrt(x) - 15*a** 
2*c**4/x - 4*a*c**5/x**(3/2) - c**6/(2*x**2), Eq(n, -1/2)), (a**6*x + 9*a* 
*5*c*x**(2/3) + 45*a**4*c**2*x**(1/3) + 20*a**3*c**3*log(x) - 45*a**2*c**4 
/x**(1/3) - 9*a*c**5/x**(2/3) - c**6/x, Eq(n, -1/3)), (a**6*x + 8*a**5*c*x 
**(3/4) + 30*a**4*c**2*sqrt(x) + 80*a**3*c**3*x**(1/4) + 60*a**2*c**4*log( 
x**(1/4)) - 24*a*c**5/x**(1/4) - 2*c**6/sqrt(x), Eq(n, -1/4)), (a**6*x + 1 
5*a**5*c*x**(4/5)/2 + 25*a**4*c**2*x**(3/5) + 50*a**3*c**3*x**(2/5) + 75*a 
**2*c**4*x**(1/5) + 30*a*c**5*log(x**(1/5)) - 5*c**6/x**(1/5), Eq(n, -1/5) 
), (a**6*x + 36*a**5*c*x**(5/6)/5 + 45*a**4*c**2*x**(2/3)/2 + 40*a**3*c**3 
*sqrt(x) + 45*a**2*c**4*x**(1/3) + 36*a*c**5*x**(1/6) + 6*c**6*log(x**(1/6 
)), Eq(n, -1/6)), (720*a**6*n**6*x/(720*n**6 + 1764*n**5 + 1624*n**4 + 735 
*n**3 + 175*n**2 + 21*n + 1) + 1764*a**6*n**5*x/(720*n**6 + 1764*n**5 + 16 
24*n**4 + 735*n**3 + 175*n**2 + 21*n + 1) + 1624*a**6*n**4*x/(720*n**6 + 1 
764*n**5 + 1624*n**4 + 735*n**3 + 175*n**2 + 21*n + 1) + 735*a**6*n**3*x/( 
720*n**6 + 1764*n**5 + 1624*n**4 + 735*n**3 + 175*n**2 + 21*n + 1) + 175*a 
**6*n**2*x/(720*n**6 + 1764*n**5 + 1624*n**4 + 735*n**3 + 175*n**2 + 21*n 
+ 1) + 21*a**6*n*x/(720*n**6 + 1764*n**5 + 1624*n**4 + 735*n**3 + 175*n**2 
 + 21*n + 1) + a**6*x/(720*n**6 + 1764*n**5 + 1624*n**4 + 735*n**3 + 17...
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.52 \[ \int \left (a^2+2 a c x^n+c^2 x^{2 n}\right )^3 \, dx=a^{6} x + \frac {c^{6} x^{6 \, n + 1}}{6 \, n + 1} + \frac {6 \, a c^{5} x^{5 \, n + 1}}{5 \, n + 1} + \frac {12 \, a^{2} c^{4} x^{4 \, n + 1}}{4 \, n + 1} + \frac {8 \, a^{3} c^{3} x^{3 \, n + 1}}{3 \, n + 1} + 3 \, {\left (\frac {c^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {2 \, a c x^{n + 1}}{n + 1}\right )} a^{4} + 3 \, {\left (\frac {c^{4} x^{4 \, n + 1}}{4 \, n + 1} + \frac {4 \, a c^{3} x^{3 \, n + 1}}{3 \, n + 1} + \frac {4 \, a^{2} c^{2} x^{2 \, n + 1}}{2 \, n + 1}\right )} a^{2} \] Input:

integrate((a^2+2*a*c*x^n+c^2*x^(2*n))^3,x, algorithm="maxima")
 

Output:

a^6*x + c^6*x^(6*n + 1)/(6*n + 1) + 6*a*c^5*x^(5*n + 1)/(5*n + 1) + 12*a^2 
*c^4*x^(4*n + 1)/(4*n + 1) + 8*a^3*c^3*x^(3*n + 1)/(3*n + 1) + 3*(c^2*x^(2 
*n + 1)/(2*n + 1) + 2*a*c*x^(n + 1)/(n + 1))*a^4 + 3*(c^4*x^(4*n + 1)/(4*n 
 + 1) + 4*a*c^3*x^(3*n + 1)/(3*n + 1) + 4*a^2*c^2*x^(2*n + 1)/(2*n + 1))*a 
^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (126) = 252\).

Time = 0.13 (sec) , antiderivative size = 618, normalized size of antiderivative = 4.90 \[ \int \left (a^2+2 a c x^n+c^2 x^{2 n}\right )^3 \, dx=\frac {720 \, a^{6} n^{6} x + 120 \, c^{6} n^{5} x x^{6 \, n} + 864 \, a c^{5} n^{5} x x^{5 \, n} + 2700 \, a^{2} c^{4} n^{5} x x^{4 \, n} + 4800 \, a^{3} c^{3} n^{5} x x^{3 \, n} + 5400 \, a^{4} c^{2} n^{5} x x^{2 \, n} + 4320 \, a^{5} c n^{5} x x^{n} + 1764 \, a^{6} n^{5} x + 274 \, c^{6} n^{4} x x^{6 \, n} + 1944 \, a c^{5} n^{4} x x^{5 \, n} + 5940 \, a^{2} c^{4} n^{4} x x^{4 \, n} + 10160 \, a^{3} c^{3} n^{4} x x^{3 \, n} + 10530 \, a^{4} c^{2} n^{4} x x^{2 \, n} + 6264 \, a^{5} c n^{4} x x^{n} + 1624 \, a^{6} n^{4} x + 225 \, c^{6} n^{3} x x^{6 \, n} + 1560 \, a c^{5} n^{3} x x^{5 \, n} + 4605 \, a^{2} c^{4} n^{3} x x^{4 \, n} + 7440 \, a^{3} c^{3} n^{3} x x^{3 \, n} + 6915 \, a^{4} c^{2} n^{3} x x^{2 \, n} + 3480 \, a^{5} c n^{3} x x^{n} + 735 \, a^{6} n^{3} x + 85 \, c^{6} n^{2} x x^{6 \, n} + 570 \, a c^{5} n^{2} x x^{5 \, n} + 1605 \, a^{2} c^{4} n^{2} x x^{4 \, n} + 2420 \, a^{3} c^{3} n^{2} x x^{3 \, n} + 2055 \, a^{4} c^{2} n^{2} x x^{2 \, n} + 930 \, a^{5} c n^{2} x x^{n} + 175 \, a^{6} n^{2} x + 15 \, c^{6} n x x^{6 \, n} + 96 \, a c^{5} n x x^{5 \, n} + 255 \, a^{2} c^{4} n x x^{4 \, n} + 360 \, a^{3} c^{3} n x x^{3 \, n} + 285 \, a^{4} c^{2} n x x^{2 \, n} + 120 \, a^{5} c n x x^{n} + 21 \, a^{6} n x + c^{6} x x^{6 \, n} + 6 \, a c^{5} x x^{5 \, n} + 15 \, a^{2} c^{4} x x^{4 \, n} + 20 \, a^{3} c^{3} x x^{3 \, n} + 15 \, a^{4} c^{2} x x^{2 \, n} + 6 \, a^{5} c x x^{n} + a^{6} x}{720 \, n^{6} + 1764 \, n^{5} + 1624 \, n^{4} + 735 \, n^{3} + 175 \, n^{2} + 21 \, n + 1} \] Input:

integrate((a^2+2*a*c*x^n+c^2*x^(2*n))^3,x, algorithm="giac")
 

Output:

(720*a^6*n^6*x + 120*c^6*n^5*x*x^(6*n) + 864*a*c^5*n^5*x*x^(5*n) + 2700*a^ 
2*c^4*n^5*x*x^(4*n) + 4800*a^3*c^3*n^5*x*x^(3*n) + 5400*a^4*c^2*n^5*x*x^(2 
*n) + 4320*a^5*c*n^5*x*x^n + 1764*a^6*n^5*x + 274*c^6*n^4*x*x^(6*n) + 1944 
*a*c^5*n^4*x*x^(5*n) + 5940*a^2*c^4*n^4*x*x^(4*n) + 10160*a^3*c^3*n^4*x*x^ 
(3*n) + 10530*a^4*c^2*n^4*x*x^(2*n) + 6264*a^5*c*n^4*x*x^n + 1624*a^6*n^4* 
x + 225*c^6*n^3*x*x^(6*n) + 1560*a*c^5*n^3*x*x^(5*n) + 4605*a^2*c^4*n^3*x* 
x^(4*n) + 7440*a^3*c^3*n^3*x*x^(3*n) + 6915*a^4*c^2*n^3*x*x^(2*n) + 3480*a 
^5*c*n^3*x*x^n + 735*a^6*n^3*x + 85*c^6*n^2*x*x^(6*n) + 570*a*c^5*n^2*x*x^ 
(5*n) + 1605*a^2*c^4*n^2*x*x^(4*n) + 2420*a^3*c^3*n^2*x*x^(3*n) + 2055*a^4 
*c^2*n^2*x*x^(2*n) + 930*a^5*c*n^2*x*x^n + 175*a^6*n^2*x + 15*c^6*n*x*x^(6 
*n) + 96*a*c^5*n*x*x^(5*n) + 255*a^2*c^4*n*x*x^(4*n) + 360*a^3*c^3*n*x*x^( 
3*n) + 285*a^4*c^2*n*x*x^(2*n) + 120*a^5*c*n*x*x^n + 21*a^6*n*x + c^6*x*x^ 
(6*n) + 6*a*c^5*x*x^(5*n) + 15*a^2*c^4*x*x^(4*n) + 20*a^3*c^3*x*x^(3*n) + 
15*a^4*c^2*x*x^(2*n) + 6*a^5*c*x*x^n + a^6*x)/(720*n^6 + 1764*n^5 + 1624*n 
^4 + 735*n^3 + 175*n^2 + 21*n + 1)
 

Mupad [B] (verification not implemented)

Time = 20.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.95 \[ \int \left (a^2+2 a c x^n+c^2 x^{2 n}\right )^3 \, dx=a^6\,x+\frac {c^6\,x\,x^{6\,n}}{6\,n+1}+\frac {6\,a\,c^5\,x\,x^{5\,n}}{5\,n+1}+\frac {6\,a^5\,c\,x\,x^n}{n+1}+\frac {15\,a^4\,c^2\,x\,x^{2\,n}}{2\,n+1}+\frac {20\,a^3\,c^3\,x\,x^{3\,n}}{3\,n+1}+\frac {15\,a^2\,c^4\,x\,x^{4\,n}}{4\,n+1} \] Input:

int((a^2 + c^2*x^(2*n) + 2*a*c*x^n)^3,x)
 

Output:

a^6*x + (c^6*x*x^(6*n))/(6*n + 1) + (6*a*c^5*x*x^(5*n))/(5*n + 1) + (6*a^5 
*c*x*x^n)/(n + 1) + (15*a^4*c^2*x*x^(2*n))/(2*n + 1) + (20*a^3*c^3*x*x^(3* 
n))/(3*n + 1) + (15*a^2*c^4*x*x^(4*n))/(4*n + 1)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 575, normalized size of antiderivative = 4.56 \[ \int \left (a^2+2 a c x^n+c^2 x^{2 n}\right )^3 \, dx=\frac {x \left (864 x^{5 n} a \,c^{5} n^{5}+1944 x^{5 n} a \,c^{5} n^{4}+1560 x^{5 n} a \,c^{5} n^{3}+570 x^{5 n} a \,c^{5} n^{2}+96 x^{5 n} a \,c^{5} n +2700 x^{4 n} a^{2} c^{4} n^{5}+5940 x^{4 n} a^{2} c^{4} n^{4}+4605 x^{4 n} a^{2} c^{4} n^{3}+1605 x^{4 n} a^{2} c^{4} n^{2}+255 x^{4 n} a^{2} c^{4} n +4800 x^{3 n} a^{3} c^{3} n^{5}+10160 x^{3 n} a^{3} c^{3} n^{4}+7440 x^{3 n} a^{3} c^{3} n^{3}+2420 x^{3 n} a^{3} c^{3} n^{2}+360 x^{3 n} a^{3} c^{3} n +5400 x^{2 n} a^{4} c^{2} n^{5}+10530 x^{2 n} a^{4} c^{2} n^{4}+120 x^{6 n} c^{6} n^{5}+274 x^{6 n} c^{6} n^{4}+225 x^{6 n} c^{6} n^{3}+85 x^{6 n} c^{6} n^{2}+15 x^{6 n} c^{6} n +6 x^{5 n} a \,c^{5}+15 x^{4 n} a^{2} c^{4}+20 x^{3 n} a^{3} c^{3}+15 x^{2 n} a^{4} c^{2}+6 x^{n} a^{5} c +a^{6}+x^{6 n} c^{6}+720 a^{6} n^{6}+1764 a^{6} n^{5}+1624 a^{6} n^{4}+735 a^{6} n^{3}+175 a^{6} n^{2}+21 a^{6} n +6915 x^{2 n} a^{4} c^{2} n^{3}+2055 x^{2 n} a^{4} c^{2} n^{2}+285 x^{2 n} a^{4} c^{2} n +4320 x^{n} a^{5} c \,n^{5}+6264 x^{n} a^{5} c \,n^{4}+3480 x^{n} a^{5} c \,n^{3}+930 x^{n} a^{5} c \,n^{2}+120 x^{n} a^{5} c n \right )}{720 n^{6}+1764 n^{5}+1624 n^{4}+735 n^{3}+175 n^{2}+21 n +1} \] Input:

int((a^2+2*a*c*x^n+c^2*x^(2*n))^3,x)
 

Output:

(x*(120*x**(6*n)*c**6*n**5 + 274*x**(6*n)*c**6*n**4 + 225*x**(6*n)*c**6*n* 
*3 + 85*x**(6*n)*c**6*n**2 + 15*x**(6*n)*c**6*n + x**(6*n)*c**6 + 864*x**( 
5*n)*a*c**5*n**5 + 1944*x**(5*n)*a*c**5*n**4 + 1560*x**(5*n)*a*c**5*n**3 + 
 570*x**(5*n)*a*c**5*n**2 + 96*x**(5*n)*a*c**5*n + 6*x**(5*n)*a*c**5 + 270 
0*x**(4*n)*a**2*c**4*n**5 + 5940*x**(4*n)*a**2*c**4*n**4 + 4605*x**(4*n)*a 
**2*c**4*n**3 + 1605*x**(4*n)*a**2*c**4*n**2 + 255*x**(4*n)*a**2*c**4*n + 
15*x**(4*n)*a**2*c**4 + 4800*x**(3*n)*a**3*c**3*n**5 + 10160*x**(3*n)*a**3 
*c**3*n**4 + 7440*x**(3*n)*a**3*c**3*n**3 + 2420*x**(3*n)*a**3*c**3*n**2 + 
 360*x**(3*n)*a**3*c**3*n + 20*x**(3*n)*a**3*c**3 + 5400*x**(2*n)*a**4*c** 
2*n**5 + 10530*x**(2*n)*a**4*c**2*n**4 + 6915*x**(2*n)*a**4*c**2*n**3 + 20 
55*x**(2*n)*a**4*c**2*n**2 + 285*x**(2*n)*a**4*c**2*n + 15*x**(2*n)*a**4*c 
**2 + 4320*x**n*a**5*c*n**5 + 6264*x**n*a**5*c*n**4 + 3480*x**n*a**5*c*n** 
3 + 930*x**n*a**5*c*n**2 + 120*x**n*a**5*c*n + 6*x**n*a**5*c + 720*a**6*n* 
*6 + 1764*a**6*n**5 + 1624*a**6*n**4 + 735*a**6*n**3 + 175*a**6*n**2 + 21* 
a**6*n + a**6))/(720*n**6 + 1764*n**5 + 1624*n**4 + 735*n**3 + 175*n**2 + 
21*n + 1)