Integrand size = 20, antiderivative size = 120 \[ \int \left (-b c^3+b \left (c+d x^n\right )^3\right )^2 \, dx=\frac {9 b^2 c^4 d^2 x^{1+2 n}}{1+2 n}+\frac {18 b^2 c^3 d^3 x^{1+3 n}}{1+3 n}+\frac {15 b^2 c^2 d^4 x^{1+4 n}}{1+4 n}+\frac {6 b^2 c d^5 x^{1+5 n}}{1+5 n}+\frac {b^2 d^6 x^{1+6 n}}{1+6 n} \] Output:
9*b^2*c^4*d^2*x^(1+2*n)/(1+2*n)+18*b^2*c^3*d^3*x^(1+3*n)/(1+3*n)+15*b^2*c^ 2*d^4*x^(1+4*n)/(1+4*n)+6*b^2*c*d^5*x^(1+5*n)/(1+5*n)+b^2*d^6*x^(1+6*n)/(1 +6*n)
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.81 \[ \int \left (-b c^3+b \left (c+d x^n\right )^3\right )^2 \, dx=b^2 d^2 x^{1+2 n} \left (\frac {9 c^4}{1+2 n}+\frac {18 c^3 d x^n}{1+3 n}+\frac {15 c^2 d^2 x^{2 n}}{1+4 n}+\frac {6 c d^3 x^{3 n}}{1+5 n}+\frac {d^4 x^{4 n}}{1+6 n}\right ) \] Input:
Integrate[(-(b*c^3) + b*(c + d*x^n)^3)^2,x]
Output:
b^2*d^2*x^(1 + 2*n)*((9*c^4)/(1 + 2*n) + (18*c^3*d*x^n)/(1 + 3*n) + (15*c^ 2*d^2*x^(2*n))/(1 + 4*n) + (6*c*d^3*x^(3*n))/(1 + 5*n) + (d^4*x^(4*n))/(1 + 6*n))
Time = 0.57 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7293, 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.
\(\displaystyle \int \left (b \left (c+d x^n\right )^3-b c^3\right )^2 \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (9 b^2 c^4 d^2 x^{2 n}+18 b^2 c^3 d^3 x^{3 n}+15 b^2 c^2 d^4 x^{4 n}+6 b^2 c d^5 x^{5 n}+b^2 d^6 x^{6 n}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {9 b^2 c^4 d^2 x^{2 n+1}}{2 n+1}+\frac {18 b^2 c^3 d^3 x^{3 n+1}}{3 n+1}+\frac {15 b^2 c^2 d^4 x^{4 n+1}}{4 n+1}+\frac {6 b^2 c d^5 x^{5 n+1}}{5 n+1}+\frac {b^2 d^6 x^{6 n+1}}{6 n+1}\) |
Input:
Int[(-(b*c^3) + b*(c + d*x^n)^3)^2,x]
Output:
(9*b^2*c^4*d^2*x^(1 + 2*n))/(1 + 2*n) + (18*b^2*c^3*d^3*x^(1 + 3*n))/(1 + 3*n) + (15*b^2*c^2*d^4*x^(1 + 4*n))/(1 + 4*n) + (6*b^2*c*d^5*x^(1 + 5*n))/ (1 + 5*n) + (b^2*d^6*x^(1 + 6*n))/(1 + 6*n)
Time = 0.78 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {b^{2} d^{6} x \,x^{6 n}}{1+6 n}+\frac {6 b^{2} c \,d^{5} x \,x^{5 n}}{1+5 n}+\frac {15 b^{2} c^{2} d^{4} x \,x^{4 n}}{1+4 n}+\frac {18 b^{2} c^{3} d^{3} x \,x^{3 n}}{1+3 n}+\frac {9 c^{4} b^{2} d^{2} x \,x^{2 n}}{1+2 n}\) | \(116\) |
norman | \(\frac {b^{2} d^{6} x \,{\mathrm e}^{6 n \ln \left (x \right )}}{1+6 n}+\frac {6 b^{2} c \,d^{5} x \,{\mathrm e}^{5 n \ln \left (x \right )}}{1+5 n}+\frac {15 b^{2} c^{2} d^{4} x \,{\mathrm e}^{4 n \ln \left (x \right )}}{1+4 n}+\frac {18 b^{2} c^{3} d^{3} x \,{\mathrm e}^{3 n \ln \left (x \right )}}{1+3 n}+\frac {9 c^{4} b^{2} d^{2} x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}\) | \(126\) |
parallelrisch | \(\frac {3240 x \,x^{4 n} b^{2} c^{2} d^{4} n^{3}+4320 x \,x^{3 n} b^{2} c^{3} d^{3} n^{4}+90 x \,x^{5 n} b^{2} c \,d^{5} n +864 x \,x^{5 n} b^{2} c \,d^{5} n^{4}+1365 x \,x^{4 n} b^{2} c^{2} d^{4} n^{2}+1872 x \,x^{3 n} b^{2} c^{3} d^{3} n^{2}+3078 x \,x^{2 n} b^{2} c^{4} d^{2} n^{3}+4824 x \,x^{3 n} b^{2} c^{3} d^{3} n^{3}+3240 x \,x^{2 n} b^{2} c^{4} d^{2} n^{4}+240 x \,x^{4 n} b^{2} c^{2} d^{4} n +1080 x \,x^{5 n} b^{2} c \,d^{5} n^{3}+2700 x \,x^{4 n} b^{2} c^{2} d^{4} n^{4}+480 x \,x^{5 n} b^{2} c \,d^{5} n^{2}+306 x \,x^{3 n} b^{2} c^{3} d^{3} n +1071 x \,x^{2 n} b^{2} c^{4} d^{2} n^{2}+162 x \,x^{2 n} b^{2} c^{4} d^{2} n +b^{2} d^{6} x \,x^{6 n}+14 x \,x^{6 n} b^{2} d^{6} n +6 b^{2} c \,d^{5} x \,x^{5 n}+154 x \,x^{6 n} b^{2} d^{6} n^{3}+120 x \,x^{6 n} b^{2} d^{6} n^{4}+71 x \,x^{6 n} b^{2} d^{6} n^{2}+15 b^{2} c^{2} d^{4} x \,x^{4 n}+18 b^{2} c^{3} d^{3} x \,x^{3 n}+9 c^{4} b^{2} d^{2} x \,x^{2 n}}{\left (1+6 n \right ) \left (1+5 n \right ) \left (1+4 n \right ) \left (1+3 n \right ) \left (1+2 n \right )}\) | \(487\) |
orering | \(\text {Expression too large to display}\) | \(1255\) |
Input:
int((-b*c^3+b*(c+d*x^n)^3)^2,x,method=_RETURNVERBOSE)
Output:
b^2*d^6/(1+6*n)*x*(x^n)^6+6*b^2*c*d^5/(1+5*n)*x*(x^n)^5+15*b^2*c^2*d^4/(1+ 4*n)*x*(x^n)^4+18*b^2*c^3*d^3/(1+3*n)*x*(x^n)^3+9*c^4*b^2*d^2/(1+2*n)*x*(x ^n)^2
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (120) = 240\).
Time = 0.11 (sec) , antiderivative size = 368, normalized size of antiderivative = 3.07 \[ \int \left (-b c^3+b \left (c+d x^n\right )^3\right )^2 \, dx=\frac {{\left (120 \, b^{2} d^{6} n^{4} + 154 \, b^{2} d^{6} n^{3} + 71 \, b^{2} d^{6} n^{2} + 14 \, b^{2} d^{6} n + b^{2} d^{6}\right )} x x^{6 \, n} + 6 \, {\left (144 \, b^{2} c d^{5} n^{4} + 180 \, b^{2} c d^{5} n^{3} + 80 \, b^{2} c d^{5} n^{2} + 15 \, b^{2} c d^{5} n + b^{2} c d^{5}\right )} x x^{5 \, n} + 15 \, {\left (180 \, b^{2} c^{2} d^{4} n^{4} + 216 \, b^{2} c^{2} d^{4} n^{3} + 91 \, b^{2} c^{2} d^{4} n^{2} + 16 \, b^{2} c^{2} d^{4} n + b^{2} c^{2} d^{4}\right )} x x^{4 \, n} + 18 \, {\left (240 \, b^{2} c^{3} d^{3} n^{4} + 268 \, b^{2} c^{3} d^{3} n^{3} + 104 \, b^{2} c^{3} d^{3} n^{2} + 17 \, b^{2} c^{3} d^{3} n + b^{2} c^{3} d^{3}\right )} x x^{3 \, n} + 9 \, {\left (360 \, b^{2} c^{4} d^{2} n^{4} + 342 \, b^{2} c^{4} d^{2} n^{3} + 119 \, b^{2} c^{4} d^{2} n^{2} + 18 \, b^{2} c^{4} d^{2} n + b^{2} c^{4} d^{2}\right )} x x^{2 \, n}}{720 \, n^{5} + 1044 \, n^{4} + 580 \, n^{3} + 155 \, n^{2} + 20 \, n + 1} \] Input:
integrate((-b*c^3+b*(c+d*x^n)^3)^2,x, algorithm="fricas")
Output:
((120*b^2*d^6*n^4 + 154*b^2*d^6*n^3 + 71*b^2*d^6*n^2 + 14*b^2*d^6*n + b^2* d^6)*x*x^(6*n) + 6*(144*b^2*c*d^5*n^4 + 180*b^2*c*d^5*n^3 + 80*b^2*c*d^5*n ^2 + 15*b^2*c*d^5*n + b^2*c*d^5)*x*x^(5*n) + 15*(180*b^2*c^2*d^4*n^4 + 216 *b^2*c^2*d^4*n^3 + 91*b^2*c^2*d^4*n^2 + 16*b^2*c^2*d^4*n + b^2*c^2*d^4)*x* x^(4*n) + 18*(240*b^2*c^3*d^3*n^4 + 268*b^2*c^3*d^3*n^3 + 104*b^2*c^3*d^3* n^2 + 17*b^2*c^3*d^3*n + b^2*c^3*d^3)*x*x^(3*n) + 9*(360*b^2*c^4*d^2*n^4 + 342*b^2*c^4*d^2*n^3 + 119*b^2*c^4*d^2*n^2 + 18*b^2*c^4*d^2*n + b^2*c^4*d^ 2)*x*x^(2*n))/(720*n^5 + 1044*n^4 + 580*n^3 + 155*n^2 + 20*n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1535 vs. \(2 (110) = 220\).
Time = 2.54 (sec) , antiderivative size = 1535, normalized size of antiderivative = 12.79 \[ \int \left (-b c^3+b \left (c+d x^n\right )^3\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((-b*c**3+b*(c+d*x**n)**3)**2,x)
Output:
Piecewise((9*b**2*c**4*d**2*log(x) - 36*b**2*c**3*d**3/sqrt(x) - 15*b**2*c **2*d**4/x - 4*b**2*c*d**5/x**(3/2) - b**2*d**6/(2*x**2), Eq(n, -1/2)), (2 7*b**2*c**4*d**2*x**(1/3) + 18*b**2*c**3*d**3*log(x) - 45*b**2*c**2*d**4/x **(1/3) - 9*b**2*c*d**5/x**(2/3) - b**2*d**6/x, Eq(n, -1/3)), (18*b**2*c** 4*d**2*sqrt(x) + 72*b**2*c**3*d**3*x**(1/4) + 15*b**2*c**2*d**4*log(x) - 2 4*b**2*c*d**5/x**(1/4) - 2*b**2*d**6/sqrt(x), Eq(n, -1/4)), (15*b**2*c**4* d**2*x**(3/5) + 45*b**2*c**3*d**3*x**(2/5) + 75*b**2*c**2*d**4*x**(1/5) + 30*b**2*c*d**5*log(x**(1/5)) - 5*b**2*d**6/x**(1/5), Eq(n, -1/5)), (27*b** 2*c**4*d**2*x**(2/3)/2 + 36*b**2*c**3*d**3*sqrt(x) + 45*b**2*c**2*d**4*x** (1/3) + 36*b**2*c*d**5*x**(1/6) + 6*b**2*d**6*log(x**(1/6)), Eq(n, -1/6)), (3240*b**2*c**4*d**2*n**4*x*x**(2*n)/(720*n**5 + 1044*n**4 + 580*n**3 + 1 55*n**2 + 20*n + 1) + 3078*b**2*c**4*d**2*n**3*x*x**(2*n)/(720*n**5 + 1044 *n**4 + 580*n**3 + 155*n**2 + 20*n + 1) + 1071*b**2*c**4*d**2*n**2*x*x**(2 *n)/(720*n**5 + 1044*n**4 + 580*n**3 + 155*n**2 + 20*n + 1) + 162*b**2*c** 4*d**2*n*x*x**(2*n)/(720*n**5 + 1044*n**4 + 580*n**3 + 155*n**2 + 20*n + 1 ) + 9*b**2*c**4*d**2*x*x**(2*n)/(720*n**5 + 1044*n**4 + 580*n**3 + 155*n** 2 + 20*n + 1) + 4320*b**2*c**3*d**3*n**4*x*x**(3*n)/(720*n**5 + 1044*n**4 + 580*n**3 + 155*n**2 + 20*n + 1) + 4824*b**2*c**3*d**3*n**3*x*x**(3*n)/(7 20*n**5 + 1044*n**4 + 580*n**3 + 155*n**2 + 20*n + 1) + 1872*b**2*c**3*d** 3*n**2*x*x**(3*n)/(720*n**5 + 1044*n**4 + 580*n**3 + 155*n**2 + 20*n + ...
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (120) = 240\).
Time = 0.04 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.11 \[ \int \left (-b c^3+b \left (c+d x^n\right )^3\right )^2 \, dx=b^{2} c^{6} x - 2 \, {\left (c^{3} x + \frac {d^{3} x^{3 \, n + 1}}{3 \, n + 1} + \frac {3 \, c d^{2} x^{2 \, n + 1}}{2 \, n + 1} + \frac {3 \, c^{2} d x^{n + 1}}{n + 1}\right )} b^{2} c^{3} + \frac {{\left ({\left (120 \, n^{5} + 274 \, n^{4} + 225 \, n^{3} + 85 \, n^{2} + 15 \, n + 1\right )} d^{6} x x^{6 \, n} + 6 \, {\left (144 \, n^{5} + 324 \, n^{4} + 260 \, n^{3} + 95 \, n^{2} + 16 \, n + 1\right )} c d^{5} x x^{5 \, n} + 15 \, {\left (180 \, n^{5} + 396 \, n^{4} + 307 \, n^{3} + 107 \, n^{2} + 17 \, n + 1\right )} c^{2} d^{4} x x^{4 \, n} + 20 \, {\left (240 \, n^{5} + 508 \, n^{4} + 372 \, n^{3} + 121 \, n^{2} + 18 \, n + 1\right )} c^{3} d^{3} x x^{3 \, n} + 15 \, {\left (360 \, n^{5} + 702 \, n^{4} + 461 \, n^{3} + 137 \, n^{2} + 19 \, n + 1\right )} c^{4} d^{2} x x^{2 \, n} + 6 \, {\left (720 \, n^{5} + 1044 \, n^{4} + 580 \, n^{3} + 155 \, n^{2} + 20 \, n + 1\right )} c^{5} d x x^{n} + {\left (720 \, n^{6} + 1764 \, n^{5} + 1624 \, n^{4} + 735 \, n^{3} + 175 \, n^{2} + 21 \, n + 1\right )} c^{6} x\right )} b^{2}}{720 \, n^{6} + 1764 \, n^{5} + 1624 \, n^{4} + 735 \, n^{3} + 175 \, n^{2} + 21 \, n + 1} \] Input:
integrate((-b*c^3+b*(c+d*x^n)^3)^2,x, algorithm="maxima")
Output:
b^2*c^6*x - 2*(c^3*x + d^3*x^(3*n + 1)/(3*n + 1) + 3*c*d^2*x^(2*n + 1)/(2* n + 1) + 3*c^2*d*x^(n + 1)/(n + 1))*b^2*c^3 + ((120*n^5 + 274*n^4 + 225*n^ 3 + 85*n^2 + 15*n + 1)*d^6*x*x^(6*n) + 6*(144*n^5 + 324*n^4 + 260*n^3 + 95 *n^2 + 16*n + 1)*c*d^5*x*x^(5*n) + 15*(180*n^5 + 396*n^4 + 307*n^3 + 107*n ^2 + 17*n + 1)*c^2*d^4*x*x^(4*n) + 20*(240*n^5 + 508*n^4 + 372*n^3 + 121*n ^2 + 18*n + 1)*c^3*d^3*x*x^(3*n) + 15*(360*n^5 + 702*n^4 + 461*n^3 + 137*n ^2 + 19*n + 1)*c^4*d^2*x*x^(2*n) + 6*(720*n^5 + 1044*n^4 + 580*n^3 + 155*n ^2 + 20*n + 1)*c^5*d*x*x^n + (720*n^6 + 1764*n^5 + 1624*n^4 + 735*n^3 + 17 5*n^2 + 21*n + 1)*c^6*x)*b^2/(720*n^6 + 1764*n^5 + 1624*n^4 + 735*n^3 + 17 5*n^2 + 21*n + 1)
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (120) = 240\).
Time = 0.14 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.98 \[ \int \left (-b c^3+b \left (c+d x^n\right )^3\right )^2 \, dx=\frac {120 \, b^{2} d^{6} n^{4} x x^{6 \, n} + 864 \, b^{2} c d^{5} n^{4} x x^{5 \, n} + 2700 \, b^{2} c^{2} d^{4} n^{4} x x^{4 \, n} + 4320 \, b^{2} c^{3} d^{3} n^{4} x x^{3 \, n} + 3240 \, b^{2} c^{4} d^{2} n^{4} x x^{2 \, n} + 154 \, b^{2} d^{6} n^{3} x x^{6 \, n} + 1080 \, b^{2} c d^{5} n^{3} x x^{5 \, n} + 3240 \, b^{2} c^{2} d^{4} n^{3} x x^{4 \, n} + 4824 \, b^{2} c^{3} d^{3} n^{3} x x^{3 \, n} + 3078 \, b^{2} c^{4} d^{2} n^{3} x x^{2 \, n} + 71 \, b^{2} d^{6} n^{2} x x^{6 \, n} + 480 \, b^{2} c d^{5} n^{2} x x^{5 \, n} + 1365 \, b^{2} c^{2} d^{4} n^{2} x x^{4 \, n} + 1872 \, b^{2} c^{3} d^{3} n^{2} x x^{3 \, n} + 1071 \, b^{2} c^{4} d^{2} n^{2} x x^{2 \, n} + 14 \, b^{2} d^{6} n x x^{6 \, n} + 90 \, b^{2} c d^{5} n x x^{5 \, n} + 240 \, b^{2} c^{2} d^{4} n x x^{4 \, n} + 306 \, b^{2} c^{3} d^{3} n x x^{3 \, n} + 162 \, b^{2} c^{4} d^{2} n x x^{2 \, n} + b^{2} d^{6} x x^{6 \, n} + 6 \, b^{2} c d^{5} x x^{5 \, n} + 15 \, b^{2} c^{2} d^{4} x x^{4 \, n} + 18 \, b^{2} c^{3} d^{3} x x^{3 \, n} + 9 \, b^{2} c^{4} d^{2} x x^{2 \, n}}{720 \, n^{5} + 1044 \, n^{4} + 580 \, n^{3} + 155 \, n^{2} + 20 \, n + 1} \] Input:
integrate((-b*c^3+b*(c+d*x^n)^3)^2,x, algorithm="giac")
Output:
(120*b^2*d^6*n^4*x*x^(6*n) + 864*b^2*c*d^5*n^4*x*x^(5*n) + 2700*b^2*c^2*d^ 4*n^4*x*x^(4*n) + 4320*b^2*c^3*d^3*n^4*x*x^(3*n) + 3240*b^2*c^4*d^2*n^4*x* x^(2*n) + 154*b^2*d^6*n^3*x*x^(6*n) + 1080*b^2*c*d^5*n^3*x*x^(5*n) + 3240* b^2*c^2*d^4*n^3*x*x^(4*n) + 4824*b^2*c^3*d^3*n^3*x*x^(3*n) + 3078*b^2*c^4* d^2*n^3*x*x^(2*n) + 71*b^2*d^6*n^2*x*x^(6*n) + 480*b^2*c*d^5*n^2*x*x^(5*n) + 1365*b^2*c^2*d^4*n^2*x*x^(4*n) + 1872*b^2*c^3*d^3*n^2*x*x^(3*n) + 1071* b^2*c^4*d^2*n^2*x*x^(2*n) + 14*b^2*d^6*n*x*x^(6*n) + 90*b^2*c*d^5*n*x*x^(5 *n) + 240*b^2*c^2*d^4*n*x*x^(4*n) + 306*b^2*c^3*d^3*n*x*x^(3*n) + 162*b^2* c^4*d^2*n*x*x^(2*n) + b^2*d^6*x*x^(6*n) + 6*b^2*c*d^5*x*x^(5*n) + 15*b^2*c ^2*d^4*x*x^(4*n) + 18*b^2*c^3*d^3*x*x^(3*n) + 9*b^2*c^4*d^2*x*x^(2*n))/(72 0*n^5 + 1044*n^4 + 580*n^3 + 155*n^2 + 20*n + 1)
Time = 8.77 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.96 \[ \int \left (-b c^3+b \left (c+d x^n\right )^3\right )^2 \, dx=\frac {b^2\,d^6\,x\,x^{6\,n}}{6\,n+1}+\frac {9\,b^2\,c^4\,d^2\,x\,x^{2\,n}}{2\,n+1}+\frac {18\,b^2\,c^3\,d^3\,x\,x^{3\,n}}{3\,n+1}+\frac {15\,b^2\,c^2\,d^4\,x\,x^{4\,n}}{4\,n+1}+\frac {6\,b^2\,c\,d^5\,x\,x^{5\,n}}{5\,n+1} \] Input:
int((b*c^3 - b*(c + d*x^n)^3)^2,x)
Output:
(b^2*d^6*x*x^(6*n))/(6*n + 1) + (9*b^2*c^4*d^2*x*x^(2*n))/(2*n + 1) + (18* b^2*c^3*d^3*x*x^(3*n))/(3*n + 1) + (15*b^2*c^2*d^4*x*x^(4*n))/(4*n + 1) + (6*b^2*c*d^5*x*x^(5*n))/(5*n + 1)
Time = 0.23 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.75 \[ \int \left (-b c^3+b \left (c+d x^n\right )^3\right )^2 \, dx=\frac {x^{2 n} b^{2} d^{2} x \left (120 x^{4 n} d^{4} n^{4}+154 x^{4 n} d^{4} n^{3}+71 x^{4 n} d^{4} n^{2}+14 x^{4 n} d^{4} n +6 x^{3 n} c \,d^{3}+15 x^{2 n} c^{2} d^{2}+18 x^{n} c^{3} d +1080 x^{3 n} c \,d^{3} n^{3}+480 x^{3 n} c \,d^{3} n^{2}+90 x^{3 n} c \,d^{3} n +2700 x^{2 n} c^{2} d^{2} n^{4}+3240 x^{2 n} c^{2} d^{2} n^{3}+1365 x^{2 n} c^{2} d^{2} n^{2}+240 x^{2 n} c^{2} d^{2} n +4320 x^{n} c^{3} d \,n^{4}+4824 x^{n} c^{3} d \,n^{3}+1872 x^{n} c^{3} d \,n^{2}+306 x^{n} c^{3} d n +864 x^{3 n} c \,d^{3} n^{4}+9 c^{4}+x^{4 n} d^{4}+3240 c^{4} n^{4}+3078 c^{4} n^{3}+1071 c^{4} n^{2}+162 c^{4} n \right )}{720 n^{5}+1044 n^{4}+580 n^{3}+155 n^{2}+20 n +1} \] Input:
int((-b*c^3+b*(c+d*x^n)^3)^2,x)
Output:
(x**(2*n)*b**2*d**2*x*(120*x**(4*n)*d**4*n**4 + 154*x**(4*n)*d**4*n**3 + 7 1*x**(4*n)*d**4*n**2 + 14*x**(4*n)*d**4*n + x**(4*n)*d**4 + 864*x**(3*n)*c *d**3*n**4 + 1080*x**(3*n)*c*d**3*n**3 + 480*x**(3*n)*c*d**3*n**2 + 90*x** (3*n)*c*d**3*n + 6*x**(3*n)*c*d**3 + 2700*x**(2*n)*c**2*d**2*n**4 + 3240*x **(2*n)*c**2*d**2*n**3 + 1365*x**(2*n)*c**2*d**2*n**2 + 240*x**(2*n)*c**2* d**2*n + 15*x**(2*n)*c**2*d**2 + 4320*x**n*c**3*d*n**4 + 4824*x**n*c**3*d* n**3 + 1872*x**n*c**3*d*n**2 + 306*x**n*c**3*d*n + 18*x**n*c**3*d + 3240*c **4*n**4 + 3078*c**4*n**3 + 1071*c**4*n**2 + 162*c**4*n + 9*c**4))/(720*n* *5 + 1044*n**4 + 580*n**3 + 155*n**2 + 20*n + 1)