Integrand size = 25, antiderivative size = 76 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3 \, dx=\frac {9 c^3 (2 c+3 d x)^7}{7 d}-\frac {9 c^2 (2 c+3 d x)^8}{8 d}+\frac {c (2 c+3 d x)^9}{3 d}-\frac {(2 c+3 d x)^{10}}{30 d} \] Output:
9/7*c^3*(3*d*x+2*c)^7/d-9/8*c^2*(3*d*x+2*c)^8/d+1/3*c*(3*d*x+2*c)^9/d-1/30 *(3*d*x+2*c)^10/d
Time = 0.01 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.29 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3 \, dx=64 c^9 x-432 c^7 d^2 x^3-324 c^6 d^3 x^4+\frac {8748}{5} c^5 d^4 x^5+2916 c^4 d^5 x^6-\frac {10935}{7} c^3 d^6 x^7-\frac {59049}{8} c^2 d^7 x^8-6561 c d^8 x^9-\frac {19683 d^9 x^{10}}{10} \] Input:
Integrate[(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^3,x]
Output:
64*c^9*x - 432*c^7*d^2*x^3 - 324*c^6*d^3*x^4 + (8748*c^5*d^4*x^5)/5 + 2916 *c^4*d^5*x^6 - (10935*c^3*d^6*x^7)/7 - (59049*c^2*d^7*x^8)/8 - 6561*c*d^8* x^9 - (19683*d^9*x^10)/10
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2464, 49, 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.
\(\displaystyle \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3 \, dx\) |
\(\Big \downarrow \) 2464 |
\(\displaystyle \int (c-3 d x)^3 (2 c+3 d x)^6dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (27 c^3 (2 c+3 d x)^6-27 c^2 (2 c+3 d x)^7-(2 c+3 d x)^9+9 c (2 c+3 d x)^8\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {9 c^3 (2 c+3 d x)^7}{7 d}-\frac {9 c^2 (2 c+3 d x)^8}{8 d}-\frac {(2 c+3 d x)^{10}}{30 d}+\frac {c (2 c+3 d x)^9}{3 d}\) |
Input:
Int[(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^3,x]
Output:
(9*c^3*(2*c + 3*d*x)^7)/(7*d) - (9*c^2*(2*c + 3*d*x)^8)/(8*d) + (c*(2*c + 3*d*x)^9)/(3*d) - (2*c + 3*d*x)^10/(30*d)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[u*Qx^p, x] / ; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && IGtQ[p, 1]
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.20
method | result | size |
default | \(-\frac {19683}{10} d^{9} x^{10}-6561 c \,d^{8} x^{9}-\frac {59049}{8} c^{2} d^{7} x^{8}-\frac {10935}{7} c^{3} d^{6} x^{7}+2916 c^{4} d^{5} x^{6}+\frac {8748}{5} c^{5} d^{4} x^{5}-324 c^{6} d^{3} x^{4}-432 c^{7} d^{2} x^{3}+64 c^{9} x\) | \(91\) |
norman | \(-\frac {19683}{10} d^{9} x^{10}-6561 c \,d^{8} x^{9}-\frac {59049}{8} c^{2} d^{7} x^{8}-\frac {10935}{7} c^{3} d^{6} x^{7}+2916 c^{4} d^{5} x^{6}+\frac {8748}{5} c^{5} d^{4} x^{5}-324 c^{6} d^{3} x^{4}-432 c^{7} d^{2} x^{3}+64 c^{9} x\) | \(91\) |
risch | \(-\frac {19683}{10} d^{9} x^{10}-6561 c \,d^{8} x^{9}-\frac {59049}{8} c^{2} d^{7} x^{8}-\frac {10935}{7} c^{3} d^{6} x^{7}+2916 c^{4} d^{5} x^{6}+\frac {8748}{5} c^{5} d^{4} x^{5}-324 c^{6} d^{3} x^{4}-432 c^{7} d^{2} x^{3}+64 c^{9} x\) | \(91\) |
parallelrisch | \(-\frac {19683}{10} d^{9} x^{10}-6561 c \,d^{8} x^{9}-\frac {59049}{8} c^{2} d^{7} x^{8}-\frac {10935}{7} c^{3} d^{6} x^{7}+2916 c^{4} d^{5} x^{6}+\frac {8748}{5} c^{5} d^{4} x^{5}-324 c^{6} d^{3} x^{4}-432 c^{7} d^{2} x^{3}+64 c^{9} x\) | \(91\) |
gosper | \(\frac {x \left (-551124 d^{9} x^{9}-1837080 c \,d^{8} x^{8}-2066715 c^{2} d^{7} x^{7}-437400 c^{3} d^{6} x^{6}+816480 c^{4} d^{5} x^{5}+489888 c^{5} d^{4} x^{4}-90720 c^{6} d^{3} x^{3}-120960 c^{7} d^{2} x^{2}+17920 c^{9}\right )}{280}\) | \(93\) |
orering | \(\frac {x \left (-551124 d^{9} x^{9}-1837080 c \,d^{8} x^{8}-2066715 c^{2} d^{7} x^{7}-437400 c^{3} d^{6} x^{6}+816480 c^{4} d^{5} x^{5}+489888 c^{5} d^{4} x^{4}-90720 c^{6} d^{3} x^{3}-120960 c^{7} d^{2} x^{2}+17920 c^{9}\right ) \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{3}}{280 \left (-3 d x +c \right )^{3} \left (3 d x +2 c \right )^{6}}\) | \(136\) |
Input:
int((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x,method=_RETURNVERBOSE)
Output:
-19683/10*d^9*x^10-6561*c*d^8*x^9-59049/8*c^2*d^7*x^8-10935/7*c^3*d^6*x^7+ 2916*c^4*d^5*x^6+8748/5*c^5*d^4*x^5-324*c^6*d^3*x^4-432*c^7*d^2*x^3+64*c^9 *x
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.18 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3 \, dx=-\frac {19683}{10} \, d^{9} x^{10} - 6561 \, c d^{8} x^{9} - \frac {59049}{8} \, c^{2} d^{7} x^{8} - \frac {10935}{7} \, c^{3} d^{6} x^{7} + 2916 \, c^{4} d^{5} x^{6} + \frac {8748}{5} \, c^{5} d^{4} x^{5} - 324 \, c^{6} d^{3} x^{4} - 432 \, c^{7} d^{2} x^{3} + 64 \, c^{9} x \] Input:
integrate((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x, algorithm="fricas")
Output:
-19683/10*d^9*x^10 - 6561*c*d^8*x^9 - 59049/8*c^2*d^7*x^8 - 10935/7*c^3*d^ 6*x^7 + 2916*c^4*d^5*x^6 + 8748/5*c^5*d^4*x^5 - 324*c^6*d^3*x^4 - 432*c^7* d^2*x^3 + 64*c^9*x
Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.34 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3 \, dx=64 c^{9} x - 432 c^{7} d^{2} x^{3} - 324 c^{6} d^{3} x^{4} + \frac {8748 c^{5} d^{4} x^{5}}{5} + 2916 c^{4} d^{5} x^{6} - \frac {10935 c^{3} d^{6} x^{7}}{7} - \frac {59049 c^{2} d^{7} x^{8}}{8} - 6561 c d^{8} x^{9} - \frac {19683 d^{9} x^{10}}{10} \] Input:
integrate((-27*d**3*x**3-27*c*d**2*x**2+4*c**3)**3,x)
Output:
64*c**9*x - 432*c**7*d**2*x**3 - 324*c**6*d**3*x**4 + 8748*c**5*d**4*x**5/ 5 + 2916*c**4*d**5*x**6 - 10935*c**3*d**6*x**7/7 - 59049*c**2*d**7*x**8/8 - 6561*c*d**8*x**9 - 19683*d**9*x**10/10
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.36 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3 \, dx=-\frac {19683}{10} \, d^{9} x^{10} - 6561 \, c d^{8} x^{9} - \frac {59049}{8} \, c^{2} d^{7} x^{8} - \frac {19683}{7} \, c^{3} d^{6} x^{7} + 64 \, c^{9} x - 108 \, {\left (3 \, d^{3} x^{4} + 4 \, c d^{2} x^{3}\right )} c^{6} + \frac {2916}{35} \, {\left (15 \, d^{6} x^{7} + 35 \, c d^{5} x^{6} + 21 \, c^{2} d^{4} x^{5}\right )} c^{3} \] Input:
integrate((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x, algorithm="maxima")
Output:
-19683/10*d^9*x^10 - 6561*c*d^8*x^9 - 59049/8*c^2*d^7*x^8 - 19683/7*c^3*d^ 6*x^7 + 64*c^9*x - 108*(3*d^3*x^4 + 4*c*d^2*x^3)*c^6 + 2916/35*(15*d^6*x^7 + 35*c*d^5*x^6 + 21*c^2*d^4*x^5)*c^3
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.18 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3 \, dx=-\frac {19683}{10} \, d^{9} x^{10} - 6561 \, c d^{8} x^{9} - \frac {59049}{8} \, c^{2} d^{7} x^{8} - \frac {10935}{7} \, c^{3} d^{6} x^{7} + 2916 \, c^{4} d^{5} x^{6} + \frac {8748}{5} \, c^{5} d^{4} x^{5} - 324 \, c^{6} d^{3} x^{4} - 432 \, c^{7} d^{2} x^{3} + 64 \, c^{9} x \] Input:
integrate((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x, algorithm="giac")
Output:
-19683/10*d^9*x^10 - 6561*c*d^8*x^9 - 59049/8*c^2*d^7*x^8 - 10935/7*c^3*d^ 6*x^7 + 2916*c^4*d^5*x^6 + 8748/5*c^5*d^4*x^5 - 324*c^6*d^3*x^4 - 432*c^7* d^2*x^3 + 64*c^9*x
Time = 12.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.18 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3 \, dx=64\,c^9\,x-432\,c^7\,d^2\,x^3-324\,c^6\,d^3\,x^4+\frac {8748\,c^5\,d^4\,x^5}{5}+2916\,c^4\,d^5\,x^6-\frac {10935\,c^3\,d^6\,x^7}{7}-\frac {59049\,c^2\,d^7\,x^8}{8}-6561\,c\,d^8\,x^9-\frac {19683\,d^9\,x^{10}}{10} \] Input:
int(-(27*d^3*x^3 - 4*c^3 + 27*c*d^2*x^2)^3,x)
Output:
64*c^9*x - (19683*d^9*x^10)/10 - 6561*c*d^8*x^9 - 432*c^7*d^2*x^3 - 324*c^ 6*d^3*x^4 + (8748*c^5*d^4*x^5)/5 + 2916*c^4*d^5*x^6 - (10935*c^3*d^6*x^7)/ 7 - (59049*c^2*d^7*x^8)/8
Time = 0.16 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.21 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^3 \, dx=\frac {x \left (-551124 d^{9} x^{9}-1837080 c \,d^{8} x^{8}-2066715 c^{2} d^{7} x^{7}-437400 c^{3} d^{6} x^{6}+816480 c^{4} d^{5} x^{5}+489888 c^{5} d^{4} x^{4}-90720 c^{6} d^{3} x^{3}-120960 c^{7} d^{2} x^{2}+17920 c^{9}\right )}{280} \] Input:
int((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^3,x)
Output:
(x*(17920*c**9 - 120960*c**7*d**2*x**2 - 90720*c**6*d**3*x**3 + 489888*c** 5*d**4*x**4 + 816480*c**4*d**5*x**5 - 437400*c**3*d**6*x**6 - 2066715*c**2 *d**7*x**7 - 1837080*c*d**8*x**8 - 551124*d**9*x**9))/280