Integrand size = 23, antiderivative size = 96 \[ \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^4 \, dx=\frac {729 a^4 (3 a+2 b x)^9}{32 b}-\frac {729 a^3 (3 a+2 b x)^{10}}{80 b}+\frac {243 a^2 (3 a+2 b x)^{11}}{176 b}-\frac {3 a (3 a+2 b x)^{12}}{32 b}+\frac {(3 a+2 b x)^{13}}{416 b} \] Output:
729/32*a^4*(2*b*x+3*a)^9/b-729/80*a^3*(2*b*x+3*a)^10/b+243/176*a^2*(2*b*x+ 3*a)^11/b-3/32*a*(2*b*x+3*a)^12/b+1/416*(2*b*x+3*a)^13/b
Time = 0.01 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.36 \[ \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^4 \, dx=531441 a^{12} x+1062882 a^{11} b x^2+1062882 a^{10} b^2 x^3+452709 a^9 b^3 x^4-\frac {413343}{5} a^8 b^4 x^5-157464 a^7 b^5 x^6-34992 a^6 b^6 x^7+17496 a^5 b^7 x^8+7776 a^4 b^8 x^9-\frac {3456}{5} a^3 b^9 x^{10}-\frac {6912}{11} a^2 b^{10} x^{11}+\frac {256 b^{12} x^{13}}{13} \] Input:
Integrate[(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^4,x]
Output:
531441*a^12*x + 1062882*a^11*b*x^2 + 1062882*a^10*b^2*x^3 + 452709*a^9*b^3 *x^4 - (413343*a^8*b^4*x^5)/5 - 157464*a^7*b^5*x^6 - 34992*a^6*b^6*x^7 + 1 7496*a^5*b^7*x^8 + 7776*a^4*b^8*x^9 - (3456*a^3*b^9*x^10)/5 - (6912*a^2*b^ 10*x^11)/11 + (256*b^12*x^13)/13
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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.
\(\displaystyle \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^4 \, dx\) |
\(\Big \downarrow \) 2464 |
\(\displaystyle \int (3 a-b x)^4 (3 a+2 b x)^8dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {6561}{16} a^4 (3 a+2 b x)^8-\frac {729}{4} a^3 (3 a+2 b x)^9+\frac {243}{8} a^2 (3 a+2 b x)^{10}+\frac {1}{16} (3 a+2 b x)^{12}-\frac {9}{4} a (3 a+2 b x)^{11}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {729 a^4 (3 a+2 b x)^9}{32 b}-\frac {729 a^3 (3 a+2 b x)^{10}}{80 b}+\frac {243 a^2 (3 a+2 b x)^{11}}{176 b}+\frac {(3 a+2 b x)^{13}}{416 b}-\frac {3 a (3 a+2 b x)^{12}}{32 b}\) |
Input:
Int[(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^4,x]
Output:
(729*a^4*(3*a + 2*b*x)^9)/(32*b) - (729*a^3*(3*a + 2*b*x)^10)/(80*b) + (24 3*a^2*(3*a + 2*b*x)^11)/(176*b) - (3*a*(3*a + 2*b*x)^12)/(32*b) + (3*a + 2 *b*x)^13/(416*b)
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 = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.29
method | result | size |
gosper | \(\frac {x \left (14080 b^{12} x^{12}-449280 a^{2} b^{10} x^{10}-494208 a^{3} b^{9} x^{9}+5559840 a^{4} b^{8} x^{8}+12509640 a^{5} b^{7} x^{7}-25019280 a^{6} b^{6} x^{6}-112586760 a^{7} b^{5} x^{5}-59108049 a^{8} b^{4} x^{4}+323686935 a^{9} b^{3} x^{3}+759960630 a^{10} b^{2} x^{2}+759960630 a^{11} b x +379980315 a^{12}\right )}{715}\) | \(124\) |
default | \(\frac {256}{13} b^{12} x^{13}-\frac {6912}{11} a^{2} b^{10} x^{11}-\frac {3456}{5} a^{3} b^{9} x^{10}+7776 a^{4} b^{8} x^{9}+17496 a^{5} b^{7} x^{8}-34992 a^{6} b^{6} x^{7}-157464 a^{7} b^{5} x^{6}-\frac {413343}{5} a^{8} b^{4} x^{5}+452709 a^{9} b^{3} x^{4}+1062882 a^{10} b^{2} x^{3}+1062882 a^{11} b \,x^{2}+531441 a^{12} x\) | \(124\) |
norman | \(\frac {256}{13} b^{12} x^{13}-\frac {6912}{11} a^{2} b^{10} x^{11}-\frac {3456}{5} a^{3} b^{9} x^{10}+7776 a^{4} b^{8} x^{9}+17496 a^{5} b^{7} x^{8}-34992 a^{6} b^{6} x^{7}-157464 a^{7} b^{5} x^{6}-\frac {413343}{5} a^{8} b^{4} x^{5}+452709 a^{9} b^{3} x^{4}+1062882 a^{10} b^{2} x^{3}+1062882 a^{11} b \,x^{2}+531441 a^{12} x\) | \(124\) |
risch | \(\frac {256}{13} b^{12} x^{13}-\frac {6912}{11} a^{2} b^{10} x^{11}-\frac {3456}{5} a^{3} b^{9} x^{10}+7776 a^{4} b^{8} x^{9}+17496 a^{5} b^{7} x^{8}-34992 a^{6} b^{6} x^{7}-157464 a^{7} b^{5} x^{6}-\frac {413343}{5} a^{8} b^{4} x^{5}+452709 a^{9} b^{3} x^{4}+1062882 a^{10} b^{2} x^{3}+1062882 a^{11} b \,x^{2}+531441 a^{12} x\) | \(124\) |
parallelrisch | \(\frac {256}{13} b^{12} x^{13}-\frac {6912}{11} a^{2} b^{10} x^{11}-\frac {3456}{5} a^{3} b^{9} x^{10}+7776 a^{4} b^{8} x^{9}+17496 a^{5} b^{7} x^{8}-34992 a^{6} b^{6} x^{7}-157464 a^{7} b^{5} x^{6}-\frac {413343}{5} a^{8} b^{4} x^{5}+452709 a^{9} b^{3} x^{4}+1062882 a^{10} b^{2} x^{3}+1062882 a^{11} b \,x^{2}+531441 a^{12} x\) | \(124\) |
orering | \(\frac {x \left (14080 b^{12} x^{12}-449280 a^{2} b^{10} x^{10}-494208 a^{3} b^{9} x^{9}+5559840 a^{4} b^{8} x^{8}+12509640 a^{5} b^{7} x^{7}-25019280 a^{6} b^{6} x^{6}-112586760 a^{7} b^{5} x^{5}-59108049 a^{8} b^{4} x^{4}+323686935 a^{9} b^{3} x^{3}+759960630 a^{10} b^{2} x^{2}+759960630 a^{11} b x +379980315 a^{12}\right ) \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{4}}{715 \left (-b x +3 a \right )^{4} \left (2 b x +3 a \right )^{8}}\) | \(167\) |
Input:
int((-4*b^3*x^3+27*a^2*b*x+27*a^3)^4,x,method=_RETURNVERBOSE)
Output:
1/715*x*(14080*b^12*x^12-449280*a^2*b^10*x^10-494208*a^3*b^9*x^9+5559840*a ^4*b^8*x^8+12509640*a^5*b^7*x^7-25019280*a^6*b^6*x^6-112586760*a^7*b^5*x^5 -59108049*a^8*b^4*x^4+323686935*a^9*b^3*x^3+759960630*a^10*b^2*x^2+7599606 30*a^11*b*x+379980315*a^12)
Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.28 \[ \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^4 \, dx=\frac {256}{13} \, b^{12} x^{13} - \frac {6912}{11} \, a^{2} b^{10} x^{11} - \frac {3456}{5} \, a^{3} b^{9} x^{10} + 7776 \, a^{4} b^{8} x^{9} + 17496 \, a^{5} b^{7} x^{8} - 34992 \, a^{6} b^{6} x^{7} - 157464 \, a^{7} b^{5} x^{6} - \frac {413343}{5} \, a^{8} b^{4} x^{5} + 452709 \, a^{9} b^{3} x^{4} + 1062882 \, a^{10} b^{2} x^{3} + 1062882 \, a^{11} b x^{2} + 531441 \, a^{12} x \] Input:
integrate((-4*b^3*x^3+27*a^2*b*x+27*a^3)^4,x, algorithm="fricas")
Output:
256/13*b^12*x^13 - 6912/11*a^2*b^10*x^11 - 3456/5*a^3*b^9*x^10 + 7776*a^4* b^8*x^9 + 17496*a^5*b^7*x^8 - 34992*a^6*b^6*x^7 - 157464*a^7*b^5*x^6 - 413 343/5*a^8*b^4*x^5 + 452709*a^9*b^3*x^4 + 1062882*a^10*b^2*x^3 + 1062882*a^ 11*b*x^2 + 531441*a^12*x
Time = 0.05 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.44 \[ \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^4 \, dx=531441 a^{12} x + 1062882 a^{11} b x^{2} + 1062882 a^{10} b^{2} x^{3} + 452709 a^{9} b^{3} x^{4} - \frac {413343 a^{8} b^{4} x^{5}}{5} - 157464 a^{7} b^{5} x^{6} - 34992 a^{6} b^{6} x^{7} + 17496 a^{5} b^{7} x^{8} + 7776 a^{4} b^{8} x^{9} - \frac {3456 a^{3} b^{9} x^{10}}{5} - \frac {6912 a^{2} b^{10} x^{11}}{11} + \frac {256 b^{12} x^{13}}{13} \] Input:
integrate((-4*b**3*x**3+27*a**2*b*x+27*a**3)**4,x)
Output:
531441*a**12*x + 1062882*a**11*b*x**2 + 1062882*a**10*b**2*x**3 + 452709*a **9*b**3*x**4 - 413343*a**8*b**4*x**5/5 - 157464*a**7*b**5*x**6 - 34992*a* *6*b**6*x**7 + 17496*a**5*b**7*x**8 + 7776*a**4*b**8*x**9 - 3456*a**3*b**9 *x**10/5 - 6912*a**2*b**10*x**11/11 + 256*b**12*x**13/13
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.72 \[ \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^4 \, dx=\frac {256}{13} \, b^{12} x^{13} - \frac {6912}{11} \, a^{2} b^{10} x^{11} + 7776 \, a^{4} b^{8} x^{9} - \frac {314928}{7} \, a^{6} b^{6} x^{7} + \frac {531441}{5} \, a^{8} b^{4} x^{5} + 531441 \, a^{12} x - 39366 \, {\left (2 \, b^{3} x^{4} - 27 \, a^{2} b x^{2}\right )} a^{9} + \frac {4374}{35} \, {\left (80 \, b^{6} x^{7} - 1512 \, a^{2} b^{4} x^{5} + 8505 \, a^{4} b^{2} x^{3}\right )} a^{6} - \frac {27}{5} \, {\left (128 \, b^{9} x^{10} - 3240 \, a^{2} b^{7} x^{8} + 29160 \, a^{4} b^{5} x^{6} - 98415 \, a^{6} b^{3} x^{4}\right )} a^{3} \] Input:
integrate((-4*b^3*x^3+27*a^2*b*x+27*a^3)^4,x, algorithm="maxima")
Output:
256/13*b^12*x^13 - 6912/11*a^2*b^10*x^11 + 7776*a^4*b^8*x^9 - 314928/7*a^6 *b^6*x^7 + 531441/5*a^8*b^4*x^5 + 531441*a^12*x - 39366*(2*b^3*x^4 - 27*a^ 2*b*x^2)*a^9 + 4374/35*(80*b^6*x^7 - 1512*a^2*b^4*x^5 + 8505*a^4*b^2*x^3)* a^6 - 27/5*(128*b^9*x^10 - 3240*a^2*b^7*x^8 + 29160*a^4*b^5*x^6 - 98415*a^ 6*b^3*x^4)*a^3
Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.28 \[ \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^4 \, dx=\frac {256}{13} \, b^{12} x^{13} - \frac {6912}{11} \, a^{2} b^{10} x^{11} - \frac {3456}{5} \, a^{3} b^{9} x^{10} + 7776 \, a^{4} b^{8} x^{9} + 17496 \, a^{5} b^{7} x^{8} - 34992 \, a^{6} b^{6} x^{7} - 157464 \, a^{7} b^{5} x^{6} - \frac {413343}{5} \, a^{8} b^{4} x^{5} + 452709 \, a^{9} b^{3} x^{4} + 1062882 \, a^{10} b^{2} x^{3} + 1062882 \, a^{11} b x^{2} + 531441 \, a^{12} x \] Input:
integrate((-4*b^3*x^3+27*a^2*b*x+27*a^3)^4,x, algorithm="giac")
Output:
256/13*b^12*x^13 - 6912/11*a^2*b^10*x^11 - 3456/5*a^3*b^9*x^10 + 7776*a^4* b^8*x^9 + 17496*a^5*b^7*x^8 - 34992*a^6*b^6*x^7 - 157464*a^7*b^5*x^6 - 413 343/5*a^8*b^4*x^5 + 452709*a^9*b^3*x^4 + 1062882*a^10*b^2*x^3 + 1062882*a^ 11*b*x^2 + 531441*a^12*x
Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.28 \[ \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^4 \, dx=531441\,a^{12}\,x+1062882\,a^{11}\,b\,x^2+1062882\,a^{10}\,b^2\,x^3+452709\,a^9\,b^3\,x^4-\frac {413343\,a^8\,b^4\,x^5}{5}-157464\,a^7\,b^5\,x^6-34992\,a^6\,b^6\,x^7+17496\,a^5\,b^7\,x^8+7776\,a^4\,b^8\,x^9-\frac {3456\,a^3\,b^9\,x^{10}}{5}-\frac {6912\,a^2\,b^{10}\,x^{11}}{11}+\frac {256\,b^{12}\,x^{13}}{13} \] Input:
int((27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^4,x)
Output:
531441*a^12*x + (256*b^12*x^13)/13 + 1062882*a^11*b*x^2 + 1062882*a^10*b^2 *x^3 + 452709*a^9*b^3*x^4 - (413343*a^8*b^4*x^5)/5 - 157464*a^7*b^5*x^6 - 34992*a^6*b^6*x^7 + 17496*a^5*b^7*x^8 + 7776*a^4*b^8*x^9 - (3456*a^3*b^9*x ^10)/5 - (6912*a^2*b^10*x^11)/11
Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.28 \[ \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^4 \, dx=\frac {x \left (14080 b^{12} x^{12}-449280 a^{2} b^{10} x^{10}-494208 a^{3} b^{9} x^{9}+5559840 a^{4} b^{8} x^{8}+12509640 a^{5} b^{7} x^{7}-25019280 a^{6} b^{6} x^{6}-112586760 a^{7} b^{5} x^{5}-59108049 a^{8} b^{4} x^{4}+323686935 a^{9} b^{3} x^{3}+759960630 a^{10} b^{2} x^{2}+759960630 a^{11} b x +379980315 a^{12}\right )}{715} \] Input:
int((-4*b^3*x^3+27*a^2*b*x+27*a^3)^4,x)
Output:
(x*(379980315*a**12 + 759960630*a**11*b*x + 759960630*a**10*b**2*x**2 + 32 3686935*a**9*b**3*x**3 - 59108049*a**8*b**4*x**4 - 112586760*a**7*b**5*x** 5 - 25019280*a**6*b**6*x**6 + 12509640*a**5*b**7*x**7 + 5559840*a**4*b**8* x**8 - 494208*a**3*b**9*x**9 - 449280*a**2*b**10*x**10 + 14080*b**12*x**12 ))/715