Integrand size = 22, antiderivative size = 139 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=\frac {8}{3} (3+a)^3 (1-x)^3-\frac {4}{5} (3-a) (3+a)^2 (1-x)^5-\frac {8}{7} (3+a) (5+3 a) (1-x)^7+\frac {2}{9} \left (37+6 a-3 a^2\right ) (1-x)^9+\frac {8}{11} (5+3 a) (1-x)^{11}-\frac {4}{13} (3-a) (1-x)^{13}-\frac {8}{15} (1-x)^{15}-\frac {1}{17} (1-x)^{17}+(3+a)^4 x \] Output:
8/3*(3+a)^3*(1-x)^3-4/5*(3-a)*(3+a)^2*(1-x)^5-8/7*(3+a)*(5+3*a)*(1-x)^7+2/ 9*(-3*a^2+6*a+37)*(1-x)^9+8/11*(5+3*a)*(1-x)^11-4/13*(3-a)*(1-x)^13-8/15*( 1-x)^15-1/17*(1-x)^17+(3+a)^4*x
Time = 0.03 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.40 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=a^4 x+16 a^3 x^2-\frac {32}{3} (-12+a) a^2 x^3+4 a \left (128-48 a+a^2\right ) x^4-\frac {4}{5} \left (-1024+1536 a-192 a^2+a^3\right ) x^5-\frac {16}{3} \left (512-288 a+15 a^2\right ) x^6+\frac {64}{7} \left (512-140 a+3 a^2\right ) x^7-6 \left (896-128 a+a^2\right ) x^8+\frac {2}{9} \left (20480-1536 a+3 a^2\right ) x^9+\frac {16}{5} (-928+35 a) x^{10}-\frac {32}{11} (-524+9 a) x^{11}+\frac {4}{3} (-464+3 a) x^{12}-\frac {4}{13} (-640+a) x^{13}-48 x^{14}+\frac {128 x^{15}}{15}-x^{16}+\frac {x^{17}}{17} \] Input:
Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^4,x]
Output:
a^4*x + 16*a^3*x^2 - (32*(-12 + a)*a^2*x^3)/3 + 4*a*(128 - 48*a + a^2)*x^4 - (4*(-1024 + 1536*a - 192*a^2 + a^3)*x^5)/5 - (16*(512 - 288*a + 15*a^2) *x^6)/3 + (64*(512 - 140*a + 3*a^2)*x^7)/7 - 6*(896 - 128*a + a^2)*x^8 + ( 2*(20480 - 1536*a + 3*a^2)*x^9)/9 + (16*(-928 + 35*a)*x^10)/5 - (32*(-524 + 9*a)*x^11)/11 + (4*(-464 + 3*a)*x^12)/3 - (4*(-640 + a)*x^13)/13 - 48*x^ 14 + (128*x^15)/15 - x^16 + x^17/17
Time = 0.36 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2458, 1403, 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.
| \(\displaystyle \int \left (a-x^4+4 x^3-8 x^2+8 x\right )^4 \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \left (a-(x-1)^4-2 (x-1)^2+3\right )^4d(x-1)\) |
| \(\Big \downarrow \) 1403 |
| \(\displaystyle \int \left (108 \left (1-\frac {1}{27} a \left (a^2+3 a-9\right )\right ) (x-1)^4+81 \left (\frac {1}{81} a \left (a^3+12 a^2+54 a+108\right )+1\right )+12 \left (1-\frac {a}{3}\right ) (x-1)^{12}-40 \left (\frac {3 a}{5}+1\right ) (x-1)^{10}-74 \left (1-\frac {3}{37} (a-2) a\right ) (x-1)^8+120 \left (\frac {1}{15} a (3 a+14)+1\right ) (x-1)^6-216 \left (a \left (\frac {1}{27} a (a+9)+1\right )+1\right ) (x-1)^2+(x-1)^{16}+8 (x-1)^{14}\right )d(x-1)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{9} \left (-3 a^2+6 a+37\right ) (x-1)^9+\frac {4}{13} (3-a) (x-1)^{13}-\frac {8}{11} (3 a+5) (x-1)^{11}+\frac {8}{7} (a+3) (3 a+5) (x-1)^7+\frac {4}{5} (3-a) (a+3)^2 (x-1)^5-\frac {8}{3} (a+3)^3 (x-1)^3+(a+3)^4 (x-1)+\frac {1}{17} (x-1)^{17}+\frac {8}{15} (x-1)^{15}\) |
Input:
Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^4,x]
Output:
(3 + a)^4*(-1 + x) - (8*(3 + a)^3*(-1 + x)^3)/3 + (4*(3 - a)*(3 + a)^2*(-1 + x)^5)/5 + (8*(3 + a)*(5 + 3*a)*(-1 + x)^7)/7 - (2*(37 + 6*a - 3*a^2)*(- 1 + x)^9)/9 - (8*(5 + 3*a)*(-1 + x)^11)/11 + (4*(3 - a)*(-1 + x)^13)/13 + (8*(-1 + x)^15)/15 + (-1 + x)^17/17
Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandInte grand[(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a *c, 0] && IGtQ[p, 0]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Time = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.29
| method | result | size |
| norman | \(a^{4} x +16 a^{3} x^{2}+\left (-\frac {32}{3} a^{3}+128 a^{2}\right ) x^{3}+\left (4 a^{3}-192 a^{2}+512 a \right ) x^{4}+\left (-\frac {4}{5} a^{3}+\frac {768}{5} a^{2}-\frac {6144}{5} a +\frac {4096}{5}\right ) x^{5}+\left (-80 a^{2}+1536 a -\frac {8192}{3}\right ) x^{6}+\left (\frac {192}{7} a^{2}-1280 a +\frac {32768}{7}\right ) x^{7}+\left (-6 a^{2}+768 a -5376\right ) x^{8}+\left (\frac {2}{3} a^{2}-\frac {1024}{3} a +\frac {40960}{9}\right ) x^{9}+\left (112 a -\frac {14848}{5}\right ) x^{10}+\left (-\frac {288 a}{11}+\frac {16768}{11}\right ) x^{11}+\left (4 a -\frac {1856}{3}\right ) x^{12}+\left (-\frac {4 a}{13}+\frac {2560}{13}\right ) x^{13}-48 x^{14}+\frac {128 x^{15}}{15}-x^{16}+\frac {x^{17}}{17}\) | \(179\) |
| gosper | \(-\frac {6144}{5} a \,x^{5}+128 a^{2} x^{3}-\frac {1856}{3} x^{12}-48 x^{14}-\frac {14848}{5} x^{10}+\frac {40960}{9} x^{9}-x^{16}+\frac {1}{17} x^{17}+\frac {128}{15} x^{15}+\frac {2560}{13} x^{13}+\frac {32768}{7} x^{7}+\frac {4096}{5} x^{5}+\frac {16768}{11} x^{11}+16 a^{3} x^{2}+4 x^{12} a -5376 x^{8}-\frac {8192}{3} x^{6}-\frac {32}{3} a^{3} x^{3}+a^{4} x -\frac {4}{13} x^{13} a +4 a^{3} x^{4}+768 a \,x^{8}+112 x^{10} a -\frac {1024}{3} x^{9} a -6 a^{2} x^{8}-\frac {4}{5} x^{5} a^{3}+1536 x^{6} a -\frac {288}{11} x^{11} a -1280 x^{7} a +\frac {2}{3} a^{2} x^{9}-192 a^{2} x^{4}+512 a \,x^{4}-80 a^{2} x^{6}+\frac {768}{5} a^{2} x^{5}+\frac {192}{7} a^{2} x^{7}\) | \(220\) |
| risch | \(-\frac {6144}{5} a \,x^{5}+128 a^{2} x^{3}-\frac {1856}{3} x^{12}-48 x^{14}-\frac {14848}{5} x^{10}+\frac {40960}{9} x^{9}-x^{16}+\frac {1}{17} x^{17}+\frac {128}{15} x^{15}+\frac {2560}{13} x^{13}+\frac {32768}{7} x^{7}+\frac {4096}{5} x^{5}+\frac {16768}{11} x^{11}+16 a^{3} x^{2}+4 x^{12} a -5376 x^{8}-\frac {8192}{3} x^{6}-\frac {32}{3} a^{3} x^{3}+a^{4} x -\frac {4}{13} x^{13} a +4 a^{3} x^{4}+768 a \,x^{8}+112 x^{10} a -\frac {1024}{3} x^{9} a -6 a^{2} x^{8}-\frac {4}{5} x^{5} a^{3}+1536 x^{6} a -\frac {288}{11} x^{11} a -1280 x^{7} a +\frac {2}{3} a^{2} x^{9}-192 a^{2} x^{4}+512 a \,x^{4}-80 a^{2} x^{6}+\frac {768}{5} a^{2} x^{5}+\frac {192}{7} a^{2} x^{7}\) | \(220\) |
| parallelrisch | \(-\frac {6144}{5} a \,x^{5}+128 a^{2} x^{3}-\frac {1856}{3} x^{12}-48 x^{14}-\frac {14848}{5} x^{10}+\frac {40960}{9} x^{9}-x^{16}+\frac {1}{17} x^{17}+\frac {128}{15} x^{15}+\frac {2560}{13} x^{13}+\frac {32768}{7} x^{7}+\frac {4096}{5} x^{5}+\frac {16768}{11} x^{11}+16 a^{3} x^{2}+4 x^{12} a -5376 x^{8}-\frac {8192}{3} x^{6}-\frac {32}{3} a^{3} x^{3}+a^{4} x -\frac {4}{13} x^{13} a +4 a^{3} x^{4}+768 a \,x^{8}+112 x^{10} a -\frac {1024}{3} x^{9} a -6 a^{2} x^{8}-\frac {4}{5} x^{5} a^{3}+1536 x^{6} a -\frac {288}{11} x^{11} a -1280 x^{7} a +\frac {2}{3} a^{2} x^{9}-192 a^{2} x^{4}+512 a \,x^{4}-80 a^{2} x^{6}+\frac {768}{5} a^{2} x^{5}+\frac {192}{7} a^{2} x^{7}\) | \(220\) |
| orering | \(\frac {x \left (45045 x^{16}-765765 x^{15}+6534528 x^{14}-235620 x^{12} a -36756720 x^{13}+3063060 x^{11} a +150796800 x^{12}-20049120 x^{10} a -473753280 x^{11}+510510 a^{2} x^{8}+85765680 x^{9} a +1167304320 x^{10}-4594590 a^{2} x^{7}-261381120 a \,x^{8}-2274015744 x^{9}+21003840 a^{2} x^{6}+588107520 x^{7} a +3485081600 x^{8}-612612 a^{3} x^{4}-61261200 a^{2} x^{5}-980179200 x^{6} a -4116752640 x^{7}+3063060 a^{3} x^{3}+117621504 a^{2} x^{4}+1176215040 a \,x^{5}+3584655360 x^{6}-8168160 a^{3} x^{2}-147026880 a^{2} x^{3}-940972032 a \,x^{4}-2091048960 x^{5}+765765 a^{4}+12252240 a^{3} x +98017920 a^{2} x^{2}+392071680 a \,x^{3}+627314688 x^{4}\right )}{765765}\) | \(221\) |
| default | \(\frac {x^{17}}{17}-x^{16}+\frac {128 x^{15}}{15}-48 x^{14}+\frac {\left (-4 a +2560\right ) x^{13}}{13}+\frac {\left (48 a -7424\right ) x^{12}}{12}+\frac {\left (-288 a +16768\right ) x^{11}}{11}+\frac {\left (1120 a -29696\right ) x^{10}}{10}+\frac {\left (2 a^{2}-2560 a +24576+\left (-2 a +128\right )^{2}\right ) x^{9}}{9}+\frac {\left (-16 a^{2}+3584 a -10240+2 \left (8 a -128\right ) \left (-2 a +128\right )\right ) x^{8}}{8}+\frac {\left (64 a^{2}-2560 a +2 \left (-16 a +64\right ) \left (-2 a +128\right )+\left (8 a -128\right )^{2}\right ) x^{7}}{7}+\frac {\left (-160 a^{2}+32 a \left (-2 a +128\right )+2 \left (-16 a +64\right ) \left (8 a -128\right )\right ) x^{6}}{6}+\frac {\left (2 a^{2} \left (-2 a +128\right )+32 a \left (8 a -128\right )+\left (-16 a +64\right )^{2}\right ) x^{5}}{5}+\frac {\left (2 a^{2} \left (8 a -128\right )+32 a \left (-16 a +64\right )\right ) x^{4}}{4}+\frac {\left (2 a^{2} \left (-16 a +64\right )+256 a^{2}\right ) x^{3}}{3}+16 a^{3} x^{2}+a^{4} x\) | \(264\) |
Input:
int((-x^4+4*x^3-8*x^2+a+8*x)^4,x,method=_RETURNVERBOSE)
Output:
a^4*x+16*a^3*x^2+(-32/3*a^3+128*a^2)*x^3+(4*a^3-192*a^2+512*a)*x^4+(-4/5*a ^3+768/5*a^2-6144/5*a+4096/5)*x^5+(-80*a^2+1536*a-8192/3)*x^6+(192/7*a^2-1 280*a+32768/7)*x^7+(-6*a^2+768*a-5376)*x^8+(2/3*a^2-1024/3*a+40960/9)*x^9+ (112*a-14848/5)*x^10+(-288/11*a+16768/11)*x^11+(4*a-1856/3)*x^12+(-4/13*a+ 2560/13)*x^13-48*x^14+128/15*x^15-x^16+1/17*x^17
Time = 0.07 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.29 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=\frac {1}{17} \, x^{17} - x^{16} + \frac {128}{15} \, x^{15} - \frac {4}{13} \, {\left (a - 640\right )} x^{13} - 48 \, x^{14} + \frac {4}{3} \, {\left (3 \, a - 464\right )} x^{12} - \frac {32}{11} \, {\left (9 \, a - 524\right )} x^{11} + \frac {16}{5} \, {\left (35 \, a - 928\right )} x^{10} + \frac {2}{9} \, {\left (3 \, a^{2} - 1536 \, a + 20480\right )} x^{9} - 6 \, {\left (a^{2} - 128 \, a + 896\right )} x^{8} + \frac {64}{7} \, {\left (3 \, a^{2} - 140 \, a + 512\right )} x^{7} - \frac {16}{3} \, {\left (15 \, a^{2} - 288 \, a + 512\right )} x^{6} - \frac {4}{5} \, {\left (a^{3} - 192 \, a^{2} + 1536 \, a - 1024\right )} x^{5} + a^{4} x + 16 \, a^{3} x^{2} + 4 \, {\left (a^{3} - 48 \, a^{2} + 128 \, a\right )} x^{4} - \frac {32}{3} \, {\left (a^{3} - 12 \, a^{2}\right )} x^{3} \] Input:
integrate((-x^4+4*x^3-8*x^2+a+8*x)^4,x, algorithm="fricas")
Output:
1/17*x^17 - x^16 + 128/15*x^15 - 4/13*(a - 640)*x^13 - 48*x^14 + 4/3*(3*a - 464)*x^12 - 32/11*(9*a - 524)*x^11 + 16/5*(35*a - 928)*x^10 + 2/9*(3*a^2 - 1536*a + 20480)*x^9 - 6*(a^2 - 128*a + 896)*x^8 + 64/7*(3*a^2 - 140*a + 512)*x^7 - 16/3*(15*a^2 - 288*a + 512)*x^6 - 4/5*(a^3 - 192*a^2 + 1536*a - 1024)*x^5 + a^4*x + 16*a^3*x^2 + 4*(a^3 - 48*a^2 + 128*a)*x^4 - 32/3*(a^ 3 - 12*a^2)*x^3
Time = 0.04 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.43 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=a^{4} x + 16 a^{3} x^{2} + \frac {x^{17}}{17} - x^{16} + \frac {128 x^{15}}{15} - 48 x^{14} + x^{13} \cdot \left (\frac {2560}{13} - \frac {4 a}{13}\right ) + x^{12} \cdot \left (4 a - \frac {1856}{3}\right ) + x^{11} \cdot \left (\frac {16768}{11} - \frac {288 a}{11}\right ) + x^{10} \cdot \left (112 a - \frac {14848}{5}\right ) + x^{9} \cdot \left (\frac {2 a^{2}}{3} - \frac {1024 a}{3} + \frac {40960}{9}\right ) + x^{8} \left (- 6 a^{2} + 768 a - 5376\right ) + x^{7} \cdot \left (\frac {192 a^{2}}{7} - 1280 a + \frac {32768}{7}\right ) + x^{6} \left (- 80 a^{2} + 1536 a - \frac {8192}{3}\right ) + x^{5} \left (- \frac {4 a^{3}}{5} + \frac {768 a^{2}}{5} - \frac {6144 a}{5} + \frac {4096}{5}\right ) + x^{4} \cdot \left (4 a^{3} - 192 a^{2} + 512 a\right ) + x^{3} \left (- \frac {32 a^{3}}{3} + 128 a^{2}\right ) \] Input:
integrate((-x**4+4*x**3-8*x**2+a+8*x)**4,x)
Output:
a**4*x + 16*a**3*x**2 + x**17/17 - x**16 + 128*x**15/15 - 48*x**14 + x**13 *(2560/13 - 4*a/13) + x**12*(4*a - 1856/3) + x**11*(16768/11 - 288*a/11) + x**10*(112*a - 14848/5) + x**9*(2*a**2/3 - 1024*a/3 + 40960/9) + x**8*(-6 *a**2 + 768*a - 5376) + x**7*(192*a**2/7 - 1280*a + 32768/7) + x**6*(-80*a **2 + 1536*a - 8192/3) + x**5*(-4*a**3/5 + 768*a**2/5 - 6144*a/5 + 4096/5) + x**4*(4*a**3 - 192*a**2 + 512*a) + x**3*(-32*a**3/3 + 128*a**2)
Time = 0.03 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.38 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=\frac {1}{17} \, x^{17} - x^{16} + \frac {128}{15} \, x^{15} - 48 \, x^{14} + \frac {2560}{13} \, x^{13} - \frac {1856}{3} \, x^{12} + \frac {16768}{11} \, x^{11} - \frac {14848}{5} \, x^{10} + \frac {40960}{9} \, x^{9} - 5376 \, x^{8} + \frac {32768}{7} \, x^{7} - \frac {8192}{3} \, x^{6} + a^{4} x + \frac {4096}{5} \, x^{5} - \frac {4}{15} \, {\left (3 \, x^{5} - 15 \, x^{4} + 40 \, x^{3} - 60 \, x^{2}\right )} a^{3} + \frac {2}{105} \, {\left (35 \, x^{9} - 315 \, x^{8} + 1440 \, x^{7} - 4200 \, x^{6} + 8064 \, x^{5} - 10080 \, x^{4} + 6720 \, x^{3}\right )} a^{2} - \frac {4}{2145} \, {\left (165 \, x^{13} - 2145 \, x^{12} + 14040 \, x^{11} - 60060 \, x^{10} + 183040 \, x^{9} - 411840 \, x^{8} + 686400 \, x^{7} - 823680 \, x^{6} + 658944 \, x^{5} - 274560 \, x^{4}\right )} a \] Input:
integrate((-x^4+4*x^3-8*x^2+a+8*x)^4,x, algorithm="maxima")
Output:
1/17*x^17 - x^16 + 128/15*x^15 - 48*x^14 + 2560/13*x^13 - 1856/3*x^12 + 16 768/11*x^11 - 14848/5*x^10 + 40960/9*x^9 - 5376*x^8 + 32768/7*x^7 - 8192/3 *x^6 + a^4*x + 4096/5*x^5 - 4/15*(3*x^5 - 15*x^4 + 40*x^3 - 60*x^2)*a^3 + 2/105*(35*x^9 - 315*x^8 + 1440*x^7 - 4200*x^6 + 8064*x^5 - 10080*x^4 + 672 0*x^3)*a^2 - 4/2145*(165*x^13 - 2145*x^12 + 14040*x^11 - 60060*x^10 + 1830 40*x^9 - 411840*x^8 + 686400*x^7 - 823680*x^6 + 658944*x^5 - 274560*x^4)*a
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (103) = 206\).
Time = 0.11 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.58 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=\frac {1}{17} \, x^{17} - x^{16} + \frac {128}{15} \, x^{15} - \frac {4}{13} \, a x^{13} - 48 \, x^{14} + 4 \, a x^{12} + \frac {2560}{13} \, x^{13} - \frac {288}{11} \, a x^{11} - \frac {1856}{3} \, x^{12} + \frac {2}{3} \, a^{2} x^{9} + 112 \, a x^{10} + \frac {16768}{11} \, x^{11} - 6 \, a^{2} x^{8} - \frac {1024}{3} \, a x^{9} - \frac {14848}{5} \, x^{10} + \frac {192}{7} \, a^{2} x^{7} + 768 \, a x^{8} + \frac {40960}{9} \, x^{9} - \frac {4}{5} \, a^{3} x^{5} - 80 \, a^{2} x^{6} - 1280 \, a x^{7} - 5376 \, x^{8} + 4 \, a^{3} x^{4} + \frac {768}{5} \, a^{2} x^{5} + 1536 \, a x^{6} + \frac {32768}{7} \, x^{7} - \frac {32}{3} \, a^{3} x^{3} - 192 \, a^{2} x^{4} - \frac {6144}{5} \, a x^{5} - \frac {8192}{3} \, x^{6} + a^{4} x + 16 \, a^{3} x^{2} + 128 \, a^{2} x^{3} + 512 \, a x^{4} + \frac {4096}{5} \, x^{5} \] Input:
integrate((-x^4+4*x^3-8*x^2+a+8*x)^4,x, algorithm="giac")
Output:
1/17*x^17 - x^16 + 128/15*x^15 - 4/13*a*x^13 - 48*x^14 + 4*a*x^12 + 2560/1 3*x^13 - 288/11*a*x^11 - 1856/3*x^12 + 2/3*a^2*x^9 + 112*a*x^10 + 16768/11 *x^11 - 6*a^2*x^8 - 1024/3*a*x^9 - 14848/5*x^10 + 192/7*a^2*x^7 + 768*a*x^ 8 + 40960/9*x^9 - 4/5*a^3*x^5 - 80*a^2*x^6 - 1280*a*x^7 - 5376*x^8 + 4*a^3 *x^4 + 768/5*a^2*x^5 + 1536*a*x^6 + 32768/7*x^7 - 32/3*a^3*x^3 - 192*a^2*x ^4 - 6144/5*a*x^5 - 8192/3*x^6 + a^4*x + 16*a^3*x^2 + 128*a^2*x^3 + 512*a* x^4 + 4096/5*x^5
Time = 0.37 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.26 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=x^{12}\,\left (4\,a-\frac {1856}{3}\right )-x^{13}\,\left (\frac {4\,a}{13}-\frac {2560}{13}\right )+x^{10}\,\left (112\,a-\frac {14848}{5}\right )-x^{11}\,\left (\frac {288\,a}{11}-\frac {16768}{11}\right )-x^8\,\left (6\,a^2-768\,a+5376\right )-x^6\,\left (80\,a^2-1536\,a+\frac {8192}{3}\right )+x^7\,\left (\frac {192\,a^2}{7}-1280\,a+\frac {32768}{7}\right )+x^9\,\left (\frac {2\,a^2}{3}-\frac {1024\,a}{3}+\frac {40960}{9}\right )-x^5\,\left (\frac {4\,a^3}{5}-\frac {768\,a^2}{5}+\frac {6144\,a}{5}-\frac {4096}{5}\right )+a^4\,x-48\,x^{14}+\frac {128\,x^{15}}{15}-x^{16}+\frac {x^{17}}{17}+16\,a^3\,x^2+4\,a\,x^4\,\left (a^2-48\,a+128\right )-\frac {32\,a^2\,x^3\,\left (a-12\right )}{3} \] Input:
int((a + 8*x - 8*x^2 + 4*x^3 - x^4)^4,x)
Output:
x^12*(4*a - 1856/3) - x^13*((4*a)/13 - 2560/13) + x^10*(112*a - 14848/5) - x^11*((288*a)/11 - 16768/11) - x^8*(6*a^2 - 768*a + 5376) - x^6*(80*a^2 - 1536*a + 8192/3) + x^7*((192*a^2)/7 - 1280*a + 32768/7) + x^9*((2*a^2)/3 - (1024*a)/3 + 40960/9) - x^5*((6144*a)/5 - (768*a^2)/5 + (4*a^3)/5 - 4096 /5) + a^4*x - 48*x^14 + (128*x^15)/15 - x^16 + x^17/17 + 16*a^3*x^2 + 4*a* x^4*(a^2 - 48*a + 128) - (32*a^2*x^3*(a - 12))/3
Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.58 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx=\frac {x \left (45045 x^{16}-765765 x^{15}+6534528 x^{14}-235620 a \,x^{12}-36756720 x^{13}+3063060 a \,x^{11}+150796800 x^{12}-20049120 a \,x^{10}-473753280 x^{11}+510510 a^{2} x^{8}+85765680 a \,x^{9}+1167304320 x^{10}-4594590 a^{2} x^{7}-261381120 a \,x^{8}-2274015744 x^{9}+21003840 a^{2} x^{6}+588107520 a \,x^{7}+3485081600 x^{8}-612612 a^{3} x^{4}-61261200 a^{2} x^{5}-980179200 a \,x^{6}-4116752640 x^{7}+3063060 a^{3} x^{3}+117621504 a^{2} x^{4}+1176215040 a \,x^{5}+3584655360 x^{6}-8168160 a^{3} x^{2}-147026880 a^{2} x^{3}-940972032 a \,x^{4}-2091048960 x^{5}+765765 a^{4}+12252240 a^{3} x +98017920 a^{2} x^{2}+392071680 a \,x^{3}+627314688 x^{4}\right )}{765765} \] Input:
int((-x^4+4*x^3-8*x^2+a+8*x)^4,x)
Output:
(x*(765765*a**4 - 612612*a**3*x**4 + 3063060*a**3*x**3 - 8168160*a**3*x**2 + 12252240*a**3*x + 510510*a**2*x**8 - 4594590*a**2*x**7 + 21003840*a**2* x**6 - 61261200*a**2*x**5 + 117621504*a**2*x**4 - 147026880*a**2*x**3 + 98 017920*a**2*x**2 - 235620*a*x**12 + 3063060*a*x**11 - 20049120*a*x**10 + 8 5765680*a*x**9 - 261381120*a*x**8 + 588107520*a*x**7 - 980179200*a*x**6 + 1176215040*a*x**5 - 940972032*a*x**4 + 392071680*a*x**3 + 45045*x**16 - 76 5765*x**15 + 6534528*x**14 - 36756720*x**13 + 150796800*x**12 - 473753280* x**11 + 1167304320*x**10 - 2274015744*x**9 + 3485081600*x**8 - 4116752640* x**7 + 3584655360*x**6 - 2091048960*x**5 + 627314688*x**4))/765765