Integrand size = 22, antiderivative size = 95 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=2 (3+a)^2 (1-x)^3-\frac {3}{5} (1-a) (3+a) (1-x)^5-\frac {4}{7} (7+3 a) (1-x)^7+\frac {1}{3} (1-a) (1-x)^9+\frac {6}{11} (1-x)^{11}+\frac {1}{13} (1-x)^{13}+(3+a)^3 x \] Output:
2*(3+a)^2*(1-x)^3-3/5*(1-a)*(3+a)*(1-x)^5-4/7*(7+3*a)*(1-x)^7+1/3*(1-a)*(1 -x)^9+6/11*(1-x)^11+1/13*(1-x)^13+(3+a)^3*x
Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.20 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=a^3 x+12 a^2 x^2-8 (-8+a) a x^3+\left (128-96 a+3 a^2\right ) x^4-\frac {3}{5} \left (512-128 a+a^2\right ) x^5-8 (-48+5 a) x^6+\frac {32}{7} (-70+3 a) x^7-3 (-64+a) x^8+\frac {1}{3} (-256+a) x^9+28 x^{10}-\frac {72 x^{11}}{11}+x^{12}-\frac {x^{13}}{13} \] Input:
Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^3,x]
Output:
a^3*x + 12*a^2*x^2 - 8*(-8 + a)*a*x^3 + (128 - 96*a + 3*a^2)*x^4 - (3*(512 - 128*a + a^2)*x^5)/5 - 8*(-48 + 5*a)*x^6 + (32*(-70 + 3*a)*x^7)/7 - 3*(- 64 + a)*x^8 + ((-256 + a)*x^9)/3 + 28*x^10 - (72*x^11)/11 + x^12 - x^13/13
Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89, 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.
\(\displaystyle \int \left (a-x^4+4 x^3-8 x^2+8 x\right )^3 \, dx\) |
\(\Big \downarrow \) 2458 |
\(\displaystyle \int \left (a-(x-1)^4-2 (x-1)^2+3\right )^3d(x-1)\) |
\(\Big \downarrow \) 1403 |
\(\displaystyle \int \left (-3 (1-a) (x-1)^8+28 \left (\frac {3 a}{7}+1\right ) (x-1)^6+9 \left (1-\frac {1}{3} a (a+2)\right ) (x-1)^4-54 \left (\frac {1}{9} a (a+6)+1\right ) (x-1)^2+27 \left (a \left (\frac {1}{27} a (a+9)+1\right )+1\right )-(x-1)^{12}-6 (x-1)^{10}\right )d(x-1)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{3} (1-a) (x-1)^9+\frac {4}{7} (3 a+7) (x-1)^7+\frac {3}{5} (1-a) (a+3) (x-1)^5-2 (a+3)^2 (x-1)^3+(a+3)^3 (x-1)-\frac {1}{13} (x-1)^{13}-\frac {6}{11} (x-1)^{11}\) |
Input:
Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^3,x]
Output:
(3 + a)^3*(-1 + x) - 2*(3 + a)^2*(-1 + x)^3 + (3*(1 - a)*(3 + a)*(-1 + x)^ 5)/5 + (4*(7 + 3*a)*(-1 + x)^7)/7 - ((1 - a)*(-1 + x)^9)/3 - (6*(-1 + x)^1 1)/11 - (-1 + x)^13/13
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 = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16
method | result | size |
norman | \(-\frac {x^{13}}{13}+x^{12}-\frac {72 x^{11}}{11}+28 x^{10}+\left (\frac {a}{3}-\frac {256}{3}\right ) x^{9}+\left (-3 a +192\right ) x^{8}+\left (\frac {96 a}{7}-320\right ) x^{7}+\left (-40 a +384\right ) x^{6}+\left (-\frac {3}{5} a^{2}+\frac {384}{5} a -\frac {1536}{5}\right ) x^{5}+\left (3 a^{2}-96 a +128\right ) x^{4}+\left (-8 a^{2}+64 a \right ) x^{3}+12 a^{2} x^{2}+a^{3} x\) | \(110\) |
gosper | \(-\frac {1}{13} x^{13}+x^{12}-\frac {72}{11} x^{11}+28 x^{10}+\frac {1}{3} x^{9} a -\frac {256}{3} x^{9}-3 a \,x^{8}+192 x^{8}+\frac {96}{7} x^{7} a -320 x^{7}-40 x^{6} a +384 x^{6}-\frac {3}{5} a^{2} x^{5}+\frac {384}{5} a \,x^{5}-\frac {1536}{5} x^{5}+3 a^{2} x^{4}-96 a \,x^{4}+128 x^{4}-8 a^{2} x^{3}+64 a \,x^{3}+12 a^{2} x^{2}+a^{3} x\) | \(129\) |
risch | \(-\frac {1}{13} x^{13}+x^{12}-\frac {72}{11} x^{11}+28 x^{10}+\frac {1}{3} x^{9} a -\frac {256}{3} x^{9}-3 a \,x^{8}+192 x^{8}+\frac {96}{7} x^{7} a -320 x^{7}-40 x^{6} a +384 x^{6}-\frac {3}{5} a^{2} x^{5}+\frac {384}{5} a \,x^{5}-\frac {1536}{5} x^{5}+3 a^{2} x^{4}-96 a \,x^{4}+128 x^{4}-8 a^{2} x^{3}+64 a \,x^{3}+12 a^{2} x^{2}+a^{3} x\) | \(129\) |
parallelrisch | \(-\frac {1}{13} x^{13}+x^{12}-\frac {72}{11} x^{11}+28 x^{10}+\frac {1}{3} x^{9} a -\frac {256}{3} x^{9}-3 a \,x^{8}+192 x^{8}+\frac {96}{7} x^{7} a -320 x^{7}-40 x^{6} a +384 x^{6}-\frac {3}{5} a^{2} x^{5}+\frac {384}{5} a \,x^{5}-\frac {1536}{5} x^{5}+3 a^{2} x^{4}-96 a \,x^{4}+128 x^{4}-8 a^{2} x^{3}+64 a \,x^{3}+12 a^{2} x^{2}+a^{3} x\) | \(129\) |
orering | \(\frac {x \left (-1155 x^{12}+15015 x^{11}-98280 x^{10}+5005 a \,x^{8}+420420 x^{9}-45045 x^{7} a -1281280 x^{8}+205920 x^{6} a +2882880 x^{7}-9009 a^{2} x^{4}-600600 a \,x^{5}-4804800 x^{6}+45045 a^{2} x^{3}+1153152 a \,x^{4}+5765760 x^{5}-120120 a^{2} x^{2}-1441440 a \,x^{3}-4612608 x^{4}+15015 a^{3}+180180 x \,a^{2}+960960 a \,x^{2}+1921920 x^{3}\right )}{15015}\) | \(132\) |
default | \(-\frac {x^{13}}{13}+x^{12}-\frac {72 x^{11}}{11}+28 x^{10}+\frac {\left (3 a -768\right ) x^{9}}{9}+\frac {\left (-24 a +1536\right ) x^{8}}{8}+\frac {\left (96 a -2240\right ) x^{7}}{7}+\frac {\left (-240 a +2304\right ) x^{6}}{6}+\frac {\left (a \left (-2 a +128\right )+256 a -1536-a^{2}\right ) x^{5}}{5}+\frac {\left (a \left (8 a -128\right )-256 a +512+4 a^{2}\right ) x^{4}}{4}+\frac {\left (a \left (-16 a +64\right )+128 a -8 a^{2}\right ) x^{3}}{3}+12 a^{2} x^{2}+a^{3} x\) | \(138\) |
Input:
int((-x^4+4*x^3-8*x^2+a+8*x)^3,x,method=_RETURNVERBOSE)
Output:
-1/13*x^13+x^12-72/11*x^11+28*x^10+(1/3*a-256/3)*x^9+(-3*a+192)*x^8+(96/7* a-320)*x^7+(-40*a+384)*x^6+(-3/5*a^2+384/5*a-1536/5)*x^5+(3*a^2-96*a+128)* x^4+(-8*a^2+64*a)*x^3+12*a^2*x^2+a^3*x
Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=-\frac {1}{13} \, x^{13} + x^{12} - \frac {72}{11} \, x^{11} + \frac {1}{3} \, {\left (a - 256\right )} x^{9} + 28 \, x^{10} - 3 \, {\left (a - 64\right )} x^{8} + \frac {32}{7} \, {\left (3 \, a - 70\right )} x^{7} - 8 \, {\left (5 \, a - 48\right )} x^{6} - \frac {3}{5} \, {\left (a^{2} - 128 \, a + 512\right )} x^{5} + {\left (3 \, a^{2} - 96 \, a + 128\right )} x^{4} + a^{3} x + 12 \, a^{2} x^{2} - 8 \, {\left (a^{2} - 8 \, a\right )} x^{3} \] Input:
integrate((-x^4+4*x^3-8*x^2+a+8*x)^3,x, algorithm="fricas")
Output:
-1/13*x^13 + x^12 - 72/11*x^11 + 1/3*(a - 256)*x^9 + 28*x^10 - 3*(a - 64)* x^8 + 32/7*(3*a - 70)*x^7 - 8*(5*a - 48)*x^6 - 3/5*(a^2 - 128*a + 512)*x^5 + (3*a^2 - 96*a + 128)*x^4 + a^3*x + 12*a^2*x^2 - 8*(a^2 - 8*a)*x^3
Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.20 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=a^{3} x + 12 a^{2} x^{2} - \frac {x^{13}}{13} + x^{12} - \frac {72 x^{11}}{11} + 28 x^{10} + x^{9} \left (\frac {a}{3} - \frac {256}{3}\right ) + x^{8} \cdot \left (192 - 3 a\right ) + x^{7} \cdot \left (\frac {96 a}{7} - 320\right ) + x^{6} \cdot \left (384 - 40 a\right ) + x^{5} \left (- \frac {3 a^{2}}{5} + \frac {384 a}{5} - \frac {1536}{5}\right ) + x^{4} \cdot \left (3 a^{2} - 96 a + 128\right ) + x^{3} \left (- 8 a^{2} + 64 a\right ) \] Input:
integrate((-x**4+4*x**3-8*x**2+a+8*x)**3,x)
Output:
a**3*x + 12*a**2*x**2 - x**13/13 + x**12 - 72*x**11/11 + 28*x**10 + x**9*( a/3 - 256/3) + x**8*(192 - 3*a) + x**7*(96*a/7 - 320) + x**6*(384 - 40*a) + x**5*(-3*a**2/5 + 384*a/5 - 1536/5) + x**4*(3*a**2 - 96*a + 128) + x**3* (-8*a**2 + 64*a)
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=-\frac {1}{13} \, x^{13} + x^{12} - \frac {72}{11} \, x^{11} + 28 \, x^{10} - \frac {256}{3} \, x^{9} + 192 \, x^{8} - 320 \, x^{7} + 384 \, x^{6} - \frac {1536}{5} \, x^{5} + a^{3} x + 128 \, x^{4} - \frac {1}{5} \, {\left (3 \, x^{5} - 15 \, x^{4} + 40 \, x^{3} - 60 \, x^{2}\right )} a^{2} + \frac {1}{105} \, {\left (35 \, x^{9} - 315 \, x^{8} + 1440 \, x^{7} - 4200 \, x^{6} + 8064 \, x^{5} - 10080 \, x^{4} + 6720 \, x^{3}\right )} a \] Input:
integrate((-x^4+4*x^3-8*x^2+a+8*x)^3,x, algorithm="maxima")
Output:
-1/13*x^13 + x^12 - 72/11*x^11 + 28*x^10 - 256/3*x^9 + 192*x^8 - 320*x^7 + 384*x^6 - 1536/5*x^5 + a^3*x + 128*x^4 - 1/5*(3*x^5 - 15*x^4 + 40*x^3 - 6 0*x^2)*a^2 + 1/105*(35*x^9 - 315*x^8 + 1440*x^7 - 4200*x^6 + 8064*x^5 - 10 080*x^4 + 6720*x^3)*a
Time = 0.11 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.35 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=-\frac {1}{13} \, x^{13} + x^{12} - \frac {72}{11} \, x^{11} + \frac {1}{3} \, a x^{9} + 28 \, x^{10} - 3 \, a x^{8} - \frac {256}{3} \, x^{9} + \frac {96}{7} \, a x^{7} + 192 \, x^{8} - \frac {3}{5} \, a^{2} x^{5} - 40 \, a x^{6} - 320 \, x^{7} + 3 \, a^{2} x^{4} + \frac {384}{5} \, a x^{5} + 384 \, x^{6} - 8 \, a^{2} x^{3} - 96 \, a x^{4} - \frac {1536}{5} \, x^{5} + a^{3} x + 12 \, a^{2} x^{2} + 64 \, a x^{3} + 128 \, x^{4} \] Input:
integrate((-x^4+4*x^3-8*x^2+a+8*x)^3,x, algorithm="giac")
Output:
-1/13*x^13 + x^12 - 72/11*x^11 + 1/3*a*x^9 + 28*x^10 - 3*a*x^8 - 256/3*x^9 + 96/7*a*x^7 + 192*x^8 - 3/5*a^2*x^5 - 40*a*x^6 - 320*x^7 + 3*a^2*x^4 + 3 84/5*a*x^5 + 384*x^6 - 8*a^2*x^3 - 96*a*x^4 - 1536/5*x^5 + a^3*x + 12*a^2* x^2 + 64*a*x^3 + 128*x^4
Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.14 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=x^9\,\left (\frac {a}{3}-\frac {256}{3}\right )-x^8\,\left (3\,a-192\right )-x^6\,\left (40\,a-384\right )+x^7\,\left (\frac {96\,a}{7}-320\right )+x^4\,\left (3\,a^2-96\,a+128\right )-x^5\,\left (\frac {3\,a^2}{5}-\frac {384\,a}{5}+\frac {1536}{5}\right )+a^3\,x+28\,x^{10}-\frac {72\,x^{11}}{11}+x^{12}-\frac {x^{13}}{13}+12\,a^2\,x^2-8\,a\,x^3\,\left (a-8\right ) \] Input:
int((a + 8*x - 8*x^2 + 4*x^3 - x^4)^3,x)
Output:
x^9*(a/3 - 256/3) - x^8*(3*a - 192) - x^6*(40*a - 384) + x^7*((96*a)/7 - 3 20) + x^4*(3*a^2 - 96*a + 128) - x^5*((3*a^2)/5 - (384*a)/5 + 1536/5) + a^ 3*x + 28*x^10 - (72*x^11)/11 + x^12 - x^13/13 + 12*a^2*x^2 - 8*a*x^3*(a - 8)
Time = 0.16 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.38 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=\frac {x \left (-1155 x^{12}+15015 x^{11}-98280 x^{10}+5005 a \,x^{8}+420420 x^{9}-45045 a \,x^{7}-1281280 x^{8}+205920 a \,x^{6}+2882880 x^{7}-9009 a^{2} x^{4}-600600 a \,x^{5}-4804800 x^{6}+45045 a^{2} x^{3}+1153152 a \,x^{4}+5765760 x^{5}-120120 a^{2} x^{2}-1441440 a \,x^{3}-4612608 x^{4}+15015 a^{3}+180180 a^{2} x +960960 a \,x^{2}+1921920 x^{3}\right )}{15015} \] Input:
int((-x^4+4*x^3-8*x^2+a+8*x)^3,x)
Output:
(x*(15015*a**3 - 9009*a**2*x**4 + 45045*a**2*x**3 - 120120*a**2*x**2 + 180 180*a**2*x + 5005*a*x**8 - 45045*a*x**7 + 205920*a*x**6 - 600600*a*x**5 + 1153152*a*x**4 - 1441440*a*x**3 + 960960*a*x**2 - 1155*x**12 + 15015*x**11 - 98280*x**10 + 420420*x**9 - 1281280*x**8 + 2882880*x**7 - 4804800*x**6 + 5765760*x**5 - 4612608*x**4 + 1921920*x**3))/15015