\(\int (a+8 x-8 x^2+4 x^3-x^4)^3 \, dx\) [117]

Optimal result

Integrand size = 22, antiderivative size = 120 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=a^3 x+12 a^2 x^2+8 (8-a) a x^3+\left (128-96 a+3 a^2\right ) x^4-\frac {3}{5} \left (512-128 a+a^2\right ) x^5+8 (48-5 a) x^6-\frac {32}{7} (70-3 a) x^7+3 (64-a) x^8-\frac {1}{3} (256-a) x^9+28 x^{10}-\frac {72 x^{11}}{11}+x^{12}-\frac {x^{13}}{13} \]

[Out]

a^3*x+12*a^2*x^2+8*(8-a)*a*x^3+(3*a^2-96*a+128)*x^4-3/5*(a^2-128*a+512)*x^5+8*(48-5*a)*x^6-32/7*(70-3*a)*x^7+3
*(64-a)*x^8-1/3*(256-a)*x^9+28*x^10-72/11*x^11+x^12-1/13*x^13

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2086} \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=a^3 x-\frac {3}{5} \left (a^2-128 a+512\right ) x^5+\left (3 a^2-96 a+128\right ) x^4+12 a^2 x^2-\frac {1}{3} (256-a) x^9+3 (64-a) x^8-\frac {32}{7} (70-3 a) x^7+8 (48-5 a) x^6+8 (8-a) a x^3-\frac {x^{13}}{13}+x^{12}-\frac {72 x^{11}}{11}+28 x^{10} \]

[In]

Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^3,x]

[Out]

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

Rule 2086

Int[(P_)^(p_), x_Symbol] :> Int[ExpandToSum[P^p, x], x] /; PolyQ[P, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^3+24 a^2 x+24 (8-a) a x^2+4 \left (128-96 a+3 a^2\right ) x^3-3 \left (512-128 a+a^2\right ) x^4+48 (48-5 a) x^5-32 (70-3 a) x^6+24 (64-a) x^7-3 (256-a) x^8+280 x^9-72 x^{10}+12 x^{11}-x^{12}\right ) \, 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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.95 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=a^3 x+12 a^2 x^2-8 (-8+a) a x^3+\left (128-96 a+3 a^2\right ) x^4-\frac {3}{5} \left (512-128 a+a^2\right ) x^5-8 (-48+5 a) x^6+\frac {32}{7} (-70+3 a) x^7-3 (-64+a) x^8+\frac {1}{3} (-256+a) x^9+28 x^{10}-\frac {72 x^{11}}{11}+x^{12}-\frac {x^{13}}{13} \]

[In]

Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.92

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	\(-\frac {1}{13} x^{13}+x^{12}-\frac {72}{11} x^{11}+28 x^{10}+\frac {1}{3} x^{9} a -\frac {256}{3} x^{9}-3 a \,x^{8}+192 x^{8}+\frac {96}{7} a \,x^{7}-320 x^{7}-40 a \,x^{6}+384 x^{6}-\frac {3}{5} a^{2} x^{5}+\frac {384}{5} a \,x^{5}-\frac {1536}{5} x^{5}+3 a^{2} x^{4}-96 a \,x^{4}+128 x^{4}-8 a^{2} x^{3}+64 a \,x^{3}+12 a^{2} x^{2}+a^{3} x\)	\(129\)
risch	\(-\frac {1}{13} x^{13}+x^{12}-\frac {72}{11} x^{11}+28 x^{10}+\frac {1}{3} x^{9} a -\frac {256}{3} x^{9}-3 a \,x^{8}+192 x^{8}+\frac {96}{7} a \,x^{7}-320 x^{7}-40 a \,x^{6}+384 x^{6}-\frac {3}{5} a^{2} x^{5}+\frac {384}{5} a \,x^{5}-\frac {1536}{5} x^{5}+3 a^{2} x^{4}-96 a \,x^{4}+128 x^{4}-8 a^{2} x^{3}+64 a \,x^{3}+12 a^{2} x^{2}+a^{3} x\)	\(129\)
parallelrisch	\(-\frac {1}{13} x^{13}+x^{12}-\frac {72}{11} x^{11}+28 x^{10}+\frac {1}{3} x^{9} a -\frac {256}{3} x^{9}-3 a \,x^{8}+192 x^{8}+\frac {96}{7} a \,x^{7}-320 x^{7}-40 a \,x^{6}+384 x^{6}-\frac {3}{5} a^{2} x^{5}+\frac {384}{5} a \,x^{5}-\frac {1536}{5} x^{5}+3 a^{2} x^{4}-96 a \,x^{4}+128 x^{4}-8 a^{2} x^{3}+64 a \,x^{3}+12 a^{2} x^{2}+a^{3} x\)	\(129\)
default	\(-\frac {x^{13}}{13}+x^{12}-\frac {72 x^{11}}{11}+28 x^{10}+\frac {\left (3 a -768\right ) x^{9}}{9}+\frac {\left (-24 a +1536\right ) x^{8}}{8}+\frac {\left (96 a -2240\right ) x^{7}}{7}+\frac {\left (-240 a +2304\right ) x^{6}}{6}+\frac {\left (a \left (-2 a +128\right )+256 a -1536-a^{2}\right ) x^{5}}{5}+\frac {\left (a \left (8 a -128\right )-256 a +512+4 a^{2}\right ) x^{4}}{4}+\frac {\left (a \left (-16 a +64\right )+128 a -8 a^{2}\right ) x^{3}}{3}+12 a^{2} x^{2}+a^{3} x\)	\(138\)

[In]

int((-x^4+4*x^3-8*x^2+a+8*x)^3,x,method=_RETURNVERBOSE)

[Out]

-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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.89 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=-\frac {1}{13} \, x^{13} + x^{12} - \frac {72}{11} \, x^{11} + \frac {1}{3} \, {\left (a - 256\right )} x^{9} + 28 \, x^{10} - 3 \, {\left (a - 64\right )} x^{8} + \frac {32}{7} \, {\left (3 \, a - 70\right )} x^{7} - 8 \, {\left (5 \, a - 48\right )} x^{6} - \frac {3}{5} \, {\left (a^{2} - 128 \, a + 512\right )} x^{5} + {\left (3 \, a^{2} - 96 \, a + 128\right )} x^{4} + a^{3} x + 12 \, a^{2} x^{2} - 8 \, {\left (a^{2} - 8 \, a\right )} x^{3} \]

[In]

integrate((-x^4+4*x^3-8*x^2+a+8*x)^3,x, algorithm="fricas")

[Out]

-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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.95 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=a^{3} x + 12 a^{2} x^{2} - \frac {x^{13}}{13} + x^{12} - \frac {72 x^{11}}{11} + 28 x^{10} + x^{9} \left (\frac {a}{3} - \frac {256}{3}\right ) + x^{8} \cdot \left (192 - 3 a\right ) + x^{7} \cdot \left (\frac {96 a}{7} - 320\right ) + x^{6} \cdot \left (384 - 40 a\right ) + x^{5} \left (- \frac {3 a^{2}}{5} + \frac {384 a}{5} - \frac {1536}{5}\right ) + x^{4} \cdot \left (3 a^{2} - 96 a + 128\right ) + x^{3} \left (- 8 a^{2} + 64 a\right ) \]

[In]

integrate((-x**4+4*x**3-8*x**2+a+8*x)**3,x)

[Out]

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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.99 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=-\frac {1}{13} \, x^{13} + x^{12} - \frac {72}{11} \, x^{11} + 28 \, x^{10} - \frac {256}{3} \, x^{9} + 192 \, x^{8} - 320 \, x^{7} + 384 \, x^{6} - \frac {1536}{5} \, x^{5} + a^{3} x + 128 \, x^{4} - \frac {1}{5} \, {\left (3 \, x^{5} - 15 \, x^{4} + 40 \, x^{3} - 60 \, x^{2}\right )} a^{2} + \frac {1}{105} \, {\left (35 \, x^{9} - 315 \, x^{8} + 1440 \, x^{7} - 4200 \, x^{6} + 8064 \, x^{5} - 10080 \, x^{4} + 6720 \, x^{3}\right )} a \]

[In]

integrate((-x^4+4*x^3-8*x^2+a+8*x)^3,x, algorithm="maxima")

[Out]

-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 - 60*x^2)*a^2 + 1/105*(35*x^9 - 315*x^8 + 1440*x^7 - 4200*x^6 + 8064*x^5 -
10080*x^4 + 6720*x^3)*a

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.07 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=-\frac {1}{13} \, x^{13} + x^{12} - \frac {72}{11} \, x^{11} + \frac {1}{3} \, a x^{9} + 28 \, x^{10} - 3 \, a x^{8} - \frac {256}{3} \, x^{9} + \frac {96}{7} \, a x^{7} + 192 \, x^{8} - \frac {3}{5} \, a^{2} x^{5} - 40 \, a x^{6} - 320 \, x^{7} + 3 \, a^{2} x^{4} + \frac {384}{5} \, a x^{5} + 384 \, x^{6} - 8 \, a^{2} x^{3} - 96 \, a x^{4} - \frac {1536}{5} \, x^{5} + a^{3} x + 12 \, a^{2} x^{2} + 64 \, a x^{3} + 128 \, x^{4} \]

[In]

integrate((-x^4+4*x^3-8*x^2+a+8*x)^3,x, algorithm="giac")

[Out]

-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 + 384/5*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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.90 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx=x^9\,\left (\frac {a}{3}-\frac {256}{3}\right )-x^8\,\left (3\,a-192\right )-x^6\,\left (40\,a-384\right )+x^7\,\left (\frac {96\,a}{7}-320\right )+x^4\,\left (3\,a^2-96\,a+128\right )-x^5\,\left (\frac {3\,a^2}{5}-\frac {384\,a}{5}+\frac {1536}{5}\right )+a^3\,x+28\,x^{10}-\frac {72\,x^{11}}{11}+x^{12}-\frac {x^{13}}{13}+12\,a^2\,x^2-8\,a\,x^3\,\left (a-8\right ) \]

[In]

int((a + 8*x - 8*x^2 + 4*x^3 - x^4)^3,x)

[Out]

x^9*(a/3 - 256/3) - x^8*(3*a - 192) - x^6*(40*a - 384) + x^7*((96*a)/7 - 320) + 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)