3.124 \(\int x (a+8 x-8 x^2+4 x^3-x^4)^3 \, dx\)

Optimal. Leaf size=134 \[ \frac {a^3 x^2}{2}-\frac {1}{2} \left (a^2-128 a+512\right ) x^6+\frac {4}{5} \left (3 a^2-96 a+128\right ) x^5+8 a^2 x^3-\frac {3}{10} (256-a) x^{10}+\frac {8}{3} (64-a) x^9-4 (70-3 a) x^8+\frac {48}{7} (48-5 a) x^7+6 (8-a) a x^4-\frac {x^{14}}{14}+\frac {12 x^{13}}{13}-6 x^{12}+\frac {280 x^{11}}{11} \]

[Out]

1/2*a^3*x^2+8*a^2*x^3+6*(8-a)*a*x^4+4/5*(3*a^2-96*a+128)*x^5-1/2*(a^2-128*a+512)*x^6+48/7*(48-5*a)*x^7-4*(70-3
*a)*x^8+8/3*(64-a)*x^9-3/10*(256-a)*x^10+280/11*x^11-6*x^12+12/13*x^13-1/14*x^14

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6742} \[ -\frac {1}{2} \left (a^2-128 a+512\right ) x^6+\frac {4}{5} \left (3 a^2-96 a+128\right ) x^5+8 a^2 x^3+\frac {a^3 x^2}{2}-\frac {3}{10} (256-a) x^{10}+\frac {8}{3} (64-a) x^9-4 (70-3 a) x^8+\frac {48}{7} (48-5 a) x^7+6 (8-a) a x^4-\frac {x^{14}}{14}+\frac {12 x^{13}}{13}-6 x^{12}+\frac {280 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*x^2)/2 + 8*a^2*x^3 + 6*(8 - a)*a*x^4 + (4*(128 - 96*a + 3*a^2)*x^5)/5 - ((512 - 128*a + a^2)*x^6)/2 + (48
*(48 - 5*a)*x^7)/7 - 4*(70 - 3*a)*x^8 + (8*(64 - a)*x^9)/3 - (3*(256 - a)*x^10)/10 + (280*x^11)/11 - 6*x^12 +
(12*x^13)/13 - x^14/14

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^3 \, dx &=\int \left (a^3 x+24 a^2 x^2-24 (-8+a) a x^3+4 \left (128-96 a+3 a^2\right ) x^4-3 \left (512-128 a+a^2\right ) x^5-48 (-48+5 a) x^6+32 (-70+3 a) x^7-24 (-64+a) x^8+3 (-256+a) x^9+280 x^{10}-72 x^{11}+12 x^{12}-x^{13}\right ) \, dx\\ &=\frac {a^3 x^2}{2}+8 a^2 x^3+6 (8-a) a x^4+\frac {4}{5} \left (128-96 a+3 a^2\right ) x^5-\frac {1}{2} \left (512-128 a+a^2\right ) x^6+\frac {48}{7} (48-5 a) x^7-4 (70-3 a) x^8+\frac {8}{3} (64-a) x^9-\frac {3}{10} (256-a) x^{10}+\frac {280 x^{11}}{11}-6 x^{12}+\frac {12 x^{13}}{13}-\frac {x^{14}}{14}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 130, normalized size = 0.97 \[ \frac {a^3 x^2}{2}+\frac {1}{2} \left (-a^2+128 a-512\right ) x^6+\frac {4}{5} \left (3 a^2-96 a+128\right ) x^5+8 a^2 x^3+\frac {3}{10} (a-256) x^{10}-\frac {8}{3} (a-64) x^9+4 (3 a-70) x^8-\frac {48}{7} (5 a-48) x^7-6 (a-8) a x^4-\frac {x^{14}}{14}+\frac {12 x^{13}}{13}-6 x^{12}+\frac {280 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*x^2)/2 + 8*a^2*x^3 - 6*(-8 + a)*a*x^4 + (4*(128 - 96*a + 3*a^2)*x^5)/5 + ((-512 + 128*a - a^2)*x^6)/2 - (
48*(-48 + 5*a)*x^7)/7 + 4*(-70 + 3*a)*x^8 - (8*(-64 + a)*x^9)/3 + (3*(-256 + a)*x^10)/10 + (280*x^11)/11 - 6*x
^12 + (12*x^13)/13 - x^14/14

________________________________________________________________________________________

fricas [A]  time = 0.35, size = 133, normalized size = 0.99 \[ -\frac {1}{14} x^{14} + \frac {12}{13} x^{13} - 6 x^{12} + \frac {280}{11} x^{11} + \frac {3}{10} x^{10} a - \frac {384}{5} x^{10} - \frac {8}{3} x^{9} a + \frac {512}{3} x^{9} + 12 x^{8} a - 280 x^{8} - \frac {240}{7} x^{7} a - \frac {1}{2} x^{6} a^{2} + \frac {2304}{7} x^{7} + 64 x^{6} a + \frac {12}{5} x^{5} a^{2} - 256 x^{6} - \frac {384}{5} x^{5} a - 6 x^{4} a^{2} + \frac {512}{5} x^{5} + 48 x^{4} a + 8 x^{3} a^{2} + \frac {1}{2} x^{2} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*x^14 + 12/13*x^13 - 6*x^12 + 280/11*x^11 + 3/10*x^10*a - 384/5*x^10 - 8/3*x^9*a + 512/3*x^9 + 12*x^8*a -
280*x^8 - 240/7*x^7*a - 1/2*x^6*a^2 + 2304/7*x^7 + 64*x^6*a + 12/5*x^5*a^2 - 256*x^6 - 384/5*x^5*a - 6*x^4*a^
2 + 512/5*x^5 + 48*x^4*a + 8*x^3*a^2 + 1/2*x^2*a^3

________________________________________________________________________________________

giac [A]  time = 0.35, size = 133, normalized size = 0.99 \[ -\frac {1}{14} \, x^{14} + \frac {12}{13} \, x^{13} - 6 \, x^{12} + \frac {3}{10} \, a x^{10} + \frac {280}{11} \, x^{11} - \frac {8}{3} \, a x^{9} - \frac {384}{5} \, x^{10} + 12 \, a x^{8} + \frac {512}{3} \, x^{9} - \frac {1}{2} \, a^{2} x^{6} - \frac {240}{7} \, a x^{7} - 280 \, x^{8} + \frac {12}{5} \, a^{2} x^{5} + 64 \, a x^{6} + \frac {2304}{7} \, x^{7} - 6 \, a^{2} x^{4} - \frac {384}{5} \, a x^{5} - 256 \, x^{6} + \frac {1}{2} \, a^{3} x^{2} + 8 \, a^{2} x^{3} + 48 \, a x^{4} + \frac {512}{5} \, x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*x^14 + 12/13*x^13 - 6*x^12 + 3/10*a*x^10 + 280/11*x^11 - 8/3*a*x^9 - 384/5*x^10 + 12*a*x^8 + 512/3*x^9 -
1/2*a^2*x^6 - 240/7*a*x^7 - 280*x^8 + 12/5*a^2*x^5 + 64*a*x^6 + 2304/7*x^7 - 6*a^2*x^4 - 384/5*a*x^5 - 256*x^
6 + 1/2*a^3*x^2 + 8*a^2*x^3 + 48*a*x^4 + 512/5*x^5

________________________________________________________________________________________

maple [A]  time = 0.00, size = 143, normalized size = 1.07 \[ -\frac {x^{14}}{14}+\frac {12 x^{13}}{13}-6 x^{12}+\frac {280 x^{11}}{11}+\frac {\left (3 a -768\right ) x^{10}}{10}+\frac {\left (-24 a +1536\right ) x^{9}}{9}+\frac {\left (96 a -2240\right ) x^{8}}{8}+\frac {\left (-240 a +2304\right ) x^{7}}{7}+\frac {\left (-a^{2}+\left (-2 a +128\right ) a +256 a -1536\right ) x^{6}}{6}+\frac {a^{3} x^{2}}{2}+8 a^{2} x^{3}+\frac {\left (4 a^{2}+\left (8 a -128\right ) a -256 a +512\right ) x^{5}}{5}+\frac {\left (-8 a^{2}+\left (-16 a +64\right ) a +128 a \right ) x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*x^14+12/13*x^13-6*x^12+280/11*x^11+1/10*(3*a-768)*x^10+1/9*(-24*a+1536)*x^9+1/8*(96*a-2240)*x^8+1/7*(-24
0*a+2304)*x^7+1/6*(-a^2+(-2*a+128)*a+256*a-1536)*x^6+1/5*(4*a^2+(8*a-128)*a-256*a+512)*x^5+1/4*(-8*a^2+(-16*a+
64)*a+128*a)*x^4+8*a^2*x^3+1/2*a^3*x^2

________________________________________________________________________________________

maxima [A]  time = 0.62, size = 113, normalized size = 0.84 \[ -\frac {1}{14} \, x^{14} + \frac {12}{13} \, x^{13} - 6 \, x^{12} + \frac {3}{10} \, {\left (a - 256\right )} x^{10} + \frac {280}{11} \, x^{11} - \frac {8}{3} \, {\left (a - 64\right )} x^{9} + 4 \, {\left (3 \, a - 70\right )} x^{8} - \frac {48}{7} \, {\left (5 \, a - 48\right )} x^{7} - \frac {1}{2} \, {\left (a^{2} - 128 \, a + 512\right )} x^{6} + \frac {4}{5} \, {\left (3 \, a^{2} - 96 \, a + 128\right )} x^{5} + \frac {1}{2} \, a^{3} x^{2} + 8 \, a^{2} x^{3} - 6 \, {\left (a^{2} - 8 \, a\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/14*x^14 + 12/13*x^13 - 6*x^12 + 3/10*(a - 256)*x^10 + 280/11*x^11 - 8/3*(a - 64)*x^9 + 4*(3*a - 70)*x^8 - 4
8/7*(5*a - 48)*x^7 - 1/2*(a^2 - 128*a + 512)*x^6 + 4/5*(3*a^2 - 96*a + 128)*x^5 + 1/2*a^3*x^2 + 8*a^2*x^3 - 6*
(a^2 - 8*a)*x^4

________________________________________________________________________________________

mupad [B]  time = 2.12, size = 113, normalized size = 0.84 \[ x^8\,\left (12\,a-280\right )+x^{10}\,\left (\frac {3\,a}{10}-\frac {384}{5}\right )-x^9\,\left (\frac {8\,a}{3}-\frac {512}{3}\right )-x^7\,\left (\frac {240\,a}{7}-\frac {2304}{7}\right )-x^6\,\left (\frac {a^2}{2}-64\,a+256\right )+x^5\,\left (\frac {12\,a^2}{5}-\frac {384\,a}{5}+\frac {512}{5}\right )+\frac {280\,x^{11}}{11}-6\,x^{12}+\frac {12\,x^{13}}{13}-\frac {x^{14}}{14}+8\,a^2\,x^3+\frac {a^3\,x^2}{2}-6\,a\,x^4\,\left (a-8\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^8*(12*a - 280) + x^10*((3*a)/10 - 384/5) - x^9*((8*a)/3 - 512/3) - x^7*((240*a)/7 - 2304/7) - x^6*(a^2/2 - 6
4*a + 256) + x^5*((12*a^2)/5 - (384*a)/5 + 512/5) + (280*x^11)/11 - 6*x^12 + (12*x^13)/13 - x^14/14 + 8*a^2*x^
3 + (a^3*x^2)/2 - 6*a*x^4*(a - 8)

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 128, normalized size = 0.96 \[ \frac {a^{3} x^{2}}{2} + 8 a^{2} x^{3} - \frac {x^{14}}{14} + \frac {12 x^{13}}{13} - 6 x^{12} + \frac {280 x^{11}}{11} + x^{10} \left (\frac {3 a}{10} - \frac {384}{5}\right ) + x^{9} \left (\frac {512}{3} - \frac {8 a}{3}\right ) + x^{8} \left (12 a - 280\right ) + x^{7} \left (\frac {2304}{7} - \frac {240 a}{7}\right ) + x^{6} \left (- \frac {a^{2}}{2} + 64 a - 256\right ) + x^{5} \left (\frac {12 a^{2}}{5} - \frac {384 a}{5} + \frac {512}{5}\right ) + x^{4} \left (- 6 a^{2} + 48 a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**2/2 + 8*a**2*x**3 - x**14/14 + 12*x**13/13 - 6*x**12 + 280*x**11/11 + x**10*(3*a/10 - 384/5) + x**9*(5
12/3 - 8*a/3) + x**8*(12*a - 280) + x**7*(2304/7 - 240*a/7) + x**6*(-a**2/2 + 64*a - 256) + x**5*(12*a**2/5 -
384*a/5 + 512/5) + x**4*(-6*a**2 + 48*a)

________________________________________________________________________________________