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

Optimal. Leaf size=210 \[ \frac{1}{5} \left (3 a^2-1536 a+20480\right ) x^{10}-\frac{16}{3} \left (a^2-128 a+896\right ) x^9-\frac{32}{7} \left (15 a^2-288 a+512\right ) x^7+\frac{2}{3} \left (-a^3+192 a^2-1536 a+1024\right ) x^6+\frac{16}{5} a \left (a^2-48 a+128\right ) x^5+8 (12-a) a^2 x^4+\frac{32 a^3 x^3}{3}+\frac{a^4 x^2}{2}+\frac{2}{7} (640-a) x^{14}-\frac{16}{13} (464-3 a) x^{13}+\frac{8}{3} (524-9 a) x^{12}-\frac{32}{11} (928-35 a) x^{11}+8 (128-3 a) (4-a) x^8+\frac{x^{18}}{18}-\frac{16 x^{17}}{17}+8 x^{16}-\frac{224 x^{15}}{5} \]

[Out]

(a^4*x^2)/2 + (32*a^3*x^3)/3 + 8*(12 - a)*a^2*x^4 + (16*a*(128 - 48*a + a^2)*x^5)/5 + (2*(1024 - 1536*a + 192*
a^2 - a^3)*x^6)/3 - (32*(512 - 288*a + 15*a^2)*x^7)/7 + 8*(128 - 3*a)*(4 - a)*x^8 - (16*(896 - 128*a + a^2)*x^
9)/3 + ((20480 - 1536*a + 3*a^2)*x^10)/5 - (32*(928 - 35*a)*x^11)/11 + (8*(524 - 9*a)*x^12)/3 - (16*(464 - 3*a
)*x^13)/13 + (2*(640 - a)*x^14)/7 - (224*x^15)/5 + 8*x^16 - (16*x^17)/17 + x^18/18

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Rubi [A]  time = 0.225637, antiderivative size = 210, normalized size of antiderivative = 1., 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} \[ \frac{1}{5} \left (3 a^2-1536 a+20480\right ) x^{10}-\frac{16}{3} \left (a^2-128 a+896\right ) x^9-\frac{32}{7} \left (15 a^2-288 a+512\right ) x^7+\frac{2}{3} \left (-a^3+192 a^2-1536 a+1024\right ) x^6+\frac{16}{5} a \left (a^2-48 a+128\right ) x^5+8 (12-a) a^2 x^4+\frac{32 a^3 x^3}{3}+\frac{a^4 x^2}{2}+\frac{2}{7} (640-a) x^{14}-\frac{16}{13} (464-3 a) x^{13}+\frac{8}{3} (524-9 a) x^{12}-\frac{32}{11} (928-35 a) x^{11}+8 (128-3 a) (4-a) x^8+\frac{x^{18}}{18}-\frac{16 x^{17}}{17}+8 x^{16}-\frac{224 x^{15}}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*x^2)/2 + (32*a^3*x^3)/3 + 8*(12 - a)*a^2*x^4 + (16*a*(128 - 48*a + a^2)*x^5)/5 + (2*(1024 - 1536*a + 192*
a^2 - a^3)*x^6)/3 - (32*(512 - 288*a + 15*a^2)*x^7)/7 + 8*(128 - 3*a)*(4 - a)*x^8 - (16*(896 - 128*a + a^2)*x^
9)/3 + ((20480 - 1536*a + 3*a^2)*x^10)/5 - (32*(928 - 35*a)*x^11)/11 + (8*(524 - 9*a)*x^12)/3 - (16*(464 - 3*a
)*x^13)/13 + (2*(640 - a)*x^14)/7 - (224*x^15)/5 + 8*x^16 - (16*x^17)/17 + x^18/18

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 )^4 \, dx &=\int \left (a^4 x+32 a^3 x^2-32 (-12+a) a^2 x^3+16 a \left (128-48 a+a^2\right ) x^4-4 \left (-1024+1536 a-192 a^2+a^3\right ) x^5-32 \left (512-288 a+15 a^2\right ) x^6+64 (128-3 a) (4-a) x^7-48 \left (896-128 a+a^2\right ) x^8+2 \left (20480-1536 a+3 a^2\right ) x^9+32 (-928+35 a) x^{10}-32 (-524+9 a) x^{11}+16 (-464+3 a) x^{12}-4 (-640+a) x^{13}-672 x^{14}+128 x^{15}-16 x^{16}+x^{17}\right ) \, dx\\ &=\frac{a^4 x^2}{2}+\frac{32 a^3 x^3}{3}+8 (12-a) a^2 x^4+\frac{16}{5} a \left (128-48 a+a^2\right ) x^5+\frac{2}{3} \left (1024-1536 a+192 a^2-a^3\right ) x^6-\frac{32}{7} \left (512-288 a+15 a^2\right ) x^7+8 (128-3 a) (4-a) x^8-\frac{16}{3} \left (896-128 a+a^2\right ) x^9+\frac{1}{5} \left (20480-1536 a+3 a^2\right ) x^{10}-\frac{32}{11} (928-35 a) x^{11}+\frac{8}{3} (524-9 a) x^{12}-\frac{16}{13} (464-3 a) x^{13}+\frac{2}{7} (640-a) x^{14}-\frac{224 x^{15}}{5}+8 x^{16}-\frac{16 x^{17}}{17}+\frac{x^{18}}{18}\\ \end{align*}

Mathematica [A]  time = 0.0258348, size = 204, normalized size = 0.97 \[ \frac{1}{5} \left (3 a^2-1536 a+20480\right ) x^{10}-\frac{16}{3} \left (a^2-128 a+896\right ) x^9+8 \left (3 a^2-140 a+512\right ) x^8-\frac{32}{7} \left (15 a^2-288 a+512\right ) x^7-\frac{2}{3} \left (a^3-192 a^2+1536 a-1024\right ) x^6+\frac{16}{5} a \left (a^2-48 a+128\right ) x^5-8 (a-12) a^2 x^4+\frac{32 a^3 x^3}{3}+\frac{a^4 x^2}{2}-\frac{2}{7} (a-640) x^{14}+\frac{16}{13} (3 a-464) x^{13}-\frac{8}{3} (9 a-524) x^{12}+\frac{32}{11} (35 a-928) x^{11}+\frac{x^{18}}{18}-\frac{16 x^{17}}{17}+8 x^{16}-\frac{224 x^{15}}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*x^2)/2 + (32*a^3*x^3)/3 - 8*(-12 + a)*a^2*x^4 + (16*a*(128 - 48*a + a^2)*x^5)/5 - (2*(-1024 + 1536*a - 19
2*a^2 + a^3)*x^6)/3 - (32*(512 - 288*a + 15*a^2)*x^7)/7 + 8*(512 - 140*a + 3*a^2)*x^8 - (16*(896 - 128*a + a^2
)*x^9)/3 + ((20480 - 1536*a + 3*a^2)*x^10)/5 + (32*(-928 + 35*a)*x^11)/11 - (8*(-524 + 9*a)*x^12)/3 + (16*(-46
4 + 3*a)*x^13)/13 - (2*(-640 + a)*x^14)/7 - (224*x^15)/5 + 8*x^16 - (16*x^17)/17 + x^18/18

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Maple [A]  time = 0.001, size = 267, normalized size = 1.3 \begin{align*}{\frac{{x}^{18}}{18}}-{\frac{16\,{x}^{17}}{17}}+8\,{x}^{16}-{\frac{224\,{x}^{15}}{5}}+{\frac{ \left ( -4\,a+2560 \right ){x}^{14}}{14}}+{\frac{ \left ( 48\,a-7424 \right ){x}^{13}}{13}}+{\frac{ \left ( -288\,a+16768 \right ){x}^{12}}{12}}+{\frac{ \left ( 1120\,a-29696 \right ){x}^{11}}{11}}+{\frac{ \left ( 2\,{a}^{2}-2560\,a+24576+ \left ( -2\,a+128 \right ) ^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( -16\,{a}^{2}+3584\,a-10240+2\, \left ( 8\,a-128 \right ) \left ( -2\,a+128 \right ) \right ){x}^{9}}{9}}+{\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}^{8}}{8}}+{\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}^{7}}{7}}+{\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}^{6}}{6}}+{\frac{ \left ( 2\,{a}^{2} \left ( 8\,a-128 \right ) +32\,a \left ( -16\,a+64 \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{a}^{2} \left ( -16\,a+64 \right ) +256\,{a}^{2} \right ){x}^{4}}{4}}+{\frac{32\,{a}^{3}{x}^{3}}{3}}+{\frac{{a}^{4}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x^18-16/17*x^17+8*x^16-224/5*x^15+1/14*(-4*a+2560)*x^14+1/13*(48*a-7424)*x^13+1/12*(-288*a+16768)*x^12+1/
11*(1120*a-29696)*x^11+1/10*(2*a^2-2560*a+24576+(-2*a+128)^2)*x^10+1/9*(-16*a^2+3584*a-10240+2*(8*a-128)*(-2*a
+128))*x^9+1/8*(64*a^2-2560*a+2*(-16*a+64)*(-2*a+128)+(8*a-128)^2)*x^8+1/7*(-160*a^2+32*a*(-2*a+128)+2*(-16*a+
64)*(8*a-128))*x^7+1/6*(2*a^2*(-2*a+128)+32*a*(8*a-128)+(-16*a+64)^2)*x^6+1/5*(2*a^2*(8*a-128)+32*a*(-16*a+64)
)*x^5+1/4*(2*a^2*(-16*a+64)+256*a^2)*x^4+32/3*a^3*x^3+1/2*a^4*x^2

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Maxima [A]  time = 1.10301, size = 246, normalized size = 1.17 \begin{align*} \frac{1}{18} \, x^{18} - \frac{16}{17} \, x^{17} + 8 \, x^{16} - \frac{2}{7} \,{\left (a - 640\right )} x^{14} - \frac{224}{5} \, x^{15} + \frac{16}{13} \,{\left (3 \, a - 464\right )} x^{13} - \frac{8}{3} \,{\left (9 \, a - 524\right )} x^{12} + \frac{32}{11} \,{\left (35 \, a - 928\right )} x^{11} + \frac{1}{5} \,{\left (3 \, a^{2} - 1536 \, a + 20480\right )} x^{10} - \frac{16}{3} \,{\left (a^{2} - 128 \, a + 896\right )} x^{9} + 8 \,{\left (3 \, a^{2} - 140 \, a + 512\right )} x^{8} - \frac{32}{7} \,{\left (15 \, a^{2} - 288 \, a + 512\right )} x^{7} - \frac{2}{3} \,{\left (a^{3} - 192 \, a^{2} + 1536 \, a - 1024\right )} x^{6} + \frac{1}{2} \, a^{4} x^{2} + \frac{32}{3} \, a^{3} x^{3} + \frac{16}{5} \,{\left (a^{3} - 48 \, a^{2} + 128 \, a\right )} x^{5} - 8 \,{\left (a^{3} - 12 \, a^{2}\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x^18 - 16/17*x^17 + 8*x^16 - 2/7*(a - 640)*x^14 - 224/5*x^15 + 16/13*(3*a - 464)*x^13 - 8/3*(9*a - 524)*x
^12 + 32/11*(35*a - 928)*x^11 + 1/5*(3*a^2 - 1536*a + 20480)*x^10 - 16/3*(a^2 - 128*a + 896)*x^9 + 8*(3*a^2 -
140*a + 512)*x^8 - 32/7*(15*a^2 - 288*a + 512)*x^7 - 2/3*(a^3 - 192*a^2 + 1536*a - 1024)*x^6 + 1/2*a^4*x^2 + 3
2/3*a^3*x^3 + 16/5*(a^3 - 48*a^2 + 128*a)*x^5 - 8*(a^3 - 12*a^2)*x^4

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Fricas [A]  time = 1.31534, size = 655, normalized size = 3.12 \begin{align*} \frac{1}{18} x^{18} - \frac{16}{17} x^{17} + 8 x^{16} - \frac{224}{5} x^{15} - \frac{2}{7} x^{14} a + \frac{1280}{7} x^{14} + \frac{48}{13} x^{13} a - \frac{7424}{13} x^{13} - 24 x^{12} a + \frac{4192}{3} x^{12} + \frac{1120}{11} x^{11} a + \frac{3}{5} x^{10} a^{2} - \frac{29696}{11} x^{11} - \frac{1536}{5} x^{10} a - \frac{16}{3} x^{9} a^{2} + 4096 x^{10} + \frac{2048}{3} x^{9} a + 24 x^{8} a^{2} - \frac{14336}{3} x^{9} - 1120 x^{8} a - \frac{480}{7} x^{7} a^{2} - \frac{2}{3} x^{6} a^{3} + 4096 x^{8} + \frac{9216}{7} x^{7} a + 128 x^{6} a^{2} + \frac{16}{5} x^{5} a^{3} - \frac{16384}{7} x^{7} - 1024 x^{6} a - \frac{768}{5} x^{5} a^{2} - 8 x^{4} a^{3} + \frac{2048}{3} x^{6} + \frac{2048}{5} x^{5} a + 96 x^{4} a^{2} + \frac{32}{3} x^{3} a^{3} + \frac{1}{2} x^{2} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x^18 - 16/17*x^17 + 8*x^16 - 224/5*x^15 - 2/7*x^14*a + 1280/7*x^14 + 48/13*x^13*a - 7424/13*x^13 - 24*x^1
2*a + 4192/3*x^12 + 1120/11*x^11*a + 3/5*x^10*a^2 - 29696/11*x^11 - 1536/5*x^10*a - 16/3*x^9*a^2 + 4096*x^10 +
2048/3*x^9*a + 24*x^8*a^2 - 14336/3*x^9 - 1120*x^8*a - 480/7*x^7*a^2 - 2/3*x^6*a^3 + 4096*x^8 + 9216/7*x^7*a
+ 128*x^6*a^2 + 16/5*x^5*a^3 - 16384/7*x^7 - 1024*x^6*a - 768/5*x^5*a^2 - 8*x^4*a^3 + 2048/3*x^6 + 2048/5*x^5*
a + 96*x^4*a^2 + 32/3*x^3*a^3 + 1/2*x^2*a^4

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Sympy [A]  time = 0.14828, size = 212, normalized size = 1.01 \begin{align*} \frac{a^{4} x^{2}}{2} + \frac{32 a^{3} x^{3}}{3} + \frac{x^{18}}{18} - \frac{16 x^{17}}{17} + 8 x^{16} - \frac{224 x^{15}}{5} + x^{14} \left (\frac{1280}{7} - \frac{2 a}{7}\right ) + x^{13} \left (\frac{48 a}{13} - \frac{7424}{13}\right ) + x^{12} \left (\frac{4192}{3} - 24 a\right ) + x^{11} \left (\frac{1120 a}{11} - \frac{29696}{11}\right ) + x^{10} \left (\frac{3 a^{2}}{5} - \frac{1536 a}{5} + 4096\right ) + x^{9} \left (- \frac{16 a^{2}}{3} + \frac{2048 a}{3} - \frac{14336}{3}\right ) + x^{8} \left (24 a^{2} - 1120 a + 4096\right ) + x^{7} \left (- \frac{480 a^{2}}{7} + \frac{9216 a}{7} - \frac{16384}{7}\right ) + x^{6} \left (- \frac{2 a^{3}}{3} + 128 a^{2} - 1024 a + \frac{2048}{3}\right ) + x^{5} \left (\frac{16 a^{3}}{5} - \frac{768 a^{2}}{5} + \frac{2048 a}{5}\right ) + x^{4} \left (- 8 a^{3} + 96 a^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*x**2/2 + 32*a**3*x**3/3 + x**18/18 - 16*x**17/17 + 8*x**16 - 224*x**15/5 + x**14*(1280/7 - 2*a/7) + x**13
*(48*a/13 - 7424/13) + x**12*(4192/3 - 24*a) + x**11*(1120*a/11 - 29696/11) + x**10*(3*a**2/5 - 1536*a/5 + 409
6) + x**9*(-16*a**2/3 + 2048*a/3 - 14336/3) + x**8*(24*a**2 - 1120*a + 4096) + x**7*(-480*a**2/7 + 9216*a/7 -
16384/7) + x**6*(-2*a**3/3 + 128*a**2 - 1024*a + 2048/3) + x**5*(16*a**3/5 - 768*a**2/5 + 2048*a/5) + x**4*(-8
*a**3 + 96*a**2)

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Giac [A]  time = 1.14848, size = 300, normalized size = 1.43 \begin{align*} \frac{1}{18} \, x^{18} - \frac{16}{17} \, x^{17} + 8 \, x^{16} - \frac{2}{7} \, a x^{14} - \frac{224}{5} \, x^{15} + \frac{48}{13} \, a x^{13} + \frac{1280}{7} \, x^{14} - 24 \, a x^{12} - \frac{7424}{13} \, x^{13} + \frac{3}{5} \, a^{2} x^{10} + \frac{1120}{11} \, a x^{11} + \frac{4192}{3} \, x^{12} - \frac{16}{3} \, a^{2} x^{9} - \frac{1536}{5} \, a x^{10} - \frac{29696}{11} \, x^{11} + 24 \, a^{2} x^{8} + \frac{2048}{3} \, a x^{9} + 4096 \, x^{10} - \frac{2}{3} \, a^{3} x^{6} - \frac{480}{7} \, a^{2} x^{7} - 1120 \, a x^{8} - \frac{14336}{3} \, x^{9} + \frac{16}{5} \, a^{3} x^{5} + 128 \, a^{2} x^{6} + \frac{9216}{7} \, a x^{7} + 4096 \, x^{8} - 8 \, a^{3} x^{4} - \frac{768}{5} \, a^{2} x^{5} - 1024 \, a x^{6} - \frac{16384}{7} \, x^{7} + \frac{1}{2} \, a^{4} x^{2} + \frac{32}{3} \, a^{3} x^{3} + 96 \, a^{2} x^{4} + \frac{2048}{5} \, a x^{5} + \frac{2048}{3} \, x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x^18 - 16/17*x^17 + 8*x^16 - 2/7*a*x^14 - 224/5*x^15 + 48/13*a*x^13 + 1280/7*x^14 - 24*a*x^12 - 7424/13*x
^13 + 3/5*a^2*x^10 + 1120/11*a*x^11 + 4192/3*x^12 - 16/3*a^2*x^9 - 1536/5*a*x^10 - 29696/11*x^11 + 24*a^2*x^8
+ 2048/3*a*x^9 + 4096*x^10 - 2/3*a^3*x^6 - 480/7*a^2*x^7 - 1120*a*x^8 - 14336/3*x^9 + 16/5*a^3*x^5 + 128*a^2*x
^6 + 9216/7*a*x^7 + 4096*x^8 - 8*a^3*x^4 - 768/5*a^2*x^5 - 1024*a*x^6 - 16384/7*x^7 + 1/2*a^4*x^2 + 32/3*a^3*x
^3 + 96*a^2*x^4 + 2048/5*a*x^5 + 2048/3*x^6