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

Optimal. Leaf size=210 \[ \frac{2}{11} \left (3 a^2-1536 a+20480\right ) x^{11}-\frac{24}{5} \left (a^2-128 a+896\right ) x^{10}-4 \left (15 a^2-288 a+512\right ) x^8+\frac{4}{7} \left (-a^3+192 a^2-1536 a+1024\right ) x^7+\frac{8}{3} a \left (a^2-48 a+128\right ) x^6+\frac{32}{5} (12-a) a^2 x^5+8 a^3 x^4+\frac{a^4 x^3}{3}+\frac{4}{15} (640-a) x^{15}-\frac{8}{7} (464-3 a) x^{14}+\frac{32}{13} (524-9 a) x^{13}-\frac{8}{3} (928-35 a) x^{12}+\frac{64}{9} (128-3 a) (4-a) x^9+\frac{x^{19}}{19}-\frac{8 x^{18}}{9}+\frac{128 x^{17}}{17}-42 x^{16} \]

[Out]

(a^4*x^3)/3 + 8*a^3*x^4 + (32*(12 - a)*a^2*x^5)/5 + (8*a*(128 - 48*a + a^2)*x^6)/3 + (4*(1024 - 1536*a + 192*a
^2 - a^3)*x^7)/7 - 4*(512 - 288*a + 15*a^2)*x^8 + (64*(128 - 3*a)*(4 - a)*x^9)/9 - (24*(896 - 128*a + a^2)*x^1
0)/5 + (2*(20480 - 1536*a + 3*a^2)*x^11)/11 - (8*(928 - 35*a)*x^12)/3 + (32*(524 - 9*a)*x^13)/13 - (8*(464 - 3
*a)*x^14)/7 + (4*(640 - a)*x^15)/15 - 42*x^16 + (128*x^17)/17 - (8*x^18)/9 + x^19/19

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Rubi [A]  time = 0.163692, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {6742} \[ \frac{2}{11} \left (3 a^2-1536 a+20480\right ) x^{11}-\frac{24}{5} \left (a^2-128 a+896\right ) x^{10}-4 \left (15 a^2-288 a+512\right ) x^8+\frac{4}{7} \left (-a^3+192 a^2-1536 a+1024\right ) x^7+\frac{8}{3} a \left (a^2-48 a+128\right ) x^6+\frac{32}{5} (12-a) a^2 x^5+8 a^3 x^4+\frac{a^4 x^3}{3}+\frac{4}{15} (640-a) x^{15}-\frac{8}{7} (464-3 a) x^{14}+\frac{32}{13} (524-9 a) x^{13}-\frac{8}{3} (928-35 a) x^{12}+\frac{64}{9} (128-3 a) (4-a) x^9+\frac{x^{19}}{19}-\frac{8 x^{18}}{9}+\frac{128 x^{17}}{17}-42 x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*x^3)/3 + 8*a^3*x^4 + (32*(12 - a)*a^2*x^5)/5 + (8*a*(128 - 48*a + a^2)*x^6)/3 + (4*(1024 - 1536*a + 192*a
^2 - a^3)*x^7)/7 - 4*(512 - 288*a + 15*a^2)*x^8 + (64*(128 - 3*a)*(4 - a)*x^9)/9 - (24*(896 - 128*a + a^2)*x^1
0)/5 + (2*(20480 - 1536*a + 3*a^2)*x^11)/11 - (8*(928 - 35*a)*x^12)/3 + (32*(524 - 9*a)*x^13)/13 - (8*(464 - 3
*a)*x^14)/7 + (4*(640 - a)*x^15)/15 - 42*x^16 + (128*x^17)/17 - (8*x^18)/9 + x^19/19

Rule 6742

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

Rubi steps

\begin{align*} \int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^4 \, dx &=\int \left (a^4 x^2+32 a^3 x^3-32 (-12+a) a^2 x^4+16 a \left (128-48 a+a^2\right ) x^5-4 \left (-1024+1536 a-192 a^2+a^3\right ) x^6-32 \left (512-288 a+15 a^2\right ) x^7+64 (128-3 a) (4-a) x^8-48 \left (896-128 a+a^2\right ) x^9+2 \left (20480-1536 a+3 a^2\right ) x^{10}+32 (-928+35 a) x^{11}-32 (-524+9 a) x^{12}+16 (-464+3 a) x^{13}-4 (-640+a) x^{14}-672 x^{15}+128 x^{16}-16 x^{17}+x^{18}\right ) \, dx\\ &=\frac{a^4 x^3}{3}+8 a^3 x^4+\frac{32}{5} (12-a) a^2 x^5+\frac{8}{3} a \left (128-48 a+a^2\right ) x^6+\frac{4}{7} \left (1024-1536 a+192 a^2-a^3\right ) x^7-4 \left (512-288 a+15 a^2\right ) x^8+\frac{64}{9} (128-3 a) (4-a) x^9-\frac{24}{5} \left (896-128 a+a^2\right ) x^{10}+\frac{2}{11} \left (20480-1536 a+3 a^2\right ) x^{11}-\frac{8}{3} (928-35 a) x^{12}+\frac{32}{13} (524-9 a) x^{13}-\frac{8}{7} (464-3 a) x^{14}+\frac{4}{15} (640-a) x^{15}-42 x^{16}+\frac{128 x^{17}}{17}-\frac{8 x^{18}}{9}+\frac{x^{19}}{19}\\ \end{align*}

Mathematica [A]  time = 0.0269374, size = 204, normalized size = 0.97 \[ \frac{2}{11} \left (3 a^2-1536 a+20480\right ) x^{11}-\frac{24}{5} \left (a^2-128 a+896\right ) x^{10}+\frac{64}{9} \left (3 a^2-140 a+512\right ) x^9-4 \left (15 a^2-288 a+512\right ) x^8-\frac{4}{7} \left (a^3-192 a^2+1536 a-1024\right ) x^7+\frac{8}{3} a \left (a^2-48 a+128\right ) x^6-\frac{32}{5} (a-12) a^2 x^5+8 a^3 x^4+\frac{a^4 x^3}{3}-\frac{4}{15} (a-640) x^{15}+\frac{8}{7} (3 a-464) x^{14}-\frac{32}{13} (9 a-524) x^{13}+\frac{8}{3} (35 a-928) x^{12}+\frac{x^{19}}{19}-\frac{8 x^{18}}{9}+\frac{128 x^{17}}{17}-42 x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*x^3)/3 + 8*a^3*x^4 - (32*(-12 + a)*a^2*x^5)/5 + (8*a*(128 - 48*a + a^2)*x^6)/3 - (4*(-1024 + 1536*a - 192
*a^2 + a^3)*x^7)/7 - 4*(512 - 288*a + 15*a^2)*x^8 + (64*(512 - 140*a + 3*a^2)*x^9)/9 - (24*(896 - 128*a + a^2)
*x^10)/5 + (2*(20480 - 1536*a + 3*a^2)*x^11)/11 + (8*(-928 + 35*a)*x^12)/3 - (32*(-524 + 9*a)*x^13)/13 + (8*(-
464 + 3*a)*x^14)/7 - (4*(-640 + a)*x^15)/15 - 42*x^16 + (128*x^17)/17 - (8*x^18)/9 + x^19/19

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Maple [A]  time = 0.002, size = 267, normalized size = 1.3 \begin{align*}{\frac{{x}^{19}}{19}}-{\frac{8\,{x}^{18}}{9}}+{\frac{128\,{x}^{17}}{17}}-42\,{x}^{16}+{\frac{ \left ( -4\,a+2560 \right ){x}^{15}}{15}}+{\frac{ \left ( 48\,a-7424 \right ){x}^{14}}{14}}+{\frac{ \left ( -288\,a+16768 \right ){x}^{13}}{13}}+{\frac{ \left ( 1120\,a-29696 \right ){x}^{12}}{12}}+{\frac{ \left ( 2\,{a}^{2}-2560\,a+24576+ \left ( -2\,a+128 \right ) ^{2} \right ){x}^{11}}{11}}+{\frac{ \left ( -16\,{a}^{2}+3584\,a-10240+2\, \left ( 8\,a-128 \right ) \left ( -2\,a+128 \right ) \right ){x}^{10}}{10}}+{\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}^{9}}{9}}+{\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}^{8}}{8}}+{\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}^{7}}{7}}+{\frac{ \left ( 2\,{a}^{2} \left ( 8\,a-128 \right ) +32\,a \left ( -16\,a+64 \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 2\,{a}^{2} \left ( -16\,a+64 \right ) +256\,{a}^{2} \right ){x}^{5}}{5}}+8\,{a}^{3}{x}^{4}+{\frac{{a}^{4}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*x^19-8/9*x^18+128/17*x^17-42*x^16+1/15*(-4*a+2560)*x^15+1/14*(48*a-7424)*x^14+1/13*(-288*a+16768)*x^13+1/
12*(1120*a-29696)*x^12+1/11*(2*a^2-2560*a+24576+(-2*a+128)^2)*x^11+1/10*(-16*a^2+3584*a-10240+2*(8*a-128)*(-2*
a+128))*x^10+1/9*(64*a^2-2560*a+2*(-16*a+64)*(-2*a+128)+(8*a-128)^2)*x^9+1/8*(-160*a^2+32*a*(-2*a+128)+2*(-16*
a+64)*(8*a-128))*x^8+1/7*(2*a^2*(-2*a+128)+32*a*(8*a-128)+(-16*a+64)^2)*x^7+1/6*(2*a^2*(8*a-128)+32*a*(-16*a+6
4))*x^6+1/5*(2*a^2*(-16*a+64)+256*a^2)*x^5+8*a^3*x^4+1/3*a^4*x^3

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Maxima [A]  time = 1.03212, size = 246, normalized size = 1.17 \begin{align*} \frac{1}{19} \, x^{19} - \frac{8}{9} \, x^{18} + \frac{128}{17} \, x^{17} - \frac{4}{15} \,{\left (a - 640\right )} x^{15} - 42 \, x^{16} + \frac{8}{7} \,{\left (3 \, a - 464\right )} x^{14} - \frac{32}{13} \,{\left (9 \, a - 524\right )} x^{13} + \frac{8}{3} \,{\left (35 \, a - 928\right )} x^{12} + \frac{2}{11} \,{\left (3 \, a^{2} - 1536 \, a + 20480\right )} x^{11} - \frac{24}{5} \,{\left (a^{2} - 128 \, a + 896\right )} x^{10} + \frac{64}{9} \,{\left (3 \, a^{2} - 140 \, a + 512\right )} x^{9} - 4 \,{\left (15 \, a^{2} - 288 \, a + 512\right )} x^{8} - \frac{4}{7} \,{\left (a^{3} - 192 \, a^{2} + 1536 \, a - 1024\right )} x^{7} + \frac{1}{3} \, a^{4} x^{3} + 8 \, a^{3} x^{4} + \frac{8}{3} \,{\left (a^{3} - 48 \, a^{2} + 128 \, a\right )} x^{6} - \frac{32}{5} \,{\left (a^{3} - 12 \, a^{2}\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*x^19 - 8/9*x^18 + 128/17*x^17 - 4/15*(a - 640)*x^15 - 42*x^16 + 8/7*(3*a - 464)*x^14 - 32/13*(9*a - 524)*
x^13 + 8/3*(35*a - 928)*x^12 + 2/11*(3*a^2 - 1536*a + 20480)*x^11 - 24/5*(a^2 - 128*a + 896)*x^10 + 64/9*(3*a^
2 - 140*a + 512)*x^9 - 4*(15*a^2 - 288*a + 512)*x^8 - 4/7*(a^3 - 192*a^2 + 1536*a - 1024)*x^7 + 1/3*a^4*x^3 +
8*a^3*x^4 + 8/3*(a^3 - 48*a^2 + 128*a)*x^6 - 32/5*(a^3 - 12*a^2)*x^5

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Fricas [A]  time = 1.30445, size = 671, normalized size = 3.2 \begin{align*} \frac{1}{19} x^{19} - \frac{8}{9} x^{18} + \frac{128}{17} x^{17} - 42 x^{16} - \frac{4}{15} x^{15} a + \frac{512}{3} x^{15} + \frac{24}{7} x^{14} a - \frac{3712}{7} x^{14} - \frac{288}{13} x^{13} a + \frac{16768}{13} x^{13} + \frac{280}{3} x^{12} a + \frac{6}{11} x^{11} a^{2} - \frac{7424}{3} x^{12} - \frac{3072}{11} x^{11} a - \frac{24}{5} x^{10} a^{2} + \frac{40960}{11} x^{11} + \frac{3072}{5} x^{10} a + \frac{64}{3} x^{9} a^{2} - \frac{21504}{5} x^{10} - \frac{8960}{9} x^{9} a - 60 x^{8} a^{2} - \frac{4}{7} x^{7} a^{3} + \frac{32768}{9} x^{9} + 1152 x^{8} a + \frac{768}{7} x^{7} a^{2} + \frac{8}{3} x^{6} a^{3} - 2048 x^{8} - \frac{6144}{7} x^{7} a - 128 x^{6} a^{2} - \frac{32}{5} x^{5} a^{3} + \frac{4096}{7} x^{7} + \frac{1024}{3} x^{6} a + \frac{384}{5} x^{5} a^{2} + 8 x^{4} a^{3} + \frac{1}{3} x^{3} a^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*x^19 - 8/9*x^18 + 128/17*x^17 - 42*x^16 - 4/15*x^15*a + 512/3*x^15 + 24/7*x^14*a - 3712/7*x^14 - 288/13*x
^13*a + 16768/13*x^13 + 280/3*x^12*a + 6/11*x^11*a^2 - 7424/3*x^12 - 3072/11*x^11*a - 24/5*x^10*a^2 + 40960/11
*x^11 + 3072/5*x^10*a + 64/3*x^9*a^2 - 21504/5*x^10 - 8960/9*x^9*a - 60*x^8*a^2 - 4/7*x^7*a^3 + 32768/9*x^9 +
1152*x^8*a + 768/7*x^7*a^2 + 8/3*x^6*a^3 - 2048*x^8 - 6144/7*x^7*a - 128*x^6*a^2 - 32/5*x^5*a^3 + 4096/7*x^7 +
1024/3*x^6*a + 384/5*x^5*a^2 + 8*x^4*a^3 + 1/3*x^3*a^4

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Sympy [A]  time = 0.117434, size = 219, normalized size = 1.04 \begin{align*} \frac{a^{4} x^{3}}{3} + 8 a^{3} x^{4} + \frac{x^{19}}{19} - \frac{8 x^{18}}{9} + \frac{128 x^{17}}{17} - 42 x^{16} + x^{15} \left (\frac{512}{3} - \frac{4 a}{15}\right ) + x^{14} \left (\frac{24 a}{7} - \frac{3712}{7}\right ) + x^{13} \left (\frac{16768}{13} - \frac{288 a}{13}\right ) + x^{12} \left (\frac{280 a}{3} - \frac{7424}{3}\right ) + x^{11} \left (\frac{6 a^{2}}{11} - \frac{3072 a}{11} + \frac{40960}{11}\right ) + x^{10} \left (- \frac{24 a^{2}}{5} + \frac{3072 a}{5} - \frac{21504}{5}\right ) + x^{9} \left (\frac{64 a^{2}}{3} - \frac{8960 a}{9} + \frac{32768}{9}\right ) + x^{8} \left (- 60 a^{2} + 1152 a - 2048\right ) + x^{7} \left (- \frac{4 a^{3}}{7} + \frac{768 a^{2}}{7} - \frac{6144 a}{7} + \frac{4096}{7}\right ) + x^{6} \left (\frac{8 a^{3}}{3} - 128 a^{2} + \frac{1024 a}{3}\right ) + x^{5} \left (- \frac{32 a^{3}}{5} + \frac{384 a^{2}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*x**3/3 + 8*a**3*x**4 + x**19/19 - 8*x**18/9 + 128*x**17/17 - 42*x**16 + x**15*(512/3 - 4*a/15) + x**14*(2
4*a/7 - 3712/7) + x**13*(16768/13 - 288*a/13) + x**12*(280*a/3 - 7424/3) + x**11*(6*a**2/11 - 3072*a/11 + 4096
0/11) + x**10*(-24*a**2/5 + 3072*a/5 - 21504/5) + x**9*(64*a**2/3 - 8960*a/9 + 32768/9) + x**8*(-60*a**2 + 115
2*a - 2048) + x**7*(-4*a**3/7 + 768*a**2/7 - 6144*a/7 + 4096/7) + x**6*(8*a**3/3 - 128*a**2 + 1024*a/3) + x**5
*(-32*a**3/5 + 384*a**2/5)

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Giac [A]  time = 1.10342, size = 300, normalized size = 1.43 \begin{align*} \frac{1}{19} \, x^{19} - \frac{8}{9} \, x^{18} + \frac{128}{17} \, x^{17} - \frac{4}{15} \, a x^{15} - 42 \, x^{16} + \frac{24}{7} \, a x^{14} + \frac{512}{3} \, x^{15} - \frac{288}{13} \, a x^{13} - \frac{3712}{7} \, x^{14} + \frac{6}{11} \, a^{2} x^{11} + \frac{280}{3} \, a x^{12} + \frac{16768}{13} \, x^{13} - \frac{24}{5} \, a^{2} x^{10} - \frac{3072}{11} \, a x^{11} - \frac{7424}{3} \, x^{12} + \frac{64}{3} \, a^{2} x^{9} + \frac{3072}{5} \, a x^{10} + \frac{40960}{11} \, x^{11} - \frac{4}{7} \, a^{3} x^{7} - 60 \, a^{2} x^{8} - \frac{8960}{9} \, a x^{9} - \frac{21504}{5} \, x^{10} + \frac{8}{3} \, a^{3} x^{6} + \frac{768}{7} \, a^{2} x^{7} + 1152 \, a x^{8} + \frac{32768}{9} \, x^{9} - \frac{32}{5} \, a^{3} x^{5} - 128 \, a^{2} x^{6} - \frac{6144}{7} \, a x^{7} - 2048 \, x^{8} + \frac{1}{3} \, a^{4} x^{3} + 8 \, a^{3} x^{4} + \frac{384}{5} \, a^{2} x^{5} + \frac{1024}{3} \, a x^{6} + \frac{4096}{7} \, x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*x^19 - 8/9*x^18 + 128/17*x^17 - 4/15*a*x^15 - 42*x^16 + 24/7*a*x^14 + 512/3*x^15 - 288/13*a*x^13 - 3712/7
*x^14 + 6/11*a^2*x^11 + 280/3*a*x^12 + 16768/13*x^13 - 24/5*a^2*x^10 - 3072/11*a*x^11 - 7424/3*x^12 + 64/3*a^2
*x^9 + 3072/5*a*x^10 + 40960/11*x^11 - 4/7*a^3*x^7 - 60*a^2*x^8 - 8960/9*a*x^9 - 21504/5*x^10 + 8/3*a^3*x^6 +
768/7*a^2*x^7 + 1152*a*x^8 + 32768/9*x^9 - 32/5*a^3*x^5 - 128*a^2*x^6 - 6144/7*a*x^7 - 2048*x^8 + 1/3*a^4*x^3
+ 8*a^3*x^4 + 384/5*a^2*x^5 + 1024/3*a*x^6 + 4096/7*x^7