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

Optimal. Leaf size=138 \[ \frac {a^3 x^3}{3}-\frac {3}{7} \left (a^2-128 a+512\right ) x^7+\frac {2}{3} \left (3 a^2-96 a+128\right ) x^6+6 a^2 x^4-\frac {3}{11} (256-a) x^{11}+\frac {12}{5} (64-a) x^{10}-\frac {32}{9} (70-3 a) x^9+6 (48-5 a) x^8+\frac {24}{5} (8-a) a x^5-\frac {x^{15}}{15}+\frac {6 x^{14}}{7}-\frac {72 x^{13}}{13}+\frac {70 x^{12}}{3} \]

[Out]

1/3*a^3*x^3+6*a^2*x^4+24/5*(8-a)*a*x^5+2/3*(3*a^2-96*a+128)*x^6-3/7*(a^2-128*a+512)*x^7+6*(48-5*a)*x^8-32/9*(7
0-3*a)*x^9+12/5*(64-a)*x^10-3/11*(256-a)*x^11+70/3*x^12-72/13*x^13+6/7*x^14-1/15*x^15

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Rubi [A]  time = 0.12, antiderivative size = 138, normalized size of antiderivative = 1.00, 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} \[ -\frac {3}{7} \left (a^2-128 a+512\right ) x^7+\frac {2}{3} \left (3 a^2-96 a+128\right ) x^6+6 a^2 x^4+\frac {a^3 x^3}{3}-\frac {3}{11} (256-a) x^{11}+\frac {12}{5} (64-a) x^{10}-\frac {32}{9} (70-3 a) x^9+6 (48-5 a) x^8+\frac {24}{5} (8-a) a x^5-\frac {x^{15}}{15}+\frac {6 x^{14}}{7}-\frac {72 x^{13}}{13}+\frac {70 x^{12}}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*x^3)/3 + 6*a^2*x^4 + (24*(8 - a)*a*x^5)/5 + (2*(128 - 96*a + 3*a^2)*x^6)/3 - (3*(512 - 128*a + a^2)*x^7)/
7 + 6*(48 - 5*a)*x^8 - (32*(70 - 3*a)*x^9)/9 + (12*(64 - a)*x^10)/5 - (3*(256 - a)*x^11)/11 + (70*x^12)/3 - (7
2*x^13)/13 + (6*x^14)/7 - x^15/15

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 )^3 \, dx &=\int \left (a^3 x^2+24 a^2 x^3-24 (-8+a) a x^4+4 \left (128-96 a+3 a^2\right ) x^5-3 \left (512-128 a+a^2\right ) x^6-48 (-48+5 a) x^7+32 (-70+3 a) x^8-24 (-64+a) x^9+3 (-256+a) x^{10}+280 x^{11}-72 x^{12}+12 x^{13}-x^{14}\right ) \, dx\\ &=\frac {a^3 x^3}{3}+6 a^2 x^4+\frac {24}{5} (8-a) a x^5+\frac {2}{3} \left (128-96 a+3 a^2\right ) x^6-\frac {3}{7} \left (512-128 a+a^2\right ) x^7+6 (48-5 a) x^8-\frac {32}{9} (70-3 a) x^9+\frac {12}{5} (64-a) x^{10}-\frac {3}{11} (256-a) x^{11}+\frac {70 x^{12}}{3}-\frac {72 x^{13}}{13}+\frac {6 x^{14}}{7}-\frac {x^{15}}{15}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 132, normalized size = 0.96 \[ \frac {a^3 x^3}{3}-\frac {3}{7} \left (a^2-128 a+512\right ) x^7+\frac {2}{3} \left (3 a^2-96 a+128\right ) x^6+6 a^2 x^4+\frac {3}{11} (a-256) x^{11}-\frac {12}{5} (a-64) x^{10}+\frac {32}{9} (3 a-70) x^9-6 (5 a-48) x^8-\frac {24}{5} (a-8) a x^5-\frac {x^{15}}{15}+\frac {6 x^{14}}{7}-\frac {72 x^{13}}{13}+\frac {70 x^{12}}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*x^3)/3 + 6*a^2*x^4 - (24*(-8 + a)*a*x^5)/5 + (2*(128 - 96*a + 3*a^2)*x^6)/3 - (3*(512 - 128*a + a^2)*x^7)
/7 - 6*(-48 + 5*a)*x^8 + (32*(-70 + 3*a)*x^9)/9 - (12*(-64 + a)*x^10)/5 + (3*(-256 + a)*x^11)/11 + (70*x^12)/3
- (72*x^13)/13 + (6*x^14)/7 - x^15/15

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fricas [A]  time = 0.34, size = 133, normalized size = 0.96 \[ -\frac {1}{15} x^{15} + \frac {6}{7} x^{14} - \frac {72}{13} x^{13} + \frac {70}{3} x^{12} + \frac {3}{11} x^{11} a - \frac {768}{11} x^{11} - \frac {12}{5} x^{10} a + \frac {768}{5} x^{10} + \frac {32}{3} x^{9} a - \frac {2240}{9} x^{9} - 30 x^{8} a - \frac {3}{7} x^{7} a^{2} + 288 x^{8} + \frac {384}{7} x^{7} a + 2 x^{6} a^{2} - \frac {1536}{7} x^{7} - 64 x^{6} a - \frac {24}{5} x^{5} a^{2} + \frac {256}{3} x^{6} + \frac {192}{5} x^{5} a + 6 x^{4} a^{2} + \frac {1}{3} x^{3} a^{3} \]

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)^3,x, algorithm="fricas")

[Out]

-1/15*x^15 + 6/7*x^14 - 72/13*x^13 + 70/3*x^12 + 3/11*x^11*a - 768/11*x^11 - 12/5*x^10*a + 768/5*x^10 + 32/3*x
^9*a - 2240/9*x^9 - 30*x^8*a - 3/7*x^7*a^2 + 288*x^8 + 384/7*x^7*a + 2*x^6*a^2 - 1536/7*x^7 - 64*x^6*a - 24/5*
x^5*a^2 + 256/3*x^6 + 192/5*x^5*a + 6*x^4*a^2 + 1/3*x^3*a^3

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giac [A]  time = 0.29, size = 133, normalized size = 0.96 \[ -\frac {1}{15} \, x^{15} + \frac {6}{7} \, x^{14} - \frac {72}{13} \, x^{13} + \frac {3}{11} \, a x^{11} + \frac {70}{3} \, x^{12} - \frac {12}{5} \, a x^{10} - \frac {768}{11} \, x^{11} + \frac {32}{3} \, a x^{9} + \frac {768}{5} \, x^{10} - \frac {3}{7} \, a^{2} x^{7} - 30 \, a x^{8} - \frac {2240}{9} \, x^{9} + 2 \, a^{2} x^{6} + \frac {384}{7} \, a x^{7} + 288 \, x^{8} - \frac {24}{5} \, a^{2} x^{5} - 64 \, a x^{6} - \frac {1536}{7} \, x^{7} + \frac {1}{3} \, a^{3} x^{3} + 6 \, a^{2} x^{4} + \frac {192}{5} \, a x^{5} + \frac {256}{3} \, x^{6} \]

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)^3,x, algorithm="giac")

[Out]

-1/15*x^15 + 6/7*x^14 - 72/13*x^13 + 3/11*a*x^11 + 70/3*x^12 - 12/5*a*x^10 - 768/11*x^11 + 32/3*a*x^9 + 768/5*
x^10 - 3/7*a^2*x^7 - 30*a*x^8 - 2240/9*x^9 + 2*a^2*x^6 + 384/7*a*x^7 + 288*x^8 - 24/5*a^2*x^5 - 64*a*x^6 - 153
6/7*x^7 + 1/3*a^3*x^3 + 6*a^2*x^4 + 192/5*a*x^5 + 256/3*x^6

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maple [A]  time = 0.00, size = 143, normalized size = 1.04 \[ -\frac {x^{15}}{15}+\frac {6 x^{14}}{7}-\frac {72 x^{13}}{13}+\frac {70 x^{12}}{3}+\frac {\left (3 a -768\right ) x^{11}}{11}+\frac {\left (-24 a +1536\right ) x^{10}}{10}+\frac {\left (96 a -2240\right ) x^{9}}{9}+\frac {\left (-240 a +2304\right ) x^{8}}{8}+\frac {\left (-a^{2}+\left (-2 a +128\right ) a +256 a -1536\right ) x^{7}}{7}+\frac {a^{3} x^{3}}{3}+6 a^{2} x^{4}+\frac {\left (4 a^{2}+\left (8 a -128\right ) a -256 a +512\right ) x^{6}}{6}+\frac {\left (-8 a^{2}+\left (-16 a +64\right ) a +128 a \right ) x^{5}}{5} \]

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)^3,x)

[Out]

-1/15*x^15+6/7*x^14-72/13*x^13+70/3*x^12+1/11*(3*a-768)*x^11+1/10*(-24*a+1536)*x^10+1/9*(96*a-2240)*x^9+1/8*(-
240*a+2304)*x^8+1/7*(-a^2+(-2*a+128)*a+256*a-1536)*x^7+1/6*(4*a^2+(8*a-128)*a-256*a+512)*x^6+1/5*(-8*a^2+(-16*
a+64)*a+128*a)*x^5+6*a^2*x^4+1/3*a^3*x^3

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maxima [A]  time = 0.56, size = 113, normalized size = 0.82 \[ -\frac {1}{15} \, x^{15} + \frac {6}{7} \, x^{14} - \frac {72}{13} \, x^{13} + \frac {3}{11} \, {\left (a - 256\right )} x^{11} + \frac {70}{3} \, x^{12} - \frac {12}{5} \, {\left (a - 64\right )} x^{10} + \frac {32}{9} \, {\left (3 \, a - 70\right )} x^{9} - 6 \, {\left (5 \, a - 48\right )} x^{8} - \frac {3}{7} \, {\left (a^{2} - 128 \, a + 512\right )} x^{7} + \frac {2}{3} \, {\left (3 \, a^{2} - 96 \, a + 128\right )} x^{6} + \frac {1}{3} \, a^{3} x^{3} + 6 \, a^{2} x^{4} - \frac {24}{5} \, {\left (a^{2} - 8 \, a\right )} x^{5} \]

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)^3,x, algorithm="maxima")

[Out]

-1/15*x^15 + 6/7*x^14 - 72/13*x^13 + 3/11*(a - 256)*x^11 + 70/3*x^12 - 12/5*(a - 64)*x^10 + 32/9*(3*a - 70)*x^
9 - 6*(5*a - 48)*x^8 - 3/7*(a^2 - 128*a + 512)*x^7 + 2/3*(3*a^2 - 96*a + 128)*x^6 + 1/3*a^3*x^3 + 6*a^2*x^4 -
24/5*(a^2 - 8*a)*x^5

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mupad [B]  time = 0.09, size = 113, normalized size = 0.82 \[ x^{11}\,\left (\frac {3\,a}{11}-\frac {768}{11}\right )-x^{10}\,\left (\frac {12\,a}{5}-\frac {768}{5}\right )-x^8\,\left (30\,a-288\right )+x^9\,\left (\frac {32\,a}{3}-\frac {2240}{9}\right )+x^6\,\left (2\,a^2-64\,a+\frac {256}{3}\right )-x^7\,\left (\frac {3\,a^2}{7}-\frac {384\,a}{7}+\frac {1536}{7}\right )+\frac {70\,x^{12}}{3}-\frac {72\,x^{13}}{13}+\frac {6\,x^{14}}{7}-\frac {x^{15}}{15}+6\,a^2\,x^4+\frac {a^3\,x^3}{3}-\frac {24\,a\,x^5\,\left (a-8\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^11*((3*a)/11 - 768/11) - x^10*((12*a)/5 - 768/5) - x^8*(30*a - 288) + x^9*((32*a)/3 - 2240/9) + x^6*(2*a^2 -
64*a + 256/3) - x^7*((3*a^2)/7 - (384*a)/7 + 1536/7) + (70*x^12)/3 - (72*x^13)/13 + (6*x^14)/7 - x^15/15 + 6*
a^2*x^4 + (a^3*x^3)/3 - (24*a*x^5*(a - 8))/5

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sympy [A]  time = 0.10, size = 134, normalized size = 0.97 \[ \frac {a^{3} x^{3}}{3} + 6 a^{2} x^{4} - \frac {x^{15}}{15} + \frac {6 x^{14}}{7} - \frac {72 x^{13}}{13} + \frac {70 x^{12}}{3} + x^{11} \left (\frac {3 a}{11} - \frac {768}{11}\right ) + x^{10} \left (\frac {768}{5} - \frac {12 a}{5}\right ) + x^{9} \left (\frac {32 a}{3} - \frac {2240}{9}\right ) + x^{8} \left (288 - 30 a\right ) + x^{7} \left (- \frac {3 a^{2}}{7} + \frac {384 a}{7} - \frac {1536}{7}\right ) + x^{6} \left (2 a^{2} - 64 a + \frac {256}{3}\right ) + x^{5} \left (- \frac {24 a^{2}}{5} + \frac {192 a}{5}\right ) \]

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)**3,x)

[Out]

a**3*x**3/3 + 6*a**2*x**4 - x**15/15 + 6*x**14/7 - 72*x**13/13 + 70*x**12/3 + x**11*(3*a/11 - 768/11) + x**10*
(768/5 - 12*a/5) + x**9*(32*a/3 - 2240/9) + x**8*(288 - 30*a) + x**7*(-3*a**2/7 + 384*a/7 - 1536/7) + x**6*(2*
a**2 - 64*a + 256/3) + x**5*(-24*a**2/5 + 192*a/5)

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