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

Optimal. Leaf size=79 \[ \frac{a^2 x^3}{3}+\frac{2}{7} (64-a) x^7-\frac{4}{3} (16-a) x^6+\frac{16}{5} (4-a) x^5+4 a x^4+\frac{x^{11}}{11}-\frac{4 x^{10}}{5}+\frac{32 x^9}{9}-10 x^8 \]

[Out]

(a^2*x^3)/3 + 4*a*x^4 + (16*(4 - a)*x^5)/5 - (4*(16 - a)*x^6)/3 + (2*(64 - a)*x^7)/7 - 10*x^8 + (32*x^9)/9 - (
4*x^10)/5 + x^11/11

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Rubi [A]  time = 0.0767837, antiderivative size = 79, 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} \[ \frac{a^2 x^3}{3}+\frac{2}{7} (64-a) x^7-\frac{4}{3} (16-a) x^6+\frac{16}{5} (4-a) x^5+4 a x^4+\frac{x^{11}}{11}-\frac{4 x^{10}}{5}+\frac{32 x^9}{9}-10 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^3)/3 + 4*a*x^4 + (16*(4 - a)*x^5)/5 - (4*(16 - a)*x^6)/3 + (2*(64 - a)*x^7)/7 - 10*x^8 + (32*x^9)/9 - (
4*x^10)/5 + x^11/11

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 )^2 \, dx &=\int \left (a^2 x^2+16 a x^3-16 (-4+a) x^4+8 (-16+a) x^5-2 (-64+a) x^6-80 x^7+32 x^8-8 x^9+x^{10}\right ) \, dx\\ &=\frac{a^2 x^3}{3}+4 a x^4+\frac{16}{5} (4-a) x^5-\frac{4}{3} (16-a) x^6+\frac{2}{7} (64-a) x^7-10 x^8+\frac{32 x^9}{9}-\frac{4 x^{10}}{5}+\frac{x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0083944, size = 73, normalized size = 0.92 \[ \frac{a^2 x^3}{3}-\frac{2}{7} (a-64) x^7+\frac{4}{3} (a-16) x^6-\frac{16}{5} (a-4) x^5+4 a x^4+\frac{x^{11}}{11}-\frac{4 x^{10}}{5}+\frac{32 x^9}{9}-10 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^3)/3 + 4*a*x^4 - (16*(-4 + a)*x^5)/5 + (4*(-16 + a)*x^6)/3 - (2*(-64 + a)*x^7)/7 - 10*x^8 + (32*x^9)/9
- (4*x^10)/5 + x^11/11

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Maple [A]  time = 0.001, size = 66, normalized size = 0.8 \begin{align*}{\frac{{x}^{11}}{11}}-{\frac{4\,{x}^{10}}{5}}+{\frac{32\,{x}^{9}}{9}}-10\,{x}^{8}+{\frac{ \left ( -2\,a+128 \right ){x}^{7}}{7}}+{\frac{ \left ( 8\,a-128 \right ){x}^{6}}{6}}+{\frac{ \left ( -16\,a+64 \right ){x}^{5}}{5}}+4\,a{x}^{4}+{\frac{{a}^{2}{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)^2,x)

[Out]

1/11*x^11-4/5*x^10+32/9*x^9-10*x^8+1/7*(-2*a+128)*x^7+1/6*(8*a-128)*x^6+1/5*(-16*a+64)*x^5+4*a*x^4+1/3*a^2*x^3

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Maxima [A]  time = 1.08814, size = 80, normalized size = 1.01 \begin{align*} \frac{1}{11} \, x^{11} - \frac{4}{5} \, x^{10} + \frac{32}{9} \, x^{9} - \frac{2}{7} \,{\left (a - 64\right )} x^{7} - 10 \, x^{8} + \frac{4}{3} \,{\left (a - 16\right )} x^{6} - \frac{16}{5} \,{\left (a - 4\right )} x^{5} + \frac{1}{3} \, a^{2} x^{3} + 4 \, a x^{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)^2,x, algorithm="maxima")

[Out]

1/11*x^11 - 4/5*x^10 + 32/9*x^9 - 2/7*(a - 64)*x^7 - 10*x^8 + 4/3*(a - 16)*x^6 - 16/5*(a - 4)*x^5 + 1/3*a^2*x^
3 + 4*a*x^4

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Fricas [A]  time = 1.27528, size = 185, normalized size = 2.34 \begin{align*} \frac{1}{11} x^{11} - \frac{4}{5} x^{10} + \frac{32}{9} x^{9} - 10 x^{8} - \frac{2}{7} x^{7} a + \frac{128}{7} x^{7} + \frac{4}{3} x^{6} a - \frac{64}{3} x^{6} - \frac{16}{5} x^{5} a + \frac{64}{5} x^{5} + 4 x^{4} a + \frac{1}{3} x^{3} a^{2} \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)^2,x, algorithm="fricas")

[Out]

1/11*x^11 - 4/5*x^10 + 32/9*x^9 - 10*x^8 - 2/7*x^7*a + 128/7*x^7 + 4/3*x^6*a - 64/3*x^6 - 16/5*x^5*a + 64/5*x^
5 + 4*x^4*a + 1/3*x^3*a^2

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Sympy [A]  time = 0.07261, size = 73, normalized size = 0.92 \begin{align*} \frac{a^{2} x^{3}}{3} + 4 a x^{4} + \frac{x^{11}}{11} - \frac{4 x^{10}}{5} + \frac{32 x^{9}}{9} - 10 x^{8} + x^{7} \left (\frac{128}{7} - \frac{2 a}{7}\right ) + x^{6} \left (\frac{4 a}{3} - \frac{64}{3}\right ) + x^{5} \left (\frac{64}{5} - \frac{16 a}{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)**2,x)

[Out]

a**2*x**3/3 + 4*a*x**4 + x**11/11 - 4*x**10/5 + 32*x**9/9 - 10*x**8 + x**7*(128/7 - 2*a/7) + x**6*(4*a/3 - 64/
3) + x**5*(64/5 - 16*a/5)

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Giac [A]  time = 1.14009, size = 92, normalized size = 1.16 \begin{align*} \frac{1}{11} \, x^{11} - \frac{4}{5} \, x^{10} + \frac{32}{9} \, x^{9} - \frac{2}{7} \, a x^{7} - 10 \, x^{8} + \frac{4}{3} \, a x^{6} + \frac{128}{7} \, x^{7} - \frac{16}{5} \, a x^{5} - \frac{64}{3} \, x^{6} + \frac{1}{3} \, a^{2} x^{3} + 4 \, a x^{4} + \frac{64}{5} \, 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)^2,x, algorithm="giac")

[Out]

1/11*x^11 - 4/5*x^10 + 32/9*x^9 - 2/7*a*x^7 - 10*x^8 + 4/3*a*x^6 + 128/7*x^7 - 16/5*a*x^5 - 64/3*x^6 + 1/3*a^2
*x^3 + 4*a*x^4 + 64/5*x^5