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

Optimal. Leaf size=79 \[ \frac{a^2 x^2}{2}+\frac{1}{3} (64-a) x^6-\frac{8}{5} (16-a) x^5+4 (4-a) x^4+\frac{16 a x^3}{3}+\frac{x^{10}}{10}-\frac{8 x^9}{9}+4 x^8-\frac{80 x^7}{7} \]

[Out]

(a^2*x^2)/2 + (16*a*x^3)/3 + 4*(4 - a)*x^4 - (8*(16 - a)*x^5)/5 + ((64 - a)*x^6)/3 - (80*x^7)/7 + 4*x^8 - (8*x
^9)/9 + x^10/10

________________________________________________________________________________________

Rubi [A]  time = 0.0759102, antiderivative size = 79, 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} \[ \frac{a^2 x^2}{2}+\frac{1}{3} (64-a) x^6-\frac{8}{5} (16-a) x^5+4 (4-a) x^4+\frac{16 a x^3}{3}+\frac{x^{10}}{10}-\frac{8 x^9}{9}+4 x^8-\frac{80 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^2)/2 + (16*a*x^3)/3 + 4*(4 - a)*x^4 - (8*(16 - a)*x^5)/5 + ((64 - a)*x^6)/3 - (80*x^7)/7 + 4*x^8 - (8*x
^9)/9 + x^10/10

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 )^2 \, dx &=\int \left (a^2 x+16 a x^2-16 (-4+a) x^3+8 (-16+a) x^4-2 (-64+a) x^5-80 x^6+32 x^7-8 x^8+x^9\right ) \, dx\\ &=\frac{a^2 x^2}{2}+\frac{16 a x^3}{3}+4 (4-a) x^4-\frac{8}{5} (16-a) x^5+\frac{1}{3} (64-a) x^6-\frac{80 x^7}{7}+4 x^8-\frac{8 x^9}{9}+\frac{x^{10}}{10}\\ \end{align*}

Mathematica [A]  time = 0.0078411, size = 75, normalized size = 0.95 \[ \frac{a^2 x^2}{2}+\frac{1}{3} (64-a) x^6+\frac{8}{5} (a-16) x^5-4 (a-4) x^4+\frac{16 a x^3}{3}+\frac{x^{10}}{10}-\frac{8 x^9}{9}+4 x^8-\frac{80 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^2)/2 + (16*a*x^3)/3 - 4*(-4 + a)*x^4 + (8*(-16 + a)*x^5)/5 + ((64 - a)*x^6)/3 - (80*x^7)/7 + 4*x^8 - (8
*x^9)/9 + x^10/10

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 66, normalized size = 0.8 \begin{align*}{\frac{{x}^{10}}{10}}-{\frac{8\,{x}^{9}}{9}}+4\,{x}^{8}-{\frac{80\,{x}^{7}}{7}}+{\frac{ \left ( -2\,a+128 \right ){x}^{6}}{6}}+{\frac{ \left ( 8\,a-128 \right ){x}^{5}}{5}}+{\frac{ \left ( -16\,a+64 \right ){x}^{4}}{4}}+{\frac{16\,a{x}^{3}}{3}}+{\frac{{a}^{2}{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)^2,x)

[Out]

1/10*x^10-8/9*x^9+4*x^8-80/7*x^7+1/6*(-2*a+128)*x^6+1/5*(8*a-128)*x^5+1/4*(-16*a+64)*x^4+16/3*a*x^3+1/2*a^2*x^
2

________________________________________________________________________________________

Maxima [A]  time = 1.07956, size = 80, normalized size = 1.01 \begin{align*} \frac{1}{10} \, x^{10} - \frac{8}{9} \, x^{9} + 4 \, x^{8} - \frac{1}{3} \,{\left (a - 64\right )} x^{6} - \frac{80}{7} \, x^{7} + \frac{8}{5} \,{\left (a - 16\right )} x^{5} - 4 \,{\left (a - 4\right )} x^{4} + \frac{1}{2} \, a^{2} x^{2} + \frac{16}{3} \, a x^{3} \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)^2,x, algorithm="maxima")

[Out]

1/10*x^10 - 8/9*x^9 + 4*x^8 - 1/3*(a - 64)*x^6 - 80/7*x^7 + 8/5*(a - 16)*x^5 - 4*(a - 4)*x^4 + 1/2*a^2*x^2 + 1
6/3*a*x^3

________________________________________________________________________________________

Fricas [A]  time = 1.34024, size = 180, normalized size = 2.28 \begin{align*} \frac{1}{10} x^{10} - \frac{8}{9} x^{9} + 4 x^{8} - \frac{80}{7} x^{7} - \frac{1}{3} x^{6} a + \frac{64}{3} x^{6} + \frac{8}{5} x^{5} a - \frac{128}{5} x^{5} - 4 x^{4} a + 16 x^{4} + \frac{16}{3} x^{3} a + \frac{1}{2} x^{2} a^{2} \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)^2,x, algorithm="fricas")

[Out]

1/10*x^10 - 8/9*x^9 + 4*x^8 - 80/7*x^7 - 1/3*x^6*a + 64/3*x^6 + 8/5*x^5*a - 128/5*x^5 - 4*x^4*a + 16*x^4 + 16/
3*x^3*a + 1/2*x^2*a^2

________________________________________________________________________________________

Sympy [A]  time = 0.072107, size = 70, normalized size = 0.89 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{16 a x^{3}}{3} + \frac{x^{10}}{10} - \frac{8 x^{9}}{9} + 4 x^{8} - \frac{80 x^{7}}{7} + x^{6} \left (\frac{64}{3} - \frac{a}{3}\right ) + x^{5} \left (\frac{8 a}{5} - \frac{128}{5}\right ) + x^{4} \left (16 - 4 a\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)**2,x)

[Out]

a**2*x**2/2 + 16*a*x**3/3 + x**10/10 - 8*x**9/9 + 4*x**8 - 80*x**7/7 + x**6*(64/3 - a/3) + x**5*(8*a/5 - 128/5
) + x**4*(16 - 4*a)

________________________________________________________________________________________

Giac [A]  time = 1.12922, size = 92, normalized size = 1.16 \begin{align*} \frac{1}{10} \, x^{10} - \frac{8}{9} \, x^{9} + 4 \, x^{8} - \frac{1}{3} \, a x^{6} - \frac{80}{7} \, x^{7} + \frac{8}{5} \, a x^{5} + \frac{64}{3} \, x^{6} - 4 \, a x^{4} - \frac{128}{5} \, x^{5} + \frac{1}{2} \, a^{2} x^{2} + \frac{16}{3} \, a x^{3} + 16 \, 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)^2,x, algorithm="giac")

[Out]

1/10*x^10 - 8/9*x^9 + 4*x^8 - 1/3*a*x^6 - 80/7*x^7 + 8/5*a*x^5 + 64/3*x^6 - 4*a*x^4 - 128/5*x^5 + 1/2*a^2*x^2
+ 16/3*a*x^3 + 16*x^4