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

Optimal. Leaf size=35 \[ \frac{a x^3}{3}-\frac{x^7}{7}+\frac{2 x^6}{3}-\frac{8 x^5}{5}+2 x^4 \]

[Out]

(a*x^3)/3 + 2*x^4 - (8*x^5)/5 + (2*x^6)/3 - x^7/7

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Rubi [A]  time = 0.0090193, antiderivative size = 35, 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 = {14} \[ \frac{a x^3}{3}-\frac{x^7}{7}+\frac{2 x^6}{3}-\frac{8 x^5}{5}+2 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^3)/3 + 2*x^4 - (8*x^5)/5 + (2*x^6)/3 - x^7/7

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx &=\int \left (a x^2+8 x^3-8 x^4+4 x^5-x^6\right ) \, dx\\ &=\frac{a x^3}{3}+2 x^4-\frac{8 x^5}{5}+\frac{2 x^6}{3}-\frac{x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0015729, size = 35, normalized size = 1. \[ \frac{a x^3}{3}-\frac{x^7}{7}+\frac{2 x^6}{3}-\frac{8 x^5}{5}+2 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^3)/3 + 2*x^4 - (8*x^5)/5 + (2*x^6)/3 - x^7/7

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Maple [A]  time = 0., size = 28, normalized size = 0.8 \begin{align*}{\frac{a{x}^{3}}{3}}+2\,{x}^{4}-{\frac{8\,{x}^{5}}{5}}+{\frac{2\,{x}^{6}}{3}}-{\frac{{x}^{7}}{7}} \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),x)

[Out]

1/3*a*x^3+2*x^4-8/5*x^5+2/3*x^6-1/7*x^7

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Maxima [A]  time = 1.13278, size = 36, normalized size = 1.03 \begin{align*} -\frac{1}{7} \, x^{7} + \frac{2}{3} \, x^{6} - \frac{8}{5} \, x^{5} + \frac{1}{3} \, a x^{3} + 2 \, 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),x, algorithm="maxima")

[Out]

-1/7*x^7 + 2/3*x^6 - 8/5*x^5 + 1/3*a*x^3 + 2*x^4

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Fricas [A]  time = 1.29859, size = 68, normalized size = 1.94 \begin{align*} -\frac{1}{7} x^{7} + \frac{2}{3} x^{6} - \frac{8}{5} x^{5} + 2 x^{4} + \frac{1}{3} x^{3} a \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),x, algorithm="fricas")

[Out]

-1/7*x^7 + 2/3*x^6 - 8/5*x^5 + 2*x^4 + 1/3*x^3*a

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Sympy [A]  time = 0.057391, size = 29, normalized size = 0.83 \begin{align*} \frac{a x^{3}}{3} - \frac{x^{7}}{7} + \frac{2 x^{6}}{3} - \frac{8 x^{5}}{5} + 2 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),x)

[Out]

a*x**3/3 - x**7/7 + 2*x**6/3 - 8*x**5/5 + 2*x**4

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Giac [A]  time = 1.14173, size = 36, normalized size = 1.03 \begin{align*} -\frac{1}{7} \, x^{7} + \frac{2}{3} \, x^{6} - \frac{8}{5} \, x^{5} + \frac{1}{3} \, a x^{3} + 2 \, 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),x, algorithm="giac")

[Out]

-1/7*x^7 + 2/3*x^6 - 8/5*x^5 + 1/3*a*x^3 + 2*x^4