3.173 \(\int x^2 (2+x)^5 (2+3 x) \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{3} x^3 (x+2)^6 \]

[Out]

(x^3*(2 + x)^6)/3

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Rubi [A]  time = 0.0012158, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {74} \[ \frac{1}{3} x^3 (x+2)^6 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(2 + x)^5*(2 + 3*x),x]

[Out]

(x^3*(2 + x)^6)/3

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

\begin{align*} \int x^2 (2+x)^5 (2+3 x) \, dx &=\frac{1}{3} x^3 (2+x)^6\\ \end{align*}

Mathematica [B]  time = 0.0019626, size = 42, normalized size = 3.5 \[ \frac{x^9}{3}+4 x^8+20 x^7+\frac{160 x^6}{3}+80 x^5+64 x^4+\frac{64 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(2 + x)^5*(2 + 3*x),x]

[Out]

(64*x^3)/3 + 64*x^4 + 80*x^5 + (160*x^6)/3 + 20*x^7 + 4*x^8 + x^9/3

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Maple [B]  time = 0.001, size = 37, normalized size = 3.1 \begin{align*}{\frac{{x}^{9}}{3}}+4\,{x}^{8}+20\,{x}^{7}+{\frac{160\,{x}^{6}}{3}}+80\,{x}^{5}+64\,{x}^{4}+{\frac{64\,{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(2+x)^5*(2+3*x),x)

[Out]

1/3*x^9+4*x^8+20*x^7+160/3*x^6+80*x^5+64*x^4+64/3*x^3

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Maxima [B]  time = 1.02013, size = 49, normalized size = 4.08 \begin{align*} \frac{1}{3} \, x^{9} + 4 \, x^{8} + 20 \, x^{7} + \frac{160}{3} \, x^{6} + 80 \, x^{5} + 64 \, x^{4} + \frac{64}{3} \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2+x)^5*(2+3*x),x, algorithm="maxima")

[Out]

1/3*x^9 + 4*x^8 + 20*x^7 + 160/3*x^6 + 80*x^5 + 64*x^4 + 64/3*x^3

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Fricas [B]  time = 1.2694, size = 90, normalized size = 7.5 \begin{align*} \frac{1}{3} x^{9} + 4 x^{8} + 20 x^{7} + \frac{160}{3} x^{6} + 80 x^{5} + 64 x^{4} + \frac{64}{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2+x)^5*(2+3*x),x, algorithm="fricas")

[Out]

1/3*x^9 + 4*x^8 + 20*x^7 + 160/3*x^6 + 80*x^5 + 64*x^4 + 64/3*x^3

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Sympy [B]  time = 0.068323, size = 37, normalized size = 3.08 \begin{align*} \frac{x^{9}}{3} + 4 x^{8} + 20 x^{7} + \frac{160 x^{6}}{3} + 80 x^{5} + 64 x^{4} + \frac{64 x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(2+x)**5*(2+3*x),x)

[Out]

x**9/3 + 4*x**8 + 20*x**7 + 160*x**6/3 + 80*x**5 + 64*x**4 + 64*x**3/3

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Giac [B]  time = 1.1728, size = 49, normalized size = 4.08 \begin{align*} \frac{1}{3} \, x^{9} + 4 \, x^{8} + 20 \, x^{7} + \frac{160}{3} \, x^{6} + 80 \, x^{5} + 64 \, x^{4} + \frac{64}{3} \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2+x)^5*(2+3*x),x, algorithm="giac")

[Out]

1/3*x^9 + 4*x^8 + 20*x^7 + 160/3*x^6 + 80*x^5 + 64*x^4 + 64/3*x^3