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]

1/3*x^3*(2+x)^6

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.00, size = 42, normalized size = 3.50 \[ \frac {x^9}{3}+4 x^8+20 x^7+\frac {160 x^6}{3}+80 x^5+64 x^4+\frac {64 x^3}{3} \]

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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fricas [B]  time = 0.78, size = 36, normalized size = 3.00 \[ \frac {1}{3} x^{9} + 4 x^{8} + 20 x^{7} + \frac {160}{3} x^{6} + 80 x^{5} + 64 x^{4} + \frac {64}{3} x^{3} \]

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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giac [B]  time = 1.04, size = 36, normalized size = 3.00 \[ \frac {1}{3} \, x^{9} + 4 \, x^{8} + 20 \, x^{7} + \frac {160}{3} \, x^{6} + 80 \, x^{5} + 64 \, x^{4} + \frac {64}{3} \, x^{3} \]

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

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maple [B]  time = 0.00, size = 37, normalized size = 3.08 \[ \frac {1}{3} x^{9}+4 x^{8}+20 x^{7}+\frac {160}{3} x^{6}+80 x^{5}+64 x^{4}+\frac {64}{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x+2)^5*(3*x+2),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 = 1.14, size = 36, normalized size = 3.00 \[ \frac {1}{3} \, x^{9} + 4 \, x^{8} + 20 \, x^{7} + \frac {160}{3} \, x^{6} + 80 \, x^{5} + 64 \, x^{4} + \frac {64}{3} \, x^{3} \]

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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mupad [B]  time = 0.03, size = 36, normalized size = 3.00 \[ \frac {x^9}{3}+4\,x^8+20\,x^7+\frac {160\,x^6}{3}+80\,x^5+64\,x^4+\frac {64\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(3*x + 2)*(x + 2)^5,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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sympy [B]  time = 0.14, size = 37, normalized size = 3.08 \[ \frac {x^{9}}{3} + 4 x^{8} + 20 x^{7} + \frac {160 x^{6}}{3} + 80 x^{5} + 64 x^{4} + \frac {64 x^{3}}{3} \]

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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