3.2 \(\int x^2 (x+2 x^2)^2 \, dx\)

Optimal. Leaf size=22 \[ \frac{4 x^7}{7}+\frac{2 x^6}{3}+\frac{x^5}{5} \]

[Out]

x^5/5 + (2*x^6)/3 + (4*x^7)/7

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Rubi [A]  time = 0.0103443, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {647, 43} \[ \frac{4 x^7}{7}+\frac{2 x^6}{3}+\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(x + 2*x^2)^2,x]

[Out]

x^5/5 + (2*x^6)/3 + (4*x^7)/7

Rule 647

Int[((e_.)*(x_))^(m_.)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e^p, Int[(e*x)^(m + p)*(b + c*x)
^p, x], x] /; FreeQ[{b, c, e, m}, x] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 \left (x+2 x^2\right )^2 \, dx &=\int x^4 (1+2 x)^2 \, dx\\ &=\int \left (x^4+4 x^5+4 x^6\right ) \, dx\\ &=\frac{x^5}{5}+\frac{2 x^6}{3}+\frac{4 x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0012019, size = 22, normalized size = 1. \[ \frac{4 x^7}{7}+\frac{2 x^6}{3}+\frac{x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(x + 2*x^2)^2,x]

[Out]

x^5/5 + (2*x^6)/3 + (4*x^7)/7

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Maple [A]  time = 0., size = 17, normalized size = 0.8 \begin{align*}{\frac{{x}^{5}}{5}}+{\frac{2\,{x}^{6}}{3}}+{\frac{4\,{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.950201, size = 22, normalized size = 1. \begin{align*} \frac{4}{7} \, x^{7} + \frac{2}{3} \, x^{6} + \frac{1}{5} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*x^7 + 2/3*x^6 + 1/5*x^5

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Fricas [A]  time = 1.60464, size = 39, normalized size = 1.77 \begin{align*} \frac{4}{7} x^{7} + \frac{2}{3} x^{6} + \frac{1}{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*x^7 + 2/3*x^6 + 1/5*x^5

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Sympy [A]  time = 0.053199, size = 17, normalized size = 0.77 \begin{align*} \frac{4 x^{7}}{7} + \frac{2 x^{6}}{3} + \frac{x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x**7/7 + 2*x**6/3 + x**5/5

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Giac [A]  time = 1.10123, size = 22, normalized size = 1. \begin{align*} \frac{4}{7} \, x^{7} + \frac{2}{3} \, x^{6} + \frac{1}{5} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*x^7 + 2/3*x^6 + 1/5*x^5