3.1.2 \(\int x^2 (x+2 x^2)^2 \, dx\) [2]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {661, 45} \begin {gather*} \frac {4 x^7}{7}+\frac {2 x^6}{3}+\frac {x^5}{5} \end {gather*}

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 45

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])

Rule 661

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]

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 = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {x^5}{5}+\frac {2 x^6}{3}+\frac {4 x^7}{7} \end {gather*}

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

________________________________________________________________________________________

Mathics [A]
time = 1.64, size = 15, normalized size = 0.68 \begin {gather*} \frac {x^5 \left (21+70 x+60 x^2\right )}{105} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^2*(2*x^2 + x)^2,x]')

[Out]

x ^ 5 (21 + 70 x + 60 x ^ 2) / 105

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 17, normalized size = 0.77

method	result	size
gosper	\(\frac {x^{5} \left (60 x^{2}+70 x +21\right )}{105}\)	\(16\)
default	\(\frac {1}{5} x^{5}+\frac {2}{3} x^{6}+\frac {4}{7} x^{7}\)	\(17\)
norman	\(\frac {1}{5} x^{5}+\frac {2}{3} x^{6}+\frac {4}{7} x^{7}\)	\(17\)
risch	\(\frac {1}{5} x^{5}+\frac {2}{3} x^{6}+\frac {4}{7} x^{7}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(2*x^2+x)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 16, normalized size = 0.73 \begin {gather*} \frac {4}{7} \, x^{7} + \frac {2}{3} \, x^{6} + \frac {1}{5} \, x^{5} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 16, normalized size = 0.73 \begin {gather*} \frac {4}{7} \, x^{7} + \frac {2}{3} \, x^{6} + \frac {1}{5} \, x^{5} \end {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 0.03, size = 17, normalized size = 0.77 \begin {gather*} \frac {4 x^{7}}{7} + \frac {2 x^{6}}{3} + \frac {x^{5}}{5} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {4}{7} x^{7}+\frac {2}{3} x^{6}+\frac {1}{5} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*x^7 + 2/3*x^6 + 1/5*x^5

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 15, normalized size = 0.68 \begin {gather*} \frac {x^5\,\left (60\,x^2+70\,x+21\right )}{105} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x + 2*x^2)^2,x)

[Out]

(x^5*(70*x + 60*x^2 + 21))/105

________________________________________________________________________________________