3.370 \(\int x^2 (1+x^3)^4 \, dx\)

Optimal. Leaf size=11 \[ \frac{1}{15} \left (x^3+1\right )^5 \]

[Out]

(1 + x^3)^5/15

________________________________________________________________________________________

Rubi [A]  time = 0.0017565, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {261} \[ \frac{1}{15} \left (x^3+1\right )^5 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(1 + x^3)^4,x]

[Out]

(1 + x^3)^5/15

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^2 \left (1+x^3\right )^4 \, dx &=\frac{1}{15} \left (1+x^3\right )^5\\ \end{align*}

Mathematica [B]  time = 0.0014124, size = 36, normalized size = 3.27 \[ \frac{x^{15}}{15}+\frac{x^{12}}{3}+\frac{2 x^9}{3}+\frac{2 x^6}{3}+\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(1 + x^3)^4,x]

[Out]

x^3/3 + (2*x^6)/3 + (2*x^9)/3 + x^12/3 + x^15/15

________________________________________________________________________________________

Maple [B]  time = 0.001, size = 27, normalized size = 2.5 \begin{align*}{\frac{{x}^{15}}{15}}+{\frac{{x}^{12}}{3}}+{\frac{2\,{x}^{9}}{3}}+{\frac{2\,{x}^{6}}{3}}+{\frac{{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^3+1)^4,x)

[Out]

1/15*x^15+1/3*x^12+2/3*x^9+2/3*x^6+1/3*x^3

________________________________________________________________________________________

Maxima [A]  time = 0.929509, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{15} \,{\left (x^{3} + 1\right )}^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3+1)^4,x, algorithm="maxima")

[Out]

1/15*(x^3 + 1)^5

________________________________________________________________________________________

Fricas [B]  time = 1.56919, size = 70, normalized size = 6.36 \begin{align*} \frac{1}{15} x^{15} + \frac{1}{3} x^{12} + \frac{2}{3} x^{9} + \frac{2}{3} x^{6} + \frac{1}{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3+1)^4,x, algorithm="fricas")

[Out]

1/15*x^15 + 1/3*x^12 + 2/3*x^9 + 2/3*x^6 + 1/3*x^3

________________________________________________________________________________________

Sympy [B]  time = 0.050945, size = 27, normalized size = 2.45 \begin{align*} \frac{x^{15}}{15} + \frac{x^{12}}{3} + \frac{2 x^{9}}{3} + \frac{2 x^{6}}{3} + \frac{x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(x**3+1)**4,x)

[Out]

x**15/15 + x**12/3 + 2*x**9/3 + 2*x**6/3 + x**3/3

________________________________________________________________________________________

Giac [A]  time = 1.05947, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{15} \,{\left (x^{3} + 1\right )}^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3+1)^4,x, algorithm="giac")

[Out]

1/15*(x^3 + 1)^5