3.463 \(\int x^{1+2 n} \, dx\)

Optimal. Leaf size=16 \[ \frac{x^{2 (n+1)}}{2 (n+1)} \]

[Out]

x^(2*(1 + n))/(2*(1 + n))

________________________________________________________________________________________

Rubi [A]  time = 0.0025427, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {30} \[ \frac{x^{2 (n+1)}}{2 (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^(1 + 2*n),x]

[Out]

x^(2*(1 + n))/(2*(1 + n))

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0022625, size = 15, normalized size = 0.94 \[ \frac{x^{2 n+2}}{2 n+2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1 + 2*n),x]

[Out]

x^(2 + 2*n)/(2 + 2*n)

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 15, normalized size = 0.9 \begin{align*}{\frac{{x}^{2+2\,n}}{2+2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1+2*n),x)

[Out]

1/2*x^(2+2*n)/(1+n)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.08276, size = 36, normalized size = 2.25 \begin{align*} \frac{x x^{2 \, n + 1}}{2 \,{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*x^(2*n + 1)/(n + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.054665, size = 19, normalized size = 1.19 \begin{align*} \begin{cases} \frac{x^{2 n + 2}}{2 n + 2} & \text{for}\: 2 n + 1 \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1+2*n),x)

[Out]

Piecewise((x**(2*n + 2)/(2*n + 2), Ne(2*n + 1, -1)), (log(x), True))

________________________________________________________________________________________

Giac [A]  time = 1.07528, size = 19, normalized size = 1.19 \begin{align*} \frac{x^{2 \, n + 2}}{2 \,{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^(2*n + 2)/(n + 1)