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

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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, 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)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{1+2 n} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[x^(1 + 2*n),x]

[Out]

Could not integrate

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fricas [A]  time = 0.86, size = 15, normalized size = 0.94 \[ \frac {x x^{2 \, n + 1}}{2 \, {\left (n + 1\right )}} \]

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)

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giac [A]  time = 0.58, size = 14, normalized size = 0.88 \[ \frac {x^{2 \, n + 2}}{2 \, {\left (n + 1\right )}} \]

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)

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maple [A]  time = 0.01, size = 15, normalized size = 0.94

method	result	size
gosper	\(\frac {x^{2+2 n}}{2+2 n}\)	\(15\)
default	\(\frac {x^{2+2 n}}{2+2 n}\)	\(16\)
risch	\(\frac {x \,x^{1+2 n}}{2+2 n}\)	\(16\)
norman	\(\frac {x \,{\mathrm e}^{\left (1+2 n \right ) \ln \relax (x )}}{2+2 n}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1+2*n),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 14, normalized size = 0.88 \[ \frac {x^{2 \, n + 2}}{2 \, {\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^(2*n + 2)/(n + 1)

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mupad [B]  time = 0.46, size = 24, normalized size = 1.50 \[ \left \{\begin {array}{cl} \ln \relax (x) & \text {\ if\ \ }n=-1\\ \frac {x^{2\,n+2}}{2\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2*n + 1),x)

[Out]

piecewise(n == -1, log(x), n ~= -1, x^(2*n + 2)/(2*(n + 1)))

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sympy [A]  time = 0.06, size = 19, normalized size = 1.19 \[ \begin {cases} \frac {x^{2 n + 2}}{2 n + 2} & \text {for}\: 2 n + 1 \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases} \]

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))

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