3.2753 \(\int x^n \sqrt{1+x^{1+n}} \, dx\)

Optimal. Leaf size=20 \[ \frac{2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]

[Out]

(2*(1 + x^(1 + n))^(3/2))/(3*(1 + n))

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Rubi [A]  time = 0.0047202, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^n*Sqrt[1 + x^(1 + n)],x]

[Out]

(2*(1 + x^(1 + n))^(3/2))/(3*(1 + n))

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^n \sqrt{1+x^{1+n}} \, dx &=\frac{2 \left (1+x^{1+n}\right )^{3/2}}{3 (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0077528, size = 20, normalized size = 1. \[ \frac{2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^n*Sqrt[1 + x^(1 + n)],x]

[Out]

(2*(1 + x^(1 + n))^(3/2))/(3*(1 + n))

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Maple [A]  time = 0.026, size = 17, normalized size = 0.9 \begin{align*}{\frac{2}{3+3\,n} \left ( 1+x{x}^{n} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(1+x*x^n)^(3/2)/(1+n)

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Maxima [A]  time = 1.07326, size = 22, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (x^{n + 1} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^(n + 1) + 1)^(3/2)/(n + 1)

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Fricas [A]  time = 1.30274, size = 47, normalized size = 2.35 \begin{align*} \frac{2 \,{\left (x^{n + 1} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^(n + 1) + 1)^(3/2)/(n + 1)

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Sympy [A]  time = 5.00389, size = 48, normalized size = 2.4 \begin{align*} \begin{cases} \frac{2 x x^{n} \sqrt{x x^{n} + 1}}{3 n + 3} + \frac{2 \sqrt{x x^{n} + 1}}{3 n + 3} & \text{for}\: n \neq -1 \\\sqrt{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*x*x**n*sqrt(x*x**n + 1)/(3*n + 3) + 2*sqrt(x*x**n + 1)/(3*n + 3), Ne(n, -1)), (sqrt(2)*log(x), Tr
ue))

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Giac [A]  time = 1.10981, size = 22, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (x^{n + 1} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^(n + 1) + 1)^(3/2)/(n + 1)