3.2754 \(\int x^n \sqrt{a^2+x^{1+n}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 \left (a^2+x^{n+1}\right )^{3/2}}{3 (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0049375, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {261} \[ \frac{2 \left (a^2+x^{n+1}\right )^{3/2}}{3 (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^n*Sqrt[a^2 + x^(1 + n)],x]

[Out]

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

Mathematica [A]  time = 0.00849, size = 22, normalized size = 1. \[ \frac{2 \left (a^2+x^{n+1}\right )^{3/2}}{3 (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^n*Sqrt[a^2 + x^(1 + n)],x]

[Out]

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

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Maple [A]  time = 0.025, size = 19, normalized size = 0.9 \begin{align*}{\frac{2}{3+3\,n} \left ({a}^{2}+x{x}^{n} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.980573, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (a^{2} + x^{n + 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*(a^2+x^(1+n))^(1/2),x, algorithm="maxima")

[Out]

2/3*(a^2 + x^(n + 1))^(3/2)/(n + 1)

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Fricas [A]  time = 1.38658, size = 50, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (a^{2} + x^{n + 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*(a^2+x^(1+n))^(1/2),x, algorithm="fricas")

[Out]

2/3*(a^2 + x^(n + 1))^(3/2)/(n + 1)

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Sympy [A]  time = 5.14821, size = 58, normalized size = 2.64 \begin{align*} \begin{cases} \frac{2 a^{2} \sqrt{a^{2} + x x^{n}}}{3 n + 3} + \frac{2 x x^{n} \sqrt{a^{2} + x x^{n}}}{3 n + 3} & \text{for}\: n \neq -1 \\\sqrt{a^{2} + 1} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16228, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (a^{2} + x^{n + 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*(a^2+x^(1+n))^(1/2),x, algorithm="giac")

[Out]

2/3*(a^2 + x^(n + 1))^(3/2)/(n + 1)