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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = 0.01, size = 22, normalized size = 1.00 \[ \frac {2 \left (a^2+x^{n+1}\right )^{3/2}}{3 (n+1)} \]

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))

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 18, normalized size = 0.82 \[ \frac {2 \, {\left (a^{2} + x^{n + 1}\right )}^{\frac {3}{2}}}{3 \, {\left (n + 1\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 18, normalized size = 0.82 \[ \frac {2 \, {\left (a^{2} + x^{n + 1}\right )}^{\frac {3}{2}}}{3 \, {\left (n + 1\right )}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 19, normalized size = 0.86 \[ \frac {2 \left (a^{2}+x \,x^{n}\right )^{\frac {3}{2}}}{3 \left (n +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 18, normalized size = 0.82 \[ \frac {2 \, {\left (a^{2} + x^{n + 1}\right )}^{\frac {3}{2}}}{3 \, {\left (n + 1\right )}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.23, size = 20, normalized size = 0.91 \[ \frac {2\,{\left (x^{n+1}+a^2\right )}^{3/2}}{3\,\left (n+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^n*(x^(n + 1) + a^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 4.39, size = 58, normalized size = 2.64 \[ \begin {cases} \frac {2 a^{2} \sqrt {a^{2} + x x^{n}}}{3 n + 3} + \frac {2 x x^{n} \sqrt {a^{2} + x x^{n}}}{3 n + 3} & \text {for}\: n \neq -1 \\\sqrt {a^{2} + 1} \log {\relax (x )} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________