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]

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

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 1.00 \[ \frac {2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]

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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fricas [A]  time = 0.65, size = 16, normalized size = 0.80 \[ \frac {2 \, {\left (x^{n + 1} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (n + 1\right )}} \]

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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giac [A]  time = 0.15, size = 16, normalized size = 0.80 \[ \frac {2 \, {\left (x^{n + 1} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (n + 1\right )}} \]

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)

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maple [A]  time = 0.03, size = 17, normalized size = 0.85 \[ \frac {2 \left (x \,x^{n}+1\right )^{\frac {3}{2}}}{3 \left (n +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 16, normalized size = 0.80 \[ \frac {2 \, {\left (x^{n + 1} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (n + 1\right )}} \]

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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mupad [B]  time = 1.25, size = 18, normalized size = 0.90 \[ \frac {2\,{\left (x^{n+1}+1\right )}^{3/2}}{3\,\left (n+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^n*(x^(n + 1) + 1)^(1/2),x)

[Out]

(2*(x^(n + 1) + 1)^(3/2))/(3*(n + 1))

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sympy [A]  time = 4.45, size = 48, normalized size = 2.40 \[ \begin {cases} \frac {2 x x^{n} \sqrt {x x^{n} + 1}}{3 n + 3} + \frac {2 \sqrt {x x^{n} + 1}}{3 n + 3} & \text {for}\: n \neq -1 \\\sqrt {2} \log {\relax (x )} & \text {otherwise} \end {cases} \]

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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