3.742 \(\int (k^{x/2}+x^{\sqrt{k}}) \, dx\)

Optimal. Leaf size=33 \[ \frac{2 k^{x/2}}{\log (k)}+\frac{x^{\sqrt{k}+1}}{\sqrt{k}+1} \]

[Out]

x^(1 + Sqrt[k])/(1 + Sqrt[k]) + (2*k^(x/2))/Log[k]

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Rubi [A]  time = 0.0090667, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2194} \[ \frac{2 k^{x/2}}{\log (k)}+\frac{x^{\sqrt{k}+1}}{\sqrt{k}+1} \]

Antiderivative was successfully verified.

[In]

Int[k^(x/2) + x^Sqrt[k],x]

[Out]

x^(1 + Sqrt[k])/(1 + Sqrt[k]) + (2*k^(x/2))/Log[k]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \left (k^{x/2}+x^{\sqrt{k}}\right ) \, dx &=\frac{x^{1+\sqrt{k}}}{1+\sqrt{k}}+\int k^{x/2} \, dx\\ &=\frac{x^{1+\sqrt{k}}}{1+\sqrt{k}}+\frac{2 k^{x/2}}{\log (k)}\\ \end{align*}

Mathematica [A]  time = 0.0160845, size = 33, normalized size = 1. \[ \frac{2 k^{x/2}}{\log (k)}+\frac{x^{\sqrt{k}+1}}{\sqrt{k}+1} \]

Antiderivative was successfully verified.

[In]

Integrate[k^(x/2) + x^Sqrt[k],x]

[Out]

x^(1 + Sqrt[k])/(1 + Sqrt[k]) + (2*k^(x/2))/Log[k]

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Maple [A]  time = 0.037, size = 28, normalized size = 0.9 \begin{align*} 2\,{\frac{{k}^{x/2}}{\ln \left ( k \right ) }}+{{x}^{1+\sqrt{k}} \left ( 1+\sqrt{k} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*k^(1/2*x)/ln(k)+x^(1+k^(1/2))/(1+k^(1/2))

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Maxima [A]  time = 0.976703, size = 36, normalized size = 1.09 \begin{align*} \frac{x^{\sqrt{k} + 1}}{\sqrt{k} + 1} + \frac{2 \, k^{\frac{1}{2} \, x}}{\log \left (k\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(sqrt(k) + 1)/(sqrt(k) + 1) + 2*k^(1/2*x)/log(k)

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Fricas [A]  time = 0.792184, size = 111, normalized size = 3.36 \begin{align*} \frac{2 \,{\left (k - 1\right )} k^{\frac{1}{2} \, x} +{\left (\sqrt{k} x \log \left (k\right ) - x \log \left (k\right )\right )} x^{\left (\sqrt{k}\right )}}{{\left (k - 1\right )} \log \left (k\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(k - 1)*k^(1/2*x) + (sqrt(k)*x*log(k) - x*log(k))*x^sqrt(k))/((k - 1)*log(k))

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Sympy [A]  time = 0.101513, size = 36, normalized size = 1.09 \begin{align*} \begin{cases} \frac{2 k^{\frac{x}{2}}}{\log{\left (k \right )}} & \text{for}\: \log{\left (k \right )} \neq 0 \\x & \text{otherwise} \end{cases} + \begin{cases} \frac{x^{\sqrt{k} + 1}}{\sqrt{k} + 1} & \text{for}\: \sqrt{k} \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*k**(x/2)/log(k), Ne(log(k), 0)), (x, True)) + Piecewise((x**(sqrt(k) + 1)/(sqrt(k) + 1), Ne(sqrt(
k), -1)), (log(x), True))

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Giac [A]  time = 1.27404, size = 36, normalized size = 1.09 \begin{align*} \frac{x^{\sqrt{k} + 1}}{\sqrt{k} + 1} + \frac{2 \, k^{\frac{1}{2} \, x}}{\log \left (k\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(sqrt(k) + 1)/(sqrt(k) + 1) + 2*k^(1/2*x)/log(k)