3.725 \(\int \left (k^{x/2}+x^{\sqrt{k}}\right ) \, dx\)

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 = 0.0188467, 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 \[ \frac{2 k^{x/2}}{\log (k)}+\frac{x^{\sqrt{k}+1}}{\sqrt{k}+1} \]

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]

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Rubi in Sympy [A]  time = 1.57341, size = 24, normalized size = 0.73 \[ \frac{2 k^{\frac{x}{2}}}{\log{\left (k \right )}} + \frac{x^{\sqrt{k} + 1}}{\sqrt{k} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(k**(1/2*x)+x**(k**(1/2)),x)

[Out]

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

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Mathematica [A]  time = 0.0233965, 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 = 0.003, size = 28, normalized size = 0.9 \[ 2\,{\frac{{k}^{x/2}}{\ln \left ( k \right ) }}+{1{x}^{1+\sqrt{k}} \left ( 1+\sqrt{k} \right ) ^{-1}} \]

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 = 0.774473, size = 36, normalized size = 1.09 \[ \frac{x^{\sqrt{k} + 1}}{\sqrt{k} + 1} + \frac{2 \, k^{\frac{1}{2} \, x}}{\log \left (k\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(k^(1/2*x) + x^sqrt(k),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 = 0.317117, size = 46, normalized size = 1.39 \[ \frac{x x^{\left (\sqrt{k}\right )} \log \left (k\right ) + 2 \, k^{\frac{1}{2} \, x}{\left (\sqrt{k} + 1\right )}}{\sqrt{k} \log \left (k\right ) + \log \left (k\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.086396, size = 36, normalized size = 1.09 \[ \begin{cases} \frac{2 k^{\frac{x}{2}}}{\log{\left (k \right )}} & \text{for}\: \log{\left (k \right )} \neq 0 \\x & \text{otherwise} \end{cases} + \begin{cases} \frac{x^{\sqrt{k} + 1}}{\sqrt{k} + 1} & \text{for}\: \sqrt{k} \neq -1 \\\log{\left (x \right )} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.225033, size = 36, normalized size = 1.09 \[ \frac{x^{\sqrt{k} + 1}}{\sqrt{k} + 1} + \frac{2 \, k^{\frac{1}{2} \, x}}{{\rm ln}\left (k\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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