Optimal. Leaf size=33 \[ \frac {2 k^{x/2}}{\log (k)}+\frac {x^{\sqrt {k}+1}}{\sqrt {k}+1} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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.
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Rule 2194
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*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ \frac {2 k^{x/2}}{\log (k)}+\frac {x^{\sqrt {k}+1}}{\sqrt {k}+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 40, normalized size = 1.21 \[ \frac {2 \, {\left (k - 1\right )} k^{\frac {1}{2} \, x} + {\left (\sqrt {k} x \log \relax (k) - x \log \relax (k)\right )} x^{\left (\sqrt {k}\right )}}{{\left (k - 1\right )} \log \relax (k)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 27, normalized size = 0.82 \[ \frac {x^{\sqrt {k} + 1}}{\sqrt {k} + 1} + \frac {2 \, \sqrt {k^{x}}}{\log \relax (k)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 28, normalized size = 0.85 \[ \frac {x^{\sqrt {k}+1}}{\sqrt {k}+1}+\frac {2 k^{\frac {x}{2}}}{\ln \relax (k )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 27, normalized size = 0.82 \[ \frac {x^{\sqrt {k} + 1}}{\sqrt {k} + 1} + \frac {2 \, k^{\frac {1}{2} \, x}}{\log \relax (k)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 26, normalized size = 0.79 \[ \frac {2\,k^{x/2}}{\ln \relax (k)}+\frac {x\,x^{\sqrt {k}}}{\sqrt {k}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 36, normalized size = 1.09 \[ \begin {cases} \frac {2 k^{\frac {x}{2}}}{\log {\relax (k )}} & \text {for}\: \log {\relax (k )} \neq 0 \\x & \text {otherwise} \end {cases} + \begin {cases} \frac {x^{\sqrt {k} + 1}}{\sqrt {k} + 1} & \text {for}\: \sqrt {k} \neq -1 \\\log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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