Integrand size = 15, antiderivative size = 33 \[ \int \left (k^{x/2}+x^{\sqrt {k}}\right ) \, dx=\frac {x^{1+\sqrt {k}}}{1+\sqrt {k}}+\frac {2 k^{x/2}}{\log (k)} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2225} \[ \int \left (k^{x/2}+x^{\sqrt {k}}\right ) \, dx=\frac {2 k^{x/2}}{\log (k)}+\frac {x^{\sqrt {k}+1}}{\sqrt {k}+1} \]
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Rule 2225
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \left (k^{x/2}+x^{\sqrt {k}}\right ) \, dx=\frac {x^{1+\sqrt {k}}}{1+\sqrt {k}}+\frac {2 k^{x/2}}{\log (k)} \]
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Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {2 k^{\frac {x}{2}}}{\ln \left (k \right )}+\frac {x^{1+\sqrt {k}}}{1+\sqrt {k}}\) | \(28\) |
parts | \(\frac {2 k^{\frac {x}{2}}}{\ln \left (k \right )}+\frac {x^{1+\sqrt {k}}}{1+\sqrt {k}}\) | \(28\) |
norman | \(\frac {x \,{\mathrm e}^{\sqrt {k}\, \ln \left (x \right )}}{1+\sqrt {k}}+\frac {2 \,{\mathrm e}^{\frac {x \ln \left (k \right )}{2}}}{\ln \left (k \right )}\) | \(30\) |
risch | \(\frac {2 k^{\frac {x}{2}}}{\ln \left (k \right )}+\frac {\left (\sqrt {k}-1\right ) x \,x^{\sqrt {k}}}{k -1}\) | \(30\) |
parallelrisch | \(\frac {x \,x^{\sqrt {k}} \ln \left (k \right )+2 k^{\frac {x}{2}} \sqrt {k}+2 k^{\frac {x}{2}}}{\left (1+\sqrt {k}\right ) \ln \left (k \right )}\) | \(40\) |
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none
Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \left (k^{x/2}+x^{\sqrt {k}}\right ) \, dx=\frac {2 \, {\left (k - 1\right )} k^{\frac {1}{2} \, x} + {\left (\sqrt {k} x \log \left (k\right ) - x \log \left (k\right )\right )} x^{\left (\sqrt {k}\right )}}{{\left (k - 1\right )} \log \left (k\right )} \]
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Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \left (k^{x/2}+x^{\sqrt {k}}\right ) \, dx=\begin {cases} \frac {2 k^{\frac {x}{2}}}{\log {\left (k \right )}} & \text {for}\: \log {\left (k \right )} \neq 0 \\x & \text {otherwise} \end {cases} + \begin {cases} \frac {x^{\sqrt {k} + 1}}{\sqrt {k} + 1} & \text {for}\: \sqrt {k} \neq -1 \\\log {\left (x \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \left (k^{x/2}+x^{\sqrt {k}}\right ) \, dx=\frac {x^{\sqrt {k} + 1}}{\sqrt {k} + 1} + \frac {2 \, k^{\frac {1}{2} \, x}}{\log \left (k\right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \left (k^{x/2}+x^{\sqrt {k}}\right ) \, dx=\frac {x^{\sqrt {k} + 1}}{\sqrt {k} + 1} + \frac {2 \, \sqrt {k^{x}}}{\log \left (k\right )} \]
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Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \left (k^{x/2}+x^{\sqrt {k}}\right ) \, dx=\frac {2\,k^{x/2}}{\ln \left (k\right )}+\frac {x\,x^{\sqrt {k}}}{\sqrt {k}+1} \]
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