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]

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

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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.

[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*}

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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.

[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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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.

[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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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.

[In]

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

[Out]

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

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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.

[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 = 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.

[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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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.

[In]

int(k^(x/2) + x^(k^(1/2)),x)

[Out]

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

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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.

[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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