3.299 \(\int (x^{a x}+x^{a x} \log (x)) \, dx\)

Optimal. Leaf size=9 \[ \frac {x^{a x}}{a} \]

[Out]

x^(a*x)/a

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Rubi [A]  time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2553} \[ \frac {x^{a x}}{a} \]

Antiderivative was successfully verified.

[In]

Int[x^(a*x) + x^(a*x)*Log[x],x]

[Out]

x^(a*x)/a

Rule 2553

Int[Log[u_]*(u_)^((a_.)*(x_)), x_Symbol] :> Simp[u^(a*x)/a, x] - Int[SimplifyIntegrand[x*u^(a*x - 1)*D[u, x],
x], x] /; FreeQ[a, x] && InverseFunctionFreeQ[u, x]

Rubi steps

\begin {align*} \int \left (x^{a x}+x^{a x} \log (x)\right ) \, dx &=\int x^{a x} \, dx+\int x^{a x} \log (x) \, dx\\ &=\frac {x^{a x}}{a}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 1.00 \[ \frac {x^{a x}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(a*x) + x^(a*x)*Log[x],x]

[Out]

x^(a*x)/a

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fricas [A]  time = 0.44, size = 9, normalized size = 1.00 \[ \frac {x^{a x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(a*x)+x^(a*x)*log(x),x, algorithm="fricas")

[Out]

x^(a*x)/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{a x} \log \relax (x) + x^{a x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(a*x)+x^(a*x)*log(x),x, algorithm="giac")

[Out]

integrate(x^(a*x)*log(x) + x^(a*x), x)

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maple [A]  time = 0.07, size = 11, normalized size = 1.22 \[ \frac {{\mathrm e}^{a x \ln \relax (x )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(a*x)+x^(a*x)*ln(x),x)

[Out]

1/a*exp(a*x*ln(x))

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maxima [A]  time = 0.82, size = 9, normalized size = 1.00 \[ \frac {x^{a x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(a*x)+x^(a*x)*log(x),x, algorithm="maxima")

[Out]

x^(a*x)/a

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mupad [B]  time = 0.37, size = 9, normalized size = 1.00 \[ \frac {x^{a\,x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(a*x) + x^(a*x)*log(x),x)

[Out]

x^(a*x)/a

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sympy [A]  time = 0.26, size = 10, normalized size = 1.11 \[ \begin {cases} \frac {x^{a x}}{a} & \text {for}\: a \neq 0 \\x \log {\relax (x )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(a*x)+x**(a*x)*ln(x),x)

[Out]

Piecewise((x**(a*x)/a, Ne(a, 0)), (x*log(x), True))

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