\(\int (x^{a x}+x^{a x} \log (x)) \, dx\) [299]

Optimal result

Integrand size = 14, antiderivative size = 9 \[ \int \left (x^{a x}+x^{a x} \log (x)\right ) \, dx=\frac {x^{a x}}{a} \]

[Out]

x^(a*x)/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2633} \[ \int \left (x^{a x}+x^{a x} \log (x)\right ) \, dx=\frac {x^{a x}}{a} \]

[In]

Int[x^(a*x) + x^(a*x)*Log[x],x]

[Out]

x^(a*x)/a

Rule 2633

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*} \text {integral}& = \int x^{a x} \, dx+\int x^{a x} \log (x) \, dx \\ & = \frac {x^{a x}}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \left (x^{a x}+x^{a x} \log (x)\right ) \, dx=\frac {x^{a x}}{a} \]

[In]

Integrate[x^(a*x) + x^(a*x)*Log[x],x]

[Out]

x^(a*x)/a

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11

method	result	size
risch	\(\frac {x^{a x}}{a}\)	\(10\)
parallelrisch	\(\frac {x^{a x}}{a}\)	\(10\)
norman	\(\frac {{\mathrm e}^{a x \ln \left (x \right )}}{a}\)	\(11\)

[In]

int(x^(a*x)+x^(a*x)*ln(x),x,method=_RETURNVERBOSE)

[Out]

x^(a*x)/a

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \left (x^{a x}+x^{a x} \log (x)\right ) \, dx=\frac {x^{a x}}{a} \]

[In]

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

[Out]

x^(a*x)/a

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.11 \[ \int \left (x^{a x}+x^{a x} \log (x)\right ) \, dx=\begin {cases} \frac {x^{a x}}{a} & \text {for}\: a \neq 0 \\x \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \left (x^{a x}+x^{a x} \log (x)\right ) \, dx=\frac {x^{a x}}{a} \]

[In]

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

[Out]

x^(a*x)/a

Giac [F]

\[ \int \left (x^{a x}+x^{a x} \log (x)\right ) \, dx=\int { x^{a x} \log \left (x\right ) + x^{a x} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.46 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \left (x^{a x}+x^{a x} \log (x)\right ) \, dx=\frac {x^{a\,x}}{a} \]

[In]

int(x^(a*x) + x^(a*x)*log(x),x)

[Out]

x^(a*x)/a