### 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 = 0.021677, antiderivative size = 9, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0133085, size = 9, normalized size = 1. $\frac{x^{a x}}{a}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^(a*x) + x^(a*x)*Log[x],x]

[Out]

x^(a*x)/a

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Maple [A]  time = 0.013, size = 11, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{ax\ln \left ( x \right ) }}}{a}} \end{align*}

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 = 1.23197, size = 12, normalized size = 1.33 \begin{align*} \frac{x^{a x}}{a} \end{align*}

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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Fricas [A]  time = 2.14332, size = 15, normalized size = 1.67 \begin{align*} \frac{x^{a x}}{a} \end{align*}

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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Sympy [A]  time = 0.298313, size = 10, normalized size = 1.11 \begin{align*} \begin{cases} \frac{x^{a x}}{a} & \text{for}\: a \neq 0 \\x \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{a x} \log \left (x\right ) + x^{a x}\,{d x} \end{align*}

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)