3.131 \(\int \frac{(1+\log (x))^5}{x} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{6} (\log (x)+1)^6 \]

[Out]

(1 + Log[x])^6/6

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Rubi [A]  time = 0.0165482, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2302, 30} \[ \frac{1}{6} (\log (x)+1)^6 \]

Antiderivative was successfully verified.

[In]

Int[(1 + Log[x])^5/x,x]

[Out]

(1 + Log[x])^6/6

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(1+\log (x))^5}{x} \, dx &=\operatorname{Subst}\left (\int x^5 \, dx,x,1+\log (x)\right )\\ &=\frac{1}{6} (1+\log (x))^6\\ \end{align*}

Mathematica [A]  time = 0.0025723, size = 10, normalized size = 1. \[ \frac{1}{6} (\log (x)+1)^6 \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Log[x])^5/x,x]

[Out]

(1 + Log[x])^6/6

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Maple [B]  time = 0.004, size = 33, normalized size = 3.3 \begin{align*}{\frac{ \left ( \ln \left ( x \right ) \right ) ^{6}}{6}}+ \left ( \ln \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \ln \left ( x \right ) \right ) ^{4}}{2}}+{\frac{10\, \left ( \ln \left ( x \right ) \right ) ^{3}}{3}}+{\frac{5\, \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}+\ln \left ( x \right ) +{\frac{1}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+ln(x))^5/x,x)

[Out]

1/6*ln(x)^6+ln(x)^5+5/2*ln(x)^4+10/3*ln(x)^3+5/2*ln(x)^2+ln(x)+1/6

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Maxima [A]  time = 1.12711, size = 11, normalized size = 1.1 \begin{align*} \frac{1}{6} \,{\left (\log \left (x\right ) + 1\right )}^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+log(x))^5/x,x, algorithm="maxima")

[Out]

1/6*(log(x) + 1)^6

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Fricas [B]  time = 2.02142, size = 108, normalized size = 10.8 \begin{align*} \frac{1}{6} \, \log \left (x\right )^{6} + \log \left (x\right )^{5} + \frac{5}{2} \, \log \left (x\right )^{4} + \frac{10}{3} \, \log \left (x\right )^{3} + \frac{5}{2} \, \log \left (x\right )^{2} + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+log(x))^5/x,x, algorithm="fricas")

[Out]

1/6*log(x)^6 + log(x)^5 + 5/2*log(x)^4 + 10/3*log(x)^3 + 5/2*log(x)^2 + log(x)

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Sympy [B]  time = 0.138367, size = 39, normalized size = 3.9 \begin{align*} \frac{\log{\left (x \right )}^{6}}{6} + \log{\left (x \right )}^{5} + \frac{5 \log{\left (x \right )}^{4}}{2} + \frac{10 \log{\left (x \right )}^{3}}{3} + \frac{5 \log{\left (x \right )}^{2}}{2} + \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+ln(x))**5/x,x)

[Out]

log(x)**6/6 + log(x)**5 + 5*log(x)**4/2 + 10*log(x)**3/3 + 5*log(x)**2/2 + log(x)

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Giac [B]  time = 1.25287, size = 42, normalized size = 4.2 \begin{align*} \frac{1}{6} \, \log \left (x\right )^{6} + \log \left (x\right )^{5} + \frac{5}{2} \, \log \left (x\right )^{4} + \frac{10}{3} \, \log \left (x\right )^{3} + \frac{5}{2} \, \log \left (x\right )^{2} + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+log(x))^5/x,x, algorithm="giac")

[Out]

1/6*log(x)^6 + log(x)^5 + 5/2*log(x)^4 + 10/3*log(x)^3 + 5/2*log(x)^2 + log(x)