3.2.31 \(\int \frac {(1+\log (x))^5}{x} \, dx\) [131]

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

[Out]

1/6*(1+ln(x))^6

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2339, 30} \begin {gather*} \frac {1}{6} (\log (x)+1)^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Log[x])^5/x,x]

[Out]

(1 + Log[x])^6/6

Rule 30

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

Rule 2339

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{6} (1+\log (x))^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Log[x])^5/x,x]

[Out]

(1 + Log[x])^6/6

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 9, normalized size = 0.90

method	result	size
derivativedivides	\(\frac {\left (1+\ln \left (x \right )\right )^{6}}{6}\)	\(9\)
default	\(\frac {\left (1+\ln \left (x \right )\right )^{6}}{6}\)	\(9\)
norman	\(\ln \left (x \right )^{5}+\ln \left (x \right )+\frac {5 \ln \left (x \right )^{2}}{2}+\frac {10 \ln \left (x \right )^{3}}{3}+\frac {5 \ln \left (x \right )^{4}}{2}+\frac {\ln \left (x \right )^{6}}{6}\)	\(32\)
risch	\(\ln \left (x \right )^{5}+\ln \left (x \right )+\frac {5 \ln \left (x \right )^{2}}{2}+\frac {10 \ln \left (x \right )^{3}}{3}+\frac {5 \ln \left (x \right )^{4}}{2}+\frac {\ln \left (x \right )^{6}}{6}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+ln(x))^5/x,x,method=_RETURNVERBOSE)

[Out]

1/6*(1+ln(x))^6

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{6} \, {\left (\log \left (x\right ) + 1\right )}^{6} \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (8) = 16\).
time = 0.36, size = 31, normalized size = 3.10 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (7) = 14\).
time = 0.14, size = 39, normalized size = 3.90 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (8) = 16\).
time = 3.49, size = 31, normalized size = 3.10 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 0.39, size = 26, normalized size = 2.60 \begin {gather*} \frac {\ln \left (x\right )\,\left (\ln \left (x\right )+2\right )\,\left ({\ln \left (x\right )}^2+\ln \left (x\right )+1\right )\,\left ({\ln \left (x\right )}^2+3\,\ln \left (x\right )+3\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x) + 1)^5/x,x)

[Out]

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

________________________________________________________________________________________