\(\int \frac {-50+x \log ^3(x)+162 \log ^4(x)}{x \log ^3(x)} \, dx\) [8554]

Optimal result

Integrand size = 22, antiderivative size = 17 \[ \int \frac {-50+x \log ^3(x)+162 \log ^4(x)}{x \log ^3(x)} \, dx=x+\frac {\left (-5-9 \log ^2(x)\right )^2}{\log ^2(x)} \]

[Out]

x+(9*ln(x)^2+5)^2/ln(x)^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6874, 2339, 30, 2338} \[ \int \frac {-50+x \log ^3(x)+162 \log ^4(x)}{x \log ^3(x)} \, dx=x+81 \log ^2(x)+\frac {25}{\log ^2(x)} \]

[In]

Int[(-50 + x*Log[x]^3 + 162*Log[x]^4)/(x*Log[x]^3),x]

[Out]

x + 25/Log[x]^2 + 81*Log[x]^2

Rule 30

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

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {50}{x \log ^3(x)}+\frac {162 \log (x)}{x}\right ) \, dx \\ & = x-50 \int \frac {1}{x \log ^3(x)} \, dx+162 \int \frac {\log (x)}{x} \, dx \\ & = x+81 \log ^2(x)-50 \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (x)\right ) \\ & = x+\frac {25}{\log ^2(x)}+81 \log ^2(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {-50+x \log ^3(x)+162 \log ^4(x)}{x \log ^3(x)} \, dx=x+\frac {25}{\log ^2(x)}+81 \log ^2(x) \]

[In]

Integrate[(-50 + x*Log[x]^3 + 162*Log[x]^4)/(x*Log[x]^3),x]

[Out]

x + 25/Log[x]^2 + 81*Log[x]^2

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88

method	result	size
default	\(81 \ln \left (x \right )^{2}+x +\frac {25}{\ln \left (x \right )^{2}}\)	\(15\)
risch	\(81 \ln \left (x \right )^{2}+x +\frac {25}{\ln \left (x \right )^{2}}\)	\(15\)
parts	\(81 \ln \left (x \right )^{2}+x +\frac {25}{\ln \left (x \right )^{2}}\)	\(15\)
norman	\(\frac {25+x \ln \left (x \right )^{2}+81 \ln \left (x \right )^{4}}{\ln \left (x \right )^{2}}\)	\(20\)
parallelrisch	\(\frac {25+x \ln \left (x \right )^{2}+81 \ln \left (x \right )^{4}}{\ln \left (x \right )^{2}}\)	\(20\)

[In]

int((162*ln(x)^4+x*ln(x)^3-50)/x/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

81*ln(x)^2+x+25/ln(x)^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {-50+x \log ^3(x)+162 \log ^4(x)}{x \log ^3(x)} \, dx=\frac {81 \, \log \left (x\right )^{4} + x \log \left (x\right )^{2} + 25}{\log \left (x\right )^{2}} \]

[In]

integrate((162*log(x)^4+x*log(x)^3-50)/x/log(x)^3,x, algorithm="fricas")

[Out]

(81*log(x)^4 + x*log(x)^2 + 25)/log(x)^2

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {-50+x \log ^3(x)+162 \log ^4(x)}{x \log ^3(x)} \, dx=x + 81 \log {\left (x \right )}^{2} + \frac {25}{\log {\left (x \right )}^{2}} \]

[In]

integrate((162*ln(x)**4+x*ln(x)**3-50)/x/ln(x)**3,x)

[Out]

x + 81*log(x)**2 + 25/log(x)**2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {-50+x \log ^3(x)+162 \log ^4(x)}{x \log ^3(x)} \, dx=81 \, \log \left (x\right )^{2} + x + \frac {25}{\log \left (x\right )^{2}} \]

[In]

integrate((162*log(x)^4+x*log(x)^3-50)/x/log(x)^3,x, algorithm="maxima")

[Out]

81*log(x)^2 + x + 25/log(x)^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {-50+x \log ^3(x)+162 \log ^4(x)}{x \log ^3(x)} \, dx=81 \, \log \left (x\right )^{2} + x + \frac {25}{\log \left (x\right )^{2}} \]

[In]

integrate((162*log(x)^4+x*log(x)^3-50)/x/log(x)^3,x, algorithm="giac")

[Out]

81*log(x)^2 + x + 25/log(x)^2

Mupad [B] (verification not implemented)

Time = 12.76 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {-50+x \log ^3(x)+162 \log ^4(x)}{x \log ^3(x)} \, dx=x+\frac {25}{{\ln \left (x\right )}^2}+81\,{\ln \left (x\right )}^2 \]

[In]

int((x*log(x)^3 + 162*log(x)^4 - 50)/(x*log(x)^3),x)

[Out]

x + 25/log(x)^2 + 81*log(x)^2