\(\int \frac {\log ^2(x)}{x^{5/2}} \, dx\) [610]

Optimal result

Integrand size = 10, antiderivative size = 34 \[ \int \frac {\log ^2(x)}{x^{5/2}} \, dx=-\frac {16}{27 x^{3/2}}-\frac {8 \log (x)}{9 x^{3/2}}-\frac {2 \log ^2(x)}{3 x^{3/2}} \]

[Out]

-16/27/x^(3/2)-8/9*ln(x)/x^(3/2)-2/3*ln(x)^2/x^(3/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2342, 2341} \[ \int \frac {\log ^2(x)}{x^{5/2}} \, dx=-\frac {16}{27 x^{3/2}}-\frac {2 \log ^2(x)}{3 x^{3/2}}-\frac {8 \log (x)}{9 x^{3/2}} \]

[In]

Int[Log[x]^2/x^(5/2),x]

[Out]

-16/(27*x^(3/2)) - (8*Log[x])/(9*x^(3/2)) - (2*Log[x]^2)/(3*x^(3/2))

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \log ^2(x)}{3 x^{3/2}}+\frac {4}{3} \int \frac {\log (x)}{x^{5/2}} \, dx \\ & = -\frac {16}{27 x^{3/2}}-\frac {8 \log (x)}{9 x^{3/2}}-\frac {2 \log ^2(x)}{3 x^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62 \[ \int \frac {\log ^2(x)}{x^{5/2}} \, dx=-\frac {2 \left (8+12 \log (x)+9 \log ^2(x)\right )}{27 x^{3/2}} \]

[In]

Integrate[Log[x]^2/x^(5/2),x]

[Out]

(-2*(8 + 12*Log[x] + 9*Log[x]^2))/(27*x^(3/2))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68

method	result	size
derivativedivides	\(-\frac {16}{27 x^{\frac {3}{2}}}-\frac {8 \ln \left (x \right )}{9 x^{\frac {3}{2}}}-\frac {2 \ln \left (x \right )^{2}}{3 x^{\frac {3}{2}}}\)	\(23\)
default	\(-\frac {16}{27 x^{\frac {3}{2}}}-\frac {8 \ln \left (x \right )}{9 x^{\frac {3}{2}}}-\frac {2 \ln \left (x \right )^{2}}{3 x^{\frac {3}{2}}}\)	\(23\)
risch	\(-\frac {16}{27 x^{\frac {3}{2}}}-\frac {8 \ln \left (x \right )}{9 x^{\frac {3}{2}}}-\frac {2 \ln \left (x \right )^{2}}{3 x^{\frac {3}{2}}}\)	\(23\)

[In]

int(ln(x)^2/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

-16/27/x^(3/2)-8/9*ln(x)/x^(3/2)-2/3*ln(x)^2/x^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.50 \[ \int \frac {\log ^2(x)}{x^{5/2}} \, dx=-\frac {2 \, {\left (9 \, \log \left (x\right )^{2} + 12 \, \log \left (x\right ) + 8\right )}}{27 \, x^{\frac {3}{2}}} \]

[In]

integrate(log(x)^2/x^(5/2),x, algorithm="fricas")

[Out]

-2/27*(9*log(x)^2 + 12*log(x) + 8)/x^(3/2)

Sympy [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^2(x)}{x^{5/2}} \, dx=- \frac {2 \log {\left (x \right )}^{2}}{3 x^{\frac {3}{2}}} - \frac {8 \log {\left (x \right )}}{9 x^{\frac {3}{2}}} - \frac {16}{27 x^{\frac {3}{2}}} \]

[In]

integrate(ln(x)**2/x**(5/2),x)

[Out]

-2*log(x)**2/(3*x**(3/2)) - 8*log(x)/(9*x**(3/2)) - 16/(27*x**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {\log ^2(x)}{x^{5/2}} \, dx=-\frac {2 \, \log \left (x\right )^{2}}{3 \, x^{\frac {3}{2}}} - \frac {8 \, \log \left (x\right )}{9 \, x^{\frac {3}{2}}} - \frac {16}{27 \, x^{\frac {3}{2}}} \]

[In]

integrate(log(x)^2/x^(5/2),x, algorithm="maxima")

[Out]

-2/3*log(x)^2/x^(3/2) - 8/9*log(x)/x^(3/2) - 16/27/x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {\log ^2(x)}{x^{5/2}} \, dx=-\frac {2 \, \log \left (x\right )^{2}}{3 \, x^{\frac {3}{2}}} - \frac {8 \, \log \left (x\right )}{9 \, x^{\frac {3}{2}}} - \frac {16}{27 \, x^{\frac {3}{2}}} \]

[In]

integrate(log(x)^2/x^(5/2),x, algorithm="giac")

[Out]

-2/3*log(x)^2/x^(3/2) - 8/9*log(x)/x^(3/2) - 16/27/x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.50 \[ \int \frac {\log ^2(x)}{x^{5/2}} \, dx=-\frac {18\,{\ln \left (x\right )}^2+24\,\ln \left (x\right )+16}{27\,x^{3/2}} \]

[In]

int(log(x)^2/x^(5/2),x)

[Out]

-(24*log(x) + 18*log(x)^2 + 16)/(27*x^(3/2))