3.610 \(\int \frac{\log ^2(x)}{x^{5/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac{16}{27 x^{3/2}}-\frac{2 \log ^2(x)}{3 x^{3/2}}-\frac{8 \log (x)}{9 x^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0202339, antiderivative size = 34, 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 = {2305, 2304} \[ -\frac{16}{27 x^{3/2}}-\frac{2 \log ^2(x)}{3 x^{3/2}}-\frac{8 \log (x)}{9 x^{3/2}} \]

Antiderivative was successfully verified.

[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 2305

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]

Rule 2304

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
]

Rubi steps

\begin{align*} \int \frac{\log ^2(x)}{x^{5/2}} \, dx &=-\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]  time = 0.0068995, size = 21, normalized size = 0.62 \[ -\frac{2 \left (9 \log ^2(x)+12 \log (x)+8\right )}{27 x^{3/2}} \]

Antiderivative was successfully verified.

[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]  time = 0.016, size = 23, normalized size = 0.7 \begin{align*} -{\frac{16}{27}{x}^{-{\frac{3}{2}}}}-{\frac{8\,\ln \left ( x \right ) }{9}{x}^{-{\frac{3}{2}}}}-{\frac{2\, \left ( \ln \left ( x \right ) \right ) ^{2}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.955466, size = 30, normalized size = 0.88 \begin{align*} -\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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 2.15775, size = 59, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (9 \, \log \left (x\right )^{2} + 12 \, \log \left (x\right ) + 8\right )}}{27 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 6.13746, size = 34, normalized size = 1. \begin{align*} - \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [A]  time = 1.16687, size = 30, normalized size = 0.88 \begin{align*} -\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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)