Optimal. Leaf size=34 \[ -\frac {16}{27 x^{3/2}}-\frac {8 \log (x)}{9 x^{3/2}}-\frac {2 \log ^2(x)}{3 x^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, 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 = {2342, 2341}
\begin {gather*} -\frac {16}{27 x^{3/2}}-\frac {2 \log ^2(x)}{3 x^{3/2}}-\frac {8 \log (x)}{9 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2342
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*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.62 \begin {gather*} -\frac {2 \left (8+12 \log (x)+9 \log ^2(x)\right )}{27 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 23, normalized size = 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 | \(-\frac {16}{27 x^{\frac {3}{2}}}-\frac {8 \ln \left (x \right )}{9 x^{\frac {3}{2}}}-\frac {2 \ln \left (x \right )^{2}}{3 x^{\frac {3}{2}}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 22, normalized size = 0.65 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.82, size = 17, normalized size = 0.50 \begin {gather*} -\frac {2 \, {\left (9 \, \log \left (x\right )^{2} + 12 \, \log \left (x\right ) + 8\right )}}{27 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.00, size = 34, normalized size = 1.00 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.01, size = 22, normalized size = 0.65 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 17, normalized size = 0.50 \begin {gather*} -\frac {18\,{\ln \left (x\right )}^2+24\,\ln \left (x\right )+16}{27\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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