Optimal. Leaf size=16 \[ \frac{x \log (x)}{b \sqrt{b x^2}} \]
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Rubi [A] time = 0.0015001, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 29} \[ \frac{x \log (x)}{b \sqrt{b x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rubi steps
\begin{align*} \int \frac{x^2}{\left (b x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{1}{x} \, dx}{b \sqrt{b x^2}}\\ &=\frac{x \log (x)}{b \sqrt{b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0013106, size = 15, normalized size = 0.94 \[ \frac{x^3 \log (x)}{\left (b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 14, normalized size = 0.9 \begin{align*}{{x}^{3}\ln \left ( x \right ) \left ( b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01693, size = 8, normalized size = 0.5 \begin{align*} \frac{\log \left (x\right )}{b^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47664, size = 38, normalized size = 2.38 \begin{align*} \frac{\sqrt{b x^{2}} \log \left (x\right )}{b^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (b x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16158, size = 28, normalized size = 1.75 \begin{align*} -\frac{\log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2}} \right |}\right )}{b^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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