Optimal. Leaf size=22 \[ \frac{x^{n/2} \log (x)}{b \sqrt{b x^n}} \]
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Rubi [A] time = 0.0018731, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 29} \[ \frac{x^{n/2} \log (x)}{b \sqrt{b x^n}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rubi steps
\begin{align*} \int \frac{x^{-1+\frac{3 n}{2}}}{\left (b x^n\right )^{3/2}} \, dx &=\frac{x^{n/2} \int \frac{1}{x} \, dx}{b \sqrt{b x^n}}\\ &=\frac{x^{n/2} \log (x)}{b \sqrt{b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0031429, size = 19, normalized size = 0.86 \[ \frac{x^{3 n/2} \log (x)}{\left (b x^n\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 23, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( x \right ) }{b}{x}^{{\frac{n}{2}}}{\frac{1}{\sqrt{b \left ({x}^{{\frac{n}{2}}} \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2} \, n - 1}}{\left (b x^{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80527, size = 49, normalized size = 2.23 \begin{align*} \frac{\sqrt{b x^{n}} \log \left (x\right )}{b^{2} x^{\frac{1}{2} \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2} \, n - 1}}{\left (b x^{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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