Optimal. Leaf size=64 \[ \frac{(d x)^{3/2} e^{-\frac{3 a}{2 b n}} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{b d n} \]
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Rubi [A] time = 0.060284, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2310, 2178} \[ \frac{(d x)^{3/2} e^{-\frac{3 a}{2 b n}} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{b d n} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{\sqrt{d x}}{a+b \log \left (c x^n\right )} \, dx &=\frac{\left ((d x)^{3/2} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{d n}\\ &=\frac{e^{-\frac{3 a}{2 b n}} (d x)^{3/2} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{b d n}\\ \end{align*}
Mathematica [A] time = 0.0685466, size = 62, normalized size = 0.97 \[ \frac{x \sqrt{d x} e^{-\frac{3 a}{2 b n}} \left (c x^n\right )^{\left .-\frac{3}{2}\right /n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b\ln \left ( c{x}^{n} \right ) }\sqrt{dx}}\, dx \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*} 2 \, b \sqrt{d} n \int \frac{\sqrt{x}}{3 \,{\left (b^{2} \log \left (c\right )^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x^{n}\right )\right )}}\,{d x} + \frac{2 \, \sqrt{d} x^{\frac{3}{2}}}{3 \,{\left (b \log \left (c\right ) + b \log \left (x^{n}\right ) + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x}}{b \log \left (c x^{n}\right ) + a}, x\right ) \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{\sqrt{d x}}{a + b \log{\left (c x^{n} \right )}}\, dx \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{\sqrt{d x}}{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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