Optimal. Leaf size=64 \[ \frac{e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n \sqrt{d x}} \]
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Rubi [A] time = 0.0607804, 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} \[ \frac{e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n \sqrt{d x}} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\left (c x^n\right )^{\left .\frac{1}{2}\right /n} \operatorname{Subst}\left (\int \frac{e^{-\frac{x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{d n \sqrt{d x}}\\ &=\frac{e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n \sqrt{d x}}\\ \end{align*}
Mathematica [A] time = 0.0712396, size = 62, normalized size = 0.97 \[ \frac{x e^{\frac{a}{2 b n}} \left (c x^n\right )^{\left .\frac{1}{2}\right /n} \text{Ei}\left (-\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b n (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.109, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b\ln \left ( c{x}^{n} \right ) } \left ( dx \right ) ^{-{\frac{3}{2}}}}\, 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 n \int \frac{1}{{\left (b^{2} d^{\frac{3}{2}} \log \left (c\right )^{2} + b^{2} d^{\frac{3}{2}} \log \left (x^{n}\right )^{2} + 2 \, a b d^{\frac{3}{2}} \log \left (c\right ) + a^{2} d^{\frac{3}{2}} + 2 \,{\left (b^{2} d^{\frac{3}{2}} \log \left (c\right ) + a b d^{\frac{3}{2}}\right )} \log \left (x^{n}\right )\right )} x^{\frac{3}{2}}}\,{d x} - \frac{2}{{\left (b d^{\frac{3}{2}} \log \left (c\right ) + b d^{\frac{3}{2}} \log \left (x^{n}\right ) + a d^{\frac{3}{2}}\right )} \sqrt{x}} \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 d^{2} x^{2} \log \left (c x^{n}\right ) + a d^{2} x^{2}}, 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{1}{\left (d x\right )^{\frac{3}{2}} \left (a + b \log{\left (c x^{n} \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28626, size = 66, normalized size = 1.03 \begin{align*} \frac{c^{\frac{1}{2 \, n}}{\rm Ei}\left (-\frac{\log \left (c\right )}{2 \, n} - \frac{a}{2 \, b n} - \frac{1}{2} \, \log \left (x\right )\right ) e^{\left (\frac{a}{2 \, b n}\right )}}{b d^{\frac{3}{2}} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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