Optimal. Leaf size=98 \[ \frac{\sqrt{d x} e^{-\frac{a}{2 b n}} \left (c x^n\right )^{\left .-\frac{1}{2}\right /n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac{\sqrt{d x}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.0879756, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2306, 2310, 2178} \[ \frac{\sqrt{d x} e^{-\frac{a}{2 b n}} \left (c x^n\right )^{\left .-\frac{1}{2}\right /n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac{\sqrt{d x}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d x} \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{\sqrt{d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac{\int \frac{1}{\sqrt{d x} \left (a+b \log \left (c x^n\right )\right )} \, dx}{2 b n}\\ &=-\frac{\sqrt{d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac{\left (\sqrt{d x} \left (c x^n\right )^{\left .-\frac{1}{2}\right /n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b d n^2}\\ &=\frac{e^{-\frac{a}{2 b n}} \sqrt{d x} \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 )}{2 b^2 d n^2}-\frac{\sqrt{d x}}{b d n \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.114841, size = 83, normalized size = 0.85 \[ \frac{x \left (e^{-\frac{a}{2 b n}} \left (c x^n\right )^{\left .-\frac{1}{2}\right /n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{2 b n}\right )-\frac{2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2 \sqrt{d x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.596, size = 427, normalized size = 4.4 \begin{align*} \text{result too large to display} \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*} 4 \, b n \int \frac{1}{{\left (b^{3} \sqrt{d} \log \left (c\right )^{3} + b^{3} \sqrt{d} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} \sqrt{d} \log \left (c\right )^{2} + 3 \, a^{2} b \sqrt{d} \log \left (c\right ) + a^{3} \sqrt{d} + 3 \,{\left (b^{3} \sqrt{d} \log \left (c\right ) + a b^{2} \sqrt{d}\right )} \log \left (x^{n}\right )^{2} + 3 \,{\left (b^{3} \sqrt{d} \log \left (c\right )^{2} + 2 \, a b^{2} \sqrt{d} \log \left (c\right ) + a^{2} b \sqrt{d}\right )} \log \left (x^{n}\right )\right )} \sqrt{x}}\,{d x} + \frac{2 \, \sqrt{x}}{b^{2} \sqrt{d} \log \left (c\right )^{2} + b^{2} \sqrt{d} \log \left (x^{n}\right )^{2} + 2 \, a b \sqrt{d} \log \left (c\right ) + a^{2} \sqrt{d} + 2 \,{\left (b^{2} \sqrt{d} \log \left (c\right ) + a b \sqrt{d}\right )} \log \left (x^{n}\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^{2} d x \log \left (c x^{n}\right )^{2} + 2 \, a b d x \log \left (c x^{n}\right ) + a^{2} d x}, 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}{\sqrt{d x} \left (a + b \log{\left (c x^{n} \right )}\right )^{2}}\, 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{1}{\sqrt{d x}{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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