Optimal. Leaf size=64 \[ \frac{\sqrt{d x} e^{-\frac{a}{2 b n}} \left (c x^n\right )^{\left .-\frac{1}{2}\right /n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0605463, 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} \[ \frac{\sqrt{d x} e^{-\frac{a}{2 b n}} \left (c x^n\right )^{\left .-\frac{1}{2}\right /n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d x} \left (a+b \log \left (c x^n\right )\right )} \, dx &=\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 )}{d n}\\ &=\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 )}{b d n}\\ \end{align*}
Mathematica [A] time = 0.0628889, size = 62, normalized size = 0.97 \[ \frac{x e^{-\frac{a}{2 b n}} \left (c x^n\right )^{\left .-\frac{1}{2}\right /n} \text{Ei}\left (\frac{a+b \log \left (c x^n\right )}{2 b n}\right )}{b n \sqrt{d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.108, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a+b\ln \left ( c{x}^{n} \right ) }{\frac{1}{\sqrt{dx}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, b n \int \frac{1}{{\left (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 )\right )} \sqrt{x}}\,{d x} + \frac{2 \, \sqrt{x}}{b \sqrt{d} \log \left (c\right ) + b \sqrt{d} \log \left (x^{n}\right ) + a \sqrt{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x}}{b d x \log \left (c x^{n}\right ) + a d x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]