Optimal. Leaf size=66 \[ \frac{(d x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0736044, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2310, 2178} \[ \frac{(d x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{(d x)^m}{a+b \log \left (c x^n\right )} \, dx &=\frac{\left ((d x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+m) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{d n}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (d x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \text{Ei}\left (\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b d n}\\ \end{align*}
Mathematica [A] time = 0.105898, size = 67, normalized size = 1.02 \[ \frac{x^{-m} (d x)^m \exp \left (-\frac{(m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \text{Ei}\left (\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.245, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{a+b\ln \left ( c{x}^{n} \right ) }}\, 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*} \int \frac{\left (d x\right )^{m}}{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.989485, size = 163, normalized size = 2.47 \begin{align*} \frac{{\rm Ei}\left (\frac{{\left (b m + b\right )} n \log \left (x\right ) + a m +{\left (b m + b\right )} \log \left (c\right ) + a}{b n}\right ) e^{\left (\frac{b m n \log \left (d\right ) - a m -{\left (b m + b\right )} \log \left (c\right ) - a}{b n}\right )}}{b n} \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{\left (d x\right )^{m}}{a + b \log{\left (c x^{n} \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{\left (d x\right )^{m}}{b \log \left (c x^{n}\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]