Optimal. Leaf size=106 \[ \frac{(f x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{f (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0671816, antiderivative size = 106, 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, 2181} \[ \frac{(f x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{f (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2310
Rule 2181
Rubi steps
\begin{align*} \int (f x)^m \left (a+b \log \left (c x^n\right )\right )^p \, dx &=\frac{\left ((f x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{(1+m) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{f n}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (f x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{f (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0668302, size = 107, normalized size = 1.01 \[ \frac{x^{-m} (f x)^m \left (a+b \log \left (c x^n\right )\right )^p \exp \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.597, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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 (\left (f x\right )^{m}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}, 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 \left (f x\right )^{m} \left (a + b \log{\left (c x^{n} \right )}\right )^{p}\, 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 \left (f x\right )^{m}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]