Optimal. Leaf size=81 \[ \frac {b \log ^{2 q}\left (c x^n\right )}{2 n q}-\frac {a x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \Gamma \left (q,-\frac {m \log \left (c x^n\right )}{n}\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2539, 2310, 2181, 2302, 30} \[ \frac {b \log ^{2 q}\left (c x^n\right )}{2 n q}-\frac {a x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \text {Gamma}\left (q,-\frac {m \log \left (c x^n\right )}{n}\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2181
Rule 2302
Rule 2310
Rule 2539
Rubi steps
\begin {align*} \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx &=\int \left (a x^{-1+m} \log ^{-1+q}\left (c x^n\right )+\frac {b \log ^{-1+2 q}\left (c x^n\right )}{x}\right ) \, dx\\ &=a \int x^{-1+m} \log ^{-1+q}\left (c x^n\right ) \, dx+b \int \frac {\log ^{-1+2 q}\left (c x^n\right )}{x} \, dx\\ &=\frac {b \operatorname {Subst}\left (\int x^{-1+2 q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac {\left (a x^m \left (c x^n\right )^{-\frac {m}{n}}\right ) \operatorname {Subst}\left (\int e^{\frac {m x}{n}} x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {b \log ^{2 q}\left (c x^n\right )}{2 n q}-\frac {a x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}}{n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 77, normalized size = 0.95 \[ \frac {\log ^q\left (c x^n\right ) \left (\frac {b \log ^q\left (c x^n\right )}{q}-2 a x^m \left (c x^n\right )^{-\frac {m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \Gamma \left (q,-\frac {m \log \left (c x^n\right )}{n}\right )\right )}{2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a x^{m} \log \left (c x^{n}\right )^{q - 1} + b \log \left (c x^{n}\right )^{q - 1} \log \left (c x^{n}\right )^{q}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} \log \left (c x^{n}\right )^{q - 1}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 21.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right ) \ln \left (c \,x^{n}\right )^{q -1}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\ln \left (c\,x^n\right )}^{q-1}\,\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x^{m} + b \log {\left (c x^{n} \right )}^{q}\right ) \log {\left (c x^{n} \right )}^{q - 1}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________