Optimal. Leaf size=22 \[ \frac{1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.109131, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.024, Rules used = {2544} \[ \frac{1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2544
Rubi steps
\begin{align*} \int \frac{\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )}{x} \, dx &=\frac{1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2\\ \end{align*}
Mathematica [A] time = 0.032603, size = 22, normalized size = 1. \[ \frac{1}{2} \left (a x^m+b \log ^q\left (c x^n\right )\right )^2 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.162, size = 135, normalized size = 6.1 \begin{align*}{\frac{{a}^{2} \left ({x}^{m} \right ) ^{2}}{2}}+{\frac{{b}^{2} \left ( \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{q} \right ) ^{2}}{2}}+ab{x}^{m} \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{q} \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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.95908, size = 115, normalized size = 5.23 \begin{align*}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} a b x^{m} + \frac{1}{2} \,{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{2 \, q} b^{2} + \frac{1}{2} \, a^{2} x^{2 \, m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (b n q \log \left (c x^{n}\right )^{q - 1} + a m x^{m}\right )}{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]