Optimal. Leaf size=16 \[ \frac{1}{2} \log \left (b x^2+c x^4\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0239123, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1584, 446, 72} \[ \frac{1}{2} \log \left (b+c x^2\right )+\log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1584
Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{x \left (b+2 c x^2\right )}{b x^2+c x^4} \, dx &=\int \frac{b+2 c x^2}{x \left (b+c x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{b+2 c x}{x (b+c x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{x}+\frac{c}{b+c x}\right ) \, dx,x,x^2\right )\\ &=\log (x)+\frac{1}{2} \log \left (b+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0061463, size = 15, normalized size = 0.94 \[ \frac{1}{2} \log \left (b+c x^2\right )+\log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 14, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) +{\frac{\ln \left ( c{x}^{2}+b \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.996828, size = 23, normalized size = 1.44 \begin{align*} \frac{1}{2} \, \log \left (c x^{2} + b\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.02086, size = 39, normalized size = 2.44 \begin{align*} \frac{1}{2} \, \log \left (c x^{2} + b\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.324457, size = 12, normalized size = 0.75 \begin{align*} \log{\left (x \right )} + \frac{\log{\left (\frac{b}{c} + x^{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09658, size = 20, normalized size = 1.25 \begin{align*} \frac{1}{2} \, \log \left ({\left | c x^{2} + b \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]