Optimal. Leaf size=19 \[ \frac{1}{2} \log \left (a-b x^2-c x^4\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0193331, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1247, 628} \[ \frac{1}{2} \log \left (a-b x^2-c x^4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1247
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (b+2 c x^2\right )}{-a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{b+2 c x}{-a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \log \left (a-b x^2-c x^4\right )\\ \end{align*}
Mathematica [A] time = 0.006661, size = 19, normalized size = 1. \[ \frac{1}{2} \log \left (-a+b x^2+c x^4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 18, normalized size = 1. \begin{align*}{\frac{\ln \left ( c{x}^{4}+b{x}^{2}-a \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01428, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.03838, size = 38, normalized size = 2. \begin{align*} \frac{1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.418333, size = 14, normalized size = 0.74 \begin{align*} \frac{\log{\left (- a + b x^{2} + c x^{4} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.158, size = 23, normalized size = 1.21 \begin{align*} \frac{1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]