Optimal. Leaf size=29 \[ \frac{x}{r \sqrt{-a^2-e^2-2 r (K-H r)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0316036, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {8} \[ \frac{x}{r \sqrt{-a^2-e^2-2 r (K-H r)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{r \sqrt{-a^2-e^2-2 K r+2 H r^2}} \, dx &=\frac{x}{r \sqrt{-a^2-e^2-2 r (K-H r)}}\\ \end{align*}
Mathematica [A] time = 0.000036, size = 30, normalized size = 1.03 \[ \frac{x}{r \sqrt{-a^2-e^2+2 H r^2-2 K r}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0., size = 29, normalized size = 1. \begin{align*}{\frac{x}{r}{\frac{1}{\sqrt{2\,H{r}^{2}-2\,Kr-{a}^{2}-{e}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.928842, size = 38, normalized size = 1.31 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.81653, size = 99, normalized size = 3.41 \begin{align*} \frac{\sqrt{2 \, H r^{2} - a^{2} - e^{2} - 2 \, K r} x}{2 \, H r^{3} - 2 \, K r^{2} -{\left (a^{2} + e^{2}\right )} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.054356, size = 24, normalized size = 0.83 \begin{align*} \frac{x}{r \sqrt{2 H r^{2} - 2 K r - a^{2} - e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.07776, size = 36, normalized size = 1.24 \begin{align*} \frac{x}{\sqrt{2 \, H r^{2} - a^{2} - 2 \, K r - e^{2}} r} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]