Optimal. Leaf size=22 \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0031639, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 261} \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 261
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^2}}{\sqrt{1+x^2}} \, dx &=\frac{\sqrt{a x^2} \int \frac{x}{\sqrt{1+x^2}} \, dx}{x}\\ &=\frac{\sqrt{a x^2} \sqrt{1+x^2}}{x}\\ \end{align*}
Mathematica [A] time = 0.0046102, size = 22, normalized size = 1. \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 19, normalized size = 0.9 \begin{align*}{\frac{1}{x}\sqrt{a{x}^{2}}\sqrt{{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.56989, size = 26, normalized size = 1.18 \begin{align*} \frac{\sqrt{a} x^{2} + \sqrt{a}}{\sqrt{x^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.27875, size = 39, normalized size = 1.77 \begin{align*} \frac{\sqrt{a x^{2}} \sqrt{x^{2} + 1}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.401803, size = 20, normalized size = 0.91 \begin{align*} \frac{\sqrt{a} \sqrt{x^{2} + 1} \sqrt{x^{2}}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14934, size = 26, normalized size = 1.18 \begin{align*}{\left (\sqrt{x^{2} + 1} \mathrm{sgn}\left (x\right ) - \mathrm{sgn}\left (x\right )\right )} \sqrt{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]