Optimal. Leaf size=27 \[ \frac{1}{8} x \sqrt{4 x^2+9}-\frac{9}{16} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0060955, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {321, 215} \[ \frac{1}{8} x \sqrt{4 x^2+9}-\frac{9}{16} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{9+4 x^2}} \, dx &=\frac{1}{8} x \sqrt{9+4 x^2}-\frac{9}{8} \int \frac{1}{\sqrt{9+4 x^2}} \, dx\\ &=\frac{1}{8} x \sqrt{9+4 x^2}-\frac{9}{16} \sinh ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0049179, size = 27, normalized size = 1. \[ \frac{1}{8} x \sqrt{4 x^2+9}-\frac{9}{16} \sinh ^{-1}\left (\frac{2 x}{3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 20, normalized size = 0.7 \begin{align*} -{\frac{9}{16}{\it Arcsinh} \left ({\frac{2\,x}{3}} \right ) }+{\frac{x}{8}\sqrt{4\,{x}^{2}+9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 3.10261, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{2} + 9} x - \frac{9}{16} \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.26852, size = 78, normalized size = 2.89 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{2} + 9} x + \frac{9}{16} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.203203, size = 22, normalized size = 0.81 \begin{align*} \frac{x \sqrt{4 x^{2} + 9}}{8} - \frac{9 \operatorname{asinh}{\left (\frac{2 x}{3} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.97447, size = 39, normalized size = 1.44 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{2} + 9} x + \frac{9}{16} \, \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]