Optimal. Leaf size=36 \[ \frac{1}{8} \sqrt{4 x^2-9} x+\frac{9}{16} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.007637, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {321, 217, 206} \[ \frac{1}{8} \sqrt{4 x^2-9} x+\frac{9}{16} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 321
Rule 217
Rule 206
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}{8} \operatorname{Subst}\left (\int \frac{1}{1-4 x^2} \, dx,x,\frac{x}{\sqrt{-9+4 x^2}}\right )\\ &=\frac{1}{8} x \sqrt{-9+4 x^2}+\frac{9}{16} \tanh ^{-1}\left (\frac{2 x}{\sqrt{-9+4 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.005273, size = 36, normalized size = 1. \[ \frac{1}{8} \sqrt{4 x^2-9} x+\frac{9}{16} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 35, normalized size = 1. \begin{align*}{\frac{x}{8}\sqrt{4\,{x}^{2}-9}}+{\frac{9\,\sqrt{4}}{32}\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.84864, size = 42, normalized size = 1.17 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{2} - 9} x + \frac{9}{16} \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} - 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.28349, size = 78, normalized size = 2.17 \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 = 0.204969, size = 22, normalized size = 0.61 \begin{align*} \frac{x \sqrt{4 x^{2} - 9}}{8} + \frac{9 \operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.15916, size = 41, normalized size = 1.14 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{2} - 9} x - \frac{9}{16} \, \log \left ({\left | -2 \, x + \sqrt{4 \, x^{2} - 9} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]