Optimal. Leaf size=31 \[ \frac{1}{48} \left (-4 x^2-9\right )^{3/2}+\frac{9}{16} \sqrt{-4 x^2-9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.013987, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{1}{48} \left (-4 x^2-9\right )^{3/2}+\frac{9}{16} \sqrt{-4 x^2-9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{-9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{-9-4 x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{9}{4 \sqrt{-9-4 x}}-\frac{1}{4} \sqrt{-9-4 x}\right ) \, dx,x,x^2\right )\\ &=\frac{9}{16} \sqrt{-9-4 x^2}+\frac{1}{48} \left (-9-4 x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0075584, size = 22, normalized size = 0.71 \[ \frac{1}{24} \sqrt{-4 x^2-9} \left (9-2 x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 19, normalized size = 0.6 \begin{align*} -{\frac{2\,{x}^{2}-9}{24}\sqrt{-4\,{x}^{2}-9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 3.19461, size = 35, normalized size = 1.13 \begin{align*} -\frac{1}{12} \, \sqrt{-4 \, x^{2} - 9} x^{2} + \frac{3}{8} \, \sqrt{-4 \, x^{2} - 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.22553, size = 49, normalized size = 1.58 \begin{align*} -\frac{1}{24} \,{\left (2 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} - 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.35704, size = 31, normalized size = 1. \begin{align*} - \frac{x^{2} \sqrt{- 4 x^{2} - 9}}{12} + \frac{3 \sqrt{- 4 x^{2} - 9}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 1.68378, size = 31, normalized size = 1. \begin{align*} -\frac{1}{48} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} + \frac{9}{16} i \, \sqrt{4 \, x^{2} + 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]