Optimal. Leaf size=46 \[ \frac{1}{448} \left (4 x^2-9\right )^{7/2}+\frac{9}{160} \left (4 x^2-9\right )^{5/2}+\frac{27}{64} \left (4 x^2-9\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.019893, antiderivative size = 46, 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} \[ \frac{1}{448} \left (4 x^2-9\right )^{7/2}+\frac{9}{160} \left (4 x^2-9\right )^{5/2}+\frac{27}{64} \left (4 x^2-9\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \sqrt{-9+4 x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 \sqrt{-9+4 x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{81}{16} \sqrt{-9+4 x}+\frac{9}{8} (-9+4 x)^{3/2}+\frac{1}{16} (-9+4 x)^{5/2}\right ) \, dx,x,x^2\right )\\ &=\frac{27}{64} \left (-9+4 x^2\right )^{3/2}+\frac{9}{160} \left (-9+4 x^2\right )^{5/2}+\frac{1}{448} \left (-9+4 x^2\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0090673, size = 27, normalized size = 0.59 \[ \frac{1}{280} \left (4 x^2-9\right )^{3/2} \left (10 x^4+18 x^2+27\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 34, normalized size = 0.7 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) \left ( 10\,{x}^{4}+18\,{x}^{2}+27 \right ) }{280}\sqrt{4\,{x}^{2}-9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 3.40864, size = 54, normalized size = 1.17 \begin{align*} \frac{1}{28} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} x^{4} + \frac{9}{140} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} x^{2} + \frac{27}{280} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.43098, size = 76, normalized size = 1.65 \begin{align*} \frac{1}{280} \,{\left (40 \, x^{6} - 18 \, x^{4} - 54 \, x^{2} - 243\right )} \sqrt{4 \, x^{2} - 9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.91672, size = 61, normalized size = 1.33 \begin{align*} \frac{x^{6} \sqrt{4 x^{2} - 9}}{7} - \frac{9 x^{4} \sqrt{4 x^{2} - 9}}{140} - \frac{27 x^{2} \sqrt{4 x^{2} - 9}}{140} - \frac{243 \sqrt{4 x^{2} - 9}}{280} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.21624, size = 46, normalized size = 1. \begin{align*} \frac{1}{448} \,{\left (4 \, x^{2} - 9\right )}^{\frac{7}{2}} + \frac{9}{160} \,{\left (4 \, x^{2} - 9\right )}^{\frac{5}{2}} + \frac{27}{64} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]