Optimal. Leaf size=72 \[ \frac{1}{6} \sqrt{4 x^2-9} x^5-\frac{3}{32} \sqrt{4 x^2-9} x^3-\frac{81}{256} \sqrt{4 x^2-9} x-\frac{729}{512} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0188935, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \[ \frac{1}{6} \sqrt{4 x^2-9} x^5-\frac{3}{32} \sqrt{4 x^2-9} x^3-\frac{81}{256} \sqrt{4 x^2-9} x-\frac{729}{512} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^4 \sqrt{-9+4 x^2} \, dx &=\frac{1}{6} x^5 \sqrt{-9+4 x^2}-\frac{3}{2} \int \frac{x^4}{\sqrt{-9+4 x^2}} \, dx\\ &=-\frac{3}{32} x^3 \sqrt{-9+4 x^2}+\frac{1}{6} x^5 \sqrt{-9+4 x^2}-\frac{81}{32} \int \frac{x^2}{\sqrt{-9+4 x^2}} \, dx\\ &=-\frac{81}{256} x \sqrt{-9+4 x^2}-\frac{3}{32} x^3 \sqrt{-9+4 x^2}+\frac{1}{6} x^5 \sqrt{-9+4 x^2}-\frac{729}{256} \int \frac{1}{\sqrt{-9+4 x^2}} \, dx\\ &=-\frac{81}{256} x \sqrt{-9+4 x^2}-\frac{3}{32} x^3 \sqrt{-9+4 x^2}+\frac{1}{6} x^5 \sqrt{-9+4 x^2}-\frac{729}{256} \operatorname{Subst}\left (\int \frac{1}{1-4 x^2} \, dx,x,\frac{x}{\sqrt{-9+4 x^2}}\right )\\ &=-\frac{81}{256} x \sqrt{-9+4 x^2}-\frac{3}{32} x^3 \sqrt{-9+4 x^2}+\frac{1}{6} x^5 \sqrt{-9+4 x^2}-\frac{729}{512} \tanh ^{-1}\left (\frac{2 x}{\sqrt{-9+4 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0110108, size = 49, normalized size = 0.68 \[ \frac{1}{768} x \sqrt{4 x^2-9} \left (128 x^4-72 x^2-243\right )-\frac{729}{512} \log \left (\sqrt{4 x^2-9}+2 x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 61, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{24} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{9\,x}{128} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{81\,x}{256}\sqrt{4\,{x}^{2}-9}}-{\frac{729\,\sqrt{4}}{1024}\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 = 2.72062, size = 77, normalized size = 1.07 \begin{align*} \frac{1}{24} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} x^{3} + \frac{9}{128} \,{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} x + \frac{81}{256} \, \sqrt{4 \, x^{2} - 9} x - \frac{729}{512} \, \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 = 1.47389, size = 119, normalized size = 1.65 \begin{align*} \frac{1}{768} \,{\left (128 \, x^{5} - 72 \, x^{3} - 243 \, x\right )} \sqrt{4 \, x^{2} - 9} + \frac{729}{512} \, \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 = 4.58974, size = 167, normalized size = 2.32 \begin{align*} \begin{cases} \frac{2 x^{7}}{3 \sqrt{4 x^{2} - 9}} - \frac{15 x^{5}}{8 \sqrt{4 x^{2} - 9}} - \frac{27 x^{3}}{64 \sqrt{4 x^{2} - 9}} + \frac{729 x}{256 \sqrt{4 x^{2} - 9}} - \frac{729 \operatorname{acosh}{\left (\frac{2 x}{3} \right )}}{512} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\- \frac{2 i x^{7}}{3 \sqrt{9 - 4 x^{2}}} + \frac{15 i x^{5}}{8 \sqrt{9 - 4 x^{2}}} + \frac{27 i x^{3}}{64 \sqrt{9 - 4 x^{2}}} - \frac{729 i x}{256 \sqrt{9 - 4 x^{2}}} + \frac{729 i \operatorname{asin}{\left (\frac{2 x}{3} \right )}}{512} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.5384, size = 59, normalized size = 0.82 \begin{align*} \frac{1}{768} \,{\left (8 \,{\left (16 \, x^{2} - 9\right )} x^{2} - 243\right )} \sqrt{4 \, x^{2} - 9} x + \frac{729}{512} \, \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]