Optimal. Leaf size=57 \[ -\frac{\sqrt{-4 x^2-9}}{54 x^2}+\frac{\sqrt{-4 x^2-9}}{36 x^4}+\frac{2}{81} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0232741, antiderivative size = 57, 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 = {266, 51, 63, 204} \[ -\frac{\sqrt{-4 x^2-9}}{54 x^2}+\frac{\sqrt{-4 x^2-9}}{36 x^4}+\frac{2}{81} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 51
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^5 \sqrt{-9-4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x^3} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{36 x^4}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x^2} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{36 x^4}-\frac{\sqrt{-9-4 x^2}}{54 x^2}+\frac{1}{27} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-9-4 x} x} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9-4 x^2}}{36 x^4}-\frac{\sqrt{-9-4 x^2}}{54 x^2}-\frac{1}{54} \operatorname{Subst}\left (\int \frac{1}{-\frac{9}{4}-\frac{x^2}{4}} \, dx,x,\sqrt{-9-4 x^2}\right )\\ &=\frac{\sqrt{-9-4 x^2}}{36 x^4}-\frac{\sqrt{-9-4 x^2}}{54 x^2}+\frac{2}{81} \tan ^{-1}\left (\frac{1}{3} \sqrt{-9-4 x^2}\right )\\ \end{align*}
Mathematica [C] time = 0.0047161, size = 32, normalized size = 0.56 \[ \frac{16}{729} \sqrt{-4 x^2-9} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{4 x^2}{9}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 44, normalized size = 0.8 \begin{align*}{\frac{1}{36\,{x}^{4}}\sqrt{-4\,{x}^{2}-9}}-{\frac{1}{54\,{x}^{2}}\sqrt{-4\,{x}^{2}-9}}-{\frac{2}{81}\arctan \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}-9}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 3.63359, size = 73, normalized size = 1.28 \begin{align*} -\frac{\sqrt{-4 \, x^{2} - 9}}{54 \, x^{2}} + \frac{\sqrt{-4 \, x^{2} - 9}}{36 \, x^{4}} - \frac{2}{81} i \, \log \left (\frac{6 \, \sqrt{4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 1.32893, size = 192, normalized size = 3.37 \begin{align*} \frac{-4 i \, x^{4} \log \left (-\frac{4 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} + 3\right )}}{81 \, x}\right ) + 4 i \, x^{4} \log \left (-\frac{4 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} + 3\right )}}{81 \, x}\right ) - 3 \,{\left (2 \, x^{2} - 3\right )} \sqrt{-4 \, x^{2} - 9}}{324 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 4.46695, size = 65, normalized size = 1.14 \begin{align*} \frac{2 i \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{81} - \frac{i}{27 x \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{i}{36 x^{3} \sqrt{1 + \frac{9}{4 x^{2}}}} + \frac{i}{8 x^{5} \sqrt{1 + \frac{9}{4 x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] time = 2.87814, size = 58, normalized size = 1.02 \begin{align*} -\frac{i \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} - 15 i \, \sqrt{4 \, x^{2} + 9}}{216 \, x^{4}} + \frac{2}{81} \, \arctan \left (\frac{1}{3} i \, \sqrt{4 \, x^{2} + 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]