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 = 0.0232267, 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, 203} \[ \frac{\sqrt{4 x^2-9}}{54 x^2}+\frac{\sqrt{4 x^2-9}}{36 x^4}+\frac{2}{81} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 51
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x^5 \sqrt{-9+4 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{-9+4 x}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{-9+4 x^2}}{36 x^4}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{-9+4 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}{27} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-9+4 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 = 0.0047386, size = 32, normalized size = 0.56 \[ \frac{16}{729} \sqrt{4 x^2-9} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};1-\frac{4 x^2}{9}\right ) \]
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 [A] time = 3.16137, size = 51, normalized size = 0.89 \begin{align*} \frac{\sqrt{4 \, x^{2} - 9}}{54 \, x^{2}} + \frac{\sqrt{4 \, x^{2} - 9}}{36 \, x^{4}} - \frac{2}{81} \, \arcsin \left (\frac{3}{2 \,{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.25057, size = 120, normalized size = 2.11 \begin{align*} \frac{16 \, x^{4} \arctan \left (-\frac{2}{3} \, x + \frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) + 3 \, \sqrt{4 \, x^{2} - 9}{\left (2 \, x^{2} + 3\right )}}{324 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.39706, size = 136, normalized size = 2.39 \begin{align*} \begin{cases} \frac{2 i \operatorname{acosh}{\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}}}} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{81} + \frac{1}{27 x \sqrt{1 - \frac{9}{4 x^{2}}}} - \frac{1}{36 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} - \frac{1}{8 x^{5} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.49245, size = 55, normalized size = 0.96 \begin{align*} \frac{{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} + 15 \, \sqrt{4 \, x^{2} - 9}}{216 \, x^{4}} + \frac{2}{81} \, \arctan \left (\frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]