Optimal. Leaf size=57 \[ \frac {\sqrt {4 x^2-9}}{18 x^2}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {4 x^2-9}\right )-\frac {\sqrt {4 x^2-9}}{4 x^4} \]
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Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 203} \[ \frac {\sqrt {4 x^2-9}}{18 x^2}-\frac {\sqrt {4 x^2-9}}{4 x^4}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {4 x^2-9}\right ) \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 203
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {-9+4 x^2}}{x^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {-9+4 x}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {-9+4 x^2}}{4 x^4}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {-9+4 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {-9+4 x^2}}{4 x^4}+\frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-9+4 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {-9+4 x^2}}{4 x^4}+\frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {1}{18} \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}}{4 x^4}+\frac {\sqrt {-9+4 x^2}}{18 x^2}+\frac {2}{27} \tan ^{-1}\left (\frac {1}{3} \sqrt {-9+4 x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 32, normalized size = 0.56 \[ \frac {16 \left (4 x^2-9\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};1-\frac {4 x^2}{9}\right )}{2187} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 45, normalized size = 0.79 \[ \frac {16 \, x^{4} \arctan \left (-\frac {2}{3} \, x + \frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) + 3 \, \sqrt {4 \, x^{2} - 9} {\left (2 \, x^{2} - 9\right )}}{108 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 41, normalized size = 0.72 \[ \frac {{\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}} - 9 \, \sqrt {4 \, x^{2} - 9}}{72 \, x^{4}} + \frac {2}{27} \, \arctan \left (\frac {1}{3} \, \sqrt {4 \, x^{2} - 9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 55, normalized size = 0.96 \[ -\frac {2 \arctan \left (\frac {3}{\sqrt {4 x^{2}-9}}\right )}{27}+\frac {\left (4 x^{2}-9\right )^{\frac {3}{2}}}{162 x^{2}}+\frac {\left (4 x^{2}-9\right )^{\frac {3}{2}}}{36 x^{4}}-\frac {2 \sqrt {4 x^{2}-9}}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 49, normalized size = 0.86 \[ -\frac {2}{81} \, \sqrt {4 \, x^{2} - 9} + \frac {{\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{162 \, x^{2}} + \frac {{\left (4 \, x^{2} - 9\right )}^{\frac {3}{2}}}{36 \, x^{4}} - \frac {2}{27} \, \arcsin \left (\frac {3}{2 \, {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.19, size = 43, normalized size = 0.75 \[ \frac {2\,\mathrm {atan}\left (\frac {\sqrt {4\,x^2-9}}{3}\right )}{27}-\frac {\frac {\sqrt {4\,x^2-9}}{8}-\frac {{\left (4\,x^2-9\right )}^{3/2}}{72}}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.36, size = 139, normalized size = 2.44 \[ \begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{27} - \frac {i}{9 x \sqrt {-1 + \frac {9}{4 x^{2}}}} + \frac {3 i}{4 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {9 i}{8 x^{5} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {9}{4 \left |{x^{2}}\right |} > 1 \\- \frac {2 \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{27} + \frac {1}{9 x \sqrt {1 - \frac {9}{4 x^{2}}}} - \frac {3}{4 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {9}{8 x^{5} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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