Integrand size = 13, antiderivative size = 95 \[ \int \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, dx=\left (a+\frac {b}{x^{3/2}}\right )^{2/3} x-\frac {2 b^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b}}{\sqrt [3]{a+\frac {b}{x^{3/2}}} \sqrt {x}}}{\sqrt {3}}\right )}{\sqrt {3}}+b^{2/3} \log \left (\sqrt [3]{a+\frac {b}{x^{3/2}}}-\frac {\sqrt [3]{b}}{\sqrt {x}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {249, 342, 283, 245} \[ \int \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, dx=-\frac {2 b^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{b}}{\sqrt {x} \sqrt [3]{a+\frac {b}{x^{3/2}}}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+b^{2/3} \log \left (\sqrt [3]{a+\frac {b}{x^{3/2}}}-\frac {\sqrt [3]{b}}{\sqrt {x}}\right )+x \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \]
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Rule 245
Rule 249
Rule 283
Rule 342
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \left (a+\frac {b}{x^3}\right )^{2/3} x \, dx,x,\sqrt {x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {\left (a+b x^3\right )^{2/3}}{x^3} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = \left (a+\frac {b}{x^{3/2}}\right )^{2/3} x-(2 b) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a+b x^3}} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = \left (a+\frac {b}{x^{3/2}}\right )^{2/3} x-\frac {2 b^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b}}{\sqrt [3]{a+\frac {b}{x^{3/2}}} \sqrt {x}}}{\sqrt {3}}\right )}{\sqrt {3}}+b^{2/3} \log \left (\sqrt [3]{a+\frac {b}{x^{3/2}}}-\frac {\sqrt [3]{b}}{\sqrt {x}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(280\) vs. \(2(95)=190\).
Time = 7.35 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.95 \[ \int \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, dx=-\frac {\left (a+\frac {b}{x^{3/2}}\right )^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{b+a x^{3/2}}\right ) \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b+a x^{3/2}}+\left (b+a x^{3/2}\right )^{2/3}\right )^2 \left (3 \left (b+a x^{3/2}\right )^{2/3}+2 \sqrt {3} b^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b+a x^{3/2}}}{\sqrt [3]{b}}}{\sqrt {3}}\right )+2 b^{2/3} \log \left (-\sqrt [3]{b}+\sqrt [3]{b+a x^{3/2}}\right )-b^{2/3} \log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b+a x^{3/2}}+\left (b+a x^{3/2}\right )^{2/3}\right )\right )}{3 a \sqrt {x} \sqrt [3]{b+a x^{3/2}} \left (b+a x^{3/2}+b^{2/3} \sqrt [3]{b+a x^{3/2}}+\sqrt [3]{b} \left (b+a x^{3/2}\right )^{2/3}\right )} \]
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\[\int \left (a +\frac {b}{x^{\frac {3}{2}}}\right )^{\frac {2}{3}}d x\]
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Timed out. \[ \int \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.48 \[ \int \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, dx=- \frac {2 a^{\frac {2}{3}} x \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{\frac {3}{2}}}} \right )}}{3 \Gamma \left (\frac {1}{3}\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.15 \[ \int \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, dx=\frac {2}{3} \, \sqrt {3} b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \sqrt {x} + b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right ) + {\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {2}{3}} x - \frac {1}{3} \, b^{\frac {2}{3}} \log \left ({\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {2}{3}} x + {\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {1}{3}} b^{\frac {1}{3}} \sqrt {x} + b^{\frac {2}{3}}\right ) + \frac {2}{3} \, b^{\frac {2}{3}} \log \left ({\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \sqrt {x} - b^{\frac {1}{3}}\right ) \]
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\[ \int \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, dx=\int { {\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {2}{3}} \,d x } \]
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Time = 6.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.39 \[ \int \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, dx=\frac {x\,{\left (a+\frac {b}{x^{3/2}}\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},-\frac {2}{3};\ \frac {1}{3};\ -\frac {b}{a\,x^{3/2}}\right )}{{\left (\frac {b}{a\,x^{3/2}}+1\right )}^{2/3}} \]
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