Optimal. Leaf size=95 \[ b^{2/3} \log \left (\sqrt [3]{a+\frac {b}{x^{3/2}}}-\frac {\sqrt [3]{b}}{\sqrt {x}}\right )-\frac {2 b^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b}}{\sqrt {x} \sqrt [3]{a+\frac {b}{x^{3/2}}}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+x \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \]
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Rubi [A] time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {243, 335, 277, 239} \[ b^{2/3} \log \left (\sqrt [3]{a+\frac {b}{x^{3/2}}}-\frac {\sqrt [3]{b}}{\sqrt {x}}\right )-\frac {2 b^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b}}{\sqrt {x} \sqrt [3]{a+\frac {b}{x^{3/2}}}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+x \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 239
Rule 243
Rule 277
Rule 335
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, dx &=2 \operatorname {Subst}\left (\int \left (a+\frac {b}{x^3}\right )^{2/3} x \, dx,x,\sqrt {x}\right )\\ &=-\left (2 \operatorname {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) \operatorname {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*}
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Mathematica [C] time = 0.03, size = 52, normalized size = 0.55 \[ \frac {x \left (a+\frac {b}{x^{3/2}}\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{3};\frac {1}{3};-\frac {b}{a x^{3/2}}\right )}{\left (\frac {b}{a x^{3/2}}+1\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \left (a +\frac {b}{x^{\frac {3}{2}}}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 109, normalized size = 1.15 \[ \frac {2}{3} \, \sqrt {3} b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \sqrt {x} + b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right ) + {\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {2}{3}} x - \frac {1}{3} \, b^{\frac {2}{3}} \log \left ({\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {2}{3}} x + {\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {1}{3}} b^{\frac {1}{3}} \sqrt {x} + b^{\frac {2}{3}}\right ) + \frac {2}{3} \, b^{\frac {2}{3}} \log \left ({\left (a + \frac {b}{x^{\frac {3}{2}}}\right )}^{\frac {1}{3}} \sqrt {x} - b^{\frac {1}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 37, normalized size = 0.39 \[ \frac {x\,{\left (a+\frac {b}{x^{3/2}}\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},-\frac {2}{3};\ \frac {1}{3};\ -\frac {b}{a\,x^{3/2}}\right )}{{\left (\frac {b}{a\,x^{3/2}}+1\right )}^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.32, size = 46, normalized size = 0.48 \[ - \frac {2 a^{\frac {2}{3}} x \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{\frac {3}{2}}}} \right )}}{3 \Gamma \left (\frac {1}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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