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 = 0.0901612, antiderivative size = 95, normalized size of antiderivative = 1., 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 243
Rule 335
Rule 277
Rule 239
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*}
Mathematica [C] time = 0.0198895, 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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Maple [F] time = 0.022, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{b{x}^{-{\frac{3}{2}}}} \right ) ^{{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.7126, size = 46, normalized size = 0.48 \begin{align*} - \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a + \frac{b}{x^{\frac{3}{2}}}\right )}^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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