Optimal. Leaf size=82 \[ -\frac{\log \left (\sqrt [3]{b} \sqrt{x}-\sqrt [3]{a+b x^{3/2}}\right )}{b^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}} \]
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Rubi [A] time = 0.0946295, antiderivative size = 140, normalized size of antiderivative = 1.71, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {243, 331, 292, 31, 634, 617, 204, 628} \[ -\frac{2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 b^{2/3}}+\frac{\log \left (\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1\right )}{3 b^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}} \]
Antiderivative was successfully verified.
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Rule 243
Rule 331
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^{3/2}\right )^{2/3}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^{2/3}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x}{1-b x^3} \, dx,x,\frac{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{b} x} \, dx,x,\frac{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 \sqrt [3]{b}}-\frac{2 \operatorname{Subst}\left (\int \frac{1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 \sqrt [3]{b}}\\ &=-\frac{2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 b^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 b^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{\sqrt [3]{b}}\\ &=-\frac{2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 b^{2/3}}+\frac{\log \left (1+\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 b^{2/3}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{b^{2/3}}\\ &=-\frac{2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}}{\sqrt{3}}\right )}{\sqrt{3} b^{2/3}}-\frac{2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 b^{2/3}}+\frac{\log \left (1+\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{3 b^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0095632, size = 52, normalized size = 0.63 \[ \frac{x \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^{3/2}}{a}\right )}{\left (a+b x^{3/2}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.021, 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 = 1.08668, size = 39, normalized size = 0.48 \begin{align*} \frac{2 x \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{\frac{3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{5}{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 \frac{1}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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