Optimal. Leaf size=82 \[ -\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a+b x^{3/2}}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3}}-\frac {\log \left (\sqrt [3]{b} \sqrt {x}-\sqrt [3]{a+b x^{3/2}}\right )}{b^{2/3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {249, 337}
\begin {gather*} -\frac {2 \text {ArcTan}\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}}-\frac {\log \left (\sqrt [3]{b} \sqrt {x}-\sqrt [3]{a+b x^{3/2}}\right )}{b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 249
Rule 337
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^{3/2}\right )^{2/3}} \, dx &=2 \text {Subst}\left (\int \frac {x}{\left (a+b x^3\right )^{2/3}} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \frac {x}{1-b x^3} \, dx,x,\frac {\sqrt {x}}{\sqrt [3]{a+b x^{3/2}}}\right )\\ &=\frac {2 \text {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 \text {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 {\text {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 {\text {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 \text {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*}
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Mathematica [A]
time = 0.19, size = 138, normalized size = 1.68 \begin {gather*} \frac {-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{b} \sqrt {x}+2 \sqrt [3]{a+b x^{3/2}}}\right )-2 \log \left (-\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a+b x^{3/2}}\right )+\log \left (b^{2/3} x+\sqrt [3]{b} \sqrt {x} \sqrt [3]{a+b x^{3/2}}+\left (a+b x^{3/2}\right )^{2/3}\right )}{3 b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 100, normalized size = 1.22 \begin {gather*} \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{\sqrt {x}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{3 \, b^{\frac {2}{3}}} + \frac {\log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {x}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}}}{x}\right )}{3 \, b^{\frac {2}{3}}} - \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{\sqrt {x}}\right )}{3 \, b^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 39, normalized size = 0.48 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 37, normalized size = 0.45 \begin {gather*} \frac {x\,{\left (\frac {b\,x^{3/2}}{a}+1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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