Optimal. Leaf size=95 \[ b^{2/3} \log \left (\sqrt [3]{a+\frac{b}{x^{3/2}}}-\frac{\sqrt [3]{b}}{\sqrt{x}}\right )-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a+\frac{b}{x^{3/2}}}}+1}{\sqrt{3}}\right )}{\sqrt{3}}+x \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.190498, 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 \[ b^{2/3} \log \left (\sqrt [3]{a+\frac{b}{x^{3/2}}}-\frac{\sqrt [3]{b}}{\sqrt{x}}\right )-\frac{2 b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a+\frac{b}{x^{3/2}}}}+1}{\sqrt{3}}\right )}{\sqrt{3}}+x \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^(3/2))^(2/3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 21.2598, size = 146, normalized size = 1.54 \[ \frac{2 b^{\frac{2}{3}} \log{\left (- \frac{\sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a + \frac{b}{x^{\frac{3}{2}}}}} + 1 \right )}}{3} - \frac{b^{\frac{2}{3}} \log{\left (\frac{b^{\frac{2}{3}}}{x \left (a + \frac{b}{x^{\frac{3}{2}}}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b}}{\sqrt{x} \sqrt [3]{a + \frac{b}{x^{\frac{3}{2}}}}} + 1 \right )}}{3} - \frac{2 \sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b}}{3 \sqrt{x} \sqrt [3]{a + \frac{b}{x^{\frac{3}{2}}}}} + \frac{1}{3}\right ) \right )}}{3} + x \left (a + \frac{b}{x^{\frac{3}{2}}}\right )^{\frac{2}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**(3/2))**(2/3),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0881688, size = 53, normalized size = 0.56 \[ \frac{x \left (a+\frac{b}{x^{3/2}}\right )^{2/3} \, _2F_1\left (-\frac{2}{3},-\frac{2}{3};\frac{1}{3};-\frac{b}{a x^{3/2}}\right )}{\left (\frac{a+\frac{b}{x^{3/2}}}{a}\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^(3/2))^(2/3),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.025, size = 0, normalized size = 0. \[ \int \left ( a+{b{x}^{-{\frac{3}{2}}}} \right ) ^{{\frac{2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^(3/2))^(2/3),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^(3/2))^(2/3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^(3/2))^(2/3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 46.9952, 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.
[In] integrate((a+b/x**(3/2))**(2/3),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (a + \frac{b}{x^{\frac{3}{2}}}\right )}^{\frac{2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^(3/2))^(2/3),x, algorithm="giac")
[Out]