Optimal. Leaf size=563 \[ \frac{55 b^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{12 \sqrt{2} \sqrt [4]{3} a^{8/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{55 \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{16\ 3^{3/4} a^{8/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{55 b^{4/3} \sqrt{a+\frac{b}{x^3}}}{24 a^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}-\frac{55 b x \sqrt{a+\frac{b}{x^3}}}{24 a^3}+\frac{11 x^4 \sqrt{a+\frac{b}{x^3}}}{12 a^2}-\frac{2 x^4}{3 a \sqrt{a+\frac{b}{x^3}}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.943052, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{55 b^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{12 \sqrt{2} \sqrt [4]{3} a^{8/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{55 \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{16\ 3^{3/4} a^{8/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{55 b^{4/3} \sqrt{a+\frac{b}{x^3}}}{24 a^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}-\frac{55 b x \sqrt{a+\frac{b}{x^3}}}{24 a^3}+\frac{11 x^4 \sqrt{a+\frac{b}{x^3}}}{12 a^2}-\frac{2 x^4}{3 a \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a + b/x^3)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 57.3059, size = 474, normalized size = 0.84 \[ - \frac{2 x^{4}}{3 a \sqrt{a + \frac{b}{x^{3}}}} + \frac{11 x^{4} \sqrt{a + \frac{b}{x^{3}}}}{12 a^{2}} + \frac{55 b^{\frac{4}{3}} \sqrt{a + \frac{b}{x^{3}}}}{24 a^{3} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )} - \frac{55 b x \sqrt{a + \frac{b}{x^{3}}}}{24 a^{3}} - \frac{55 \sqrt [4]{3} b^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{48 a^{\frac{8}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} + \frac{55 \sqrt{2} \cdot 3^{\frac{3}{4}} b^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{72 a^{\frac{8}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(a+b/x**3)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.76578, size = 370, normalized size = 0.66 \[ \frac{\left (a x^3+b\right ) \left (-55 \left (-a^{2/3} b^{4/3} x^2+\sqrt [3]{a} b^{5/3} x+a b x^3\right )-\frac{55 (-1)^{2/3} b^{4/3} \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )^2 \sqrt{\frac{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} x \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\left (\sqrt [3]{a} x+\sqrt [3]{b}\right )^2}} \sqrt{\frac{(-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}}{\sqrt [3]{a} x+\sqrt [3]{b}}} \left (\left (1+i \sqrt{3}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )+\left (-3-i \sqrt{3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )\right )}{2 \left ((-1)^{2/3}-1\right )}+16 a b x^3+6 a x^3 \left (a x^3+b\right )\right )}{24 a^3 x^5 \left (a+\frac{b}{x^3}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^3/(a + b/x^3)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.025, size = 2936, normalized size = 5.2 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(a+b/x^3)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a + b/x^3)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a + b/x^3)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.63922, size = 46, normalized size = 0.08 \[ - \frac{x^{4} \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{3}{2} \\ - \frac{1}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (- \frac{1}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(a+b/x**3)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(a + b/x^3)^(3/2),x, algorithm="giac")
[Out]