Optimal. Leaf size=293 \[ -\frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{55 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{27 a x \sqrt [3]{a+b x^2}}{55 b^2}+\frac{3 x^3 \sqrt [3]{a+b x^2}}{11 b} \]
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Rubi [A] time = 0.419613, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{55 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{27 a x \sqrt [3]{a+b x^2}}{55 b^2}+\frac{3 x^3 \sqrt [3]{a+b x^2}}{11 b} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b*x^2)^(2/3),x]
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Rubi in Sympy [A] time = 15.5465, size = 243, normalized size = 0.83 \[ - \frac{27 \cdot 3^{\frac{3}{4}} a^{2} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{55 b^{3} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} - \frac{27 a x \sqrt [3]{a + b x^{2}}}{55 b^{2}} + \frac{3 x^{3} \sqrt [3]{a + b x^{2}}}{11 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**2+a)**(2/3),x)
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Mathematica [C] time = 0.0574078, size = 79, normalized size = 0.27 \[ \frac{3 \left (9 a^2 x \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right )-9 a^2 x-4 a b x^3+5 b^2 x^5\right )}{55 b^2 \left (a+b x^2\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b*x^2)^(2/3),x]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^2+a)^(2/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^2 + a)^(2/3),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^2 + a)^(2/3),x, algorithm="fricas")
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Sympy [A] time = 2.53838, size = 27, normalized size = 0.09 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**2+a)**(2/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^2 + a)^(2/3),x, algorithm="giac")
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