Optimal. Leaf size=107 \[ \frac{1}{3} x^3 \sqrt{a+\frac{b}{x^4}}-\frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a+\frac{b}{x^4}}} \]
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Rubi [A] time = 0.137301, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{3} x^3 \sqrt{a+\frac{b}{x^4}}-\frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x^4]*x^2,x]
[Out]
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Rubi in Sympy [A] time = 8.71533, size = 94, normalized size = 0.88 \[ \frac{x^{3} \sqrt{a + \frac{b}{x^{4}}}}{3} - \frac{b^{\frac{3}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt [4]{a} \sqrt{a + \frac{b}{x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(a+b/x**4)**(1/2),x)
[Out]
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Mathematica [C] time = 0.303075, size = 93, normalized size = 0.87 \[ \frac{1}{3} x^2 \sqrt{a+\frac{b}{x^4}} \left (x-\frac{2 i b \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x^4]*x^2,x]
[Out]
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Maple [C] time = 0.052, size = 130, normalized size = 1.2 \[{\frac{{x}^{2}}{3\,a{x}^{4}+3\,b}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( \sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{x}^{5}a+2\,b\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ) +\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}xb \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(a+b/x^4)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + \frac{b}{x^{4}}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^4)*x^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^4)*x^2,x, algorithm="fricas")
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Sympy [A] time = 3.79714, size = 44, normalized size = 0.41 \[ - \frac{\sqrt{a} x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac{1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(a+b/x**4)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + \frac{b}{x^{4}}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^4)*x^2,x, algorithm="giac")
[Out]