Optimal. Leaf size=110 \[ \frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x^3 \sqrt{a+\frac{b}{x^4}}}{3 a} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.140947, antiderivative size = 110, 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 \[ \frac{b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{6 a^{5/4} \sqrt{a+\frac{b}{x^4}}}+\frac{x^3 \sqrt{a+\frac{b}{x^4}}}{3 a} \]
Antiderivative was successfully verified.
[In] Int[x^2/Sqrt[a + b/x^4],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.18552, size = 95, normalized size = 0.86 \[ \frac{x^{3} \sqrt{a + \frac{b}{x^{4}}}}{3 a} + \frac{b^{\frac{3}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{6 a^{\frac{5}{4}} \sqrt{a + \frac{b}{x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b/x**4)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.129054, size = 113, normalized size = 1.03 \[ \frac{x \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )+i b \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )}{3 a x^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/Sqrt[a + b/x^4],x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.016, size = 124, normalized size = 1.1 \[{\frac{1}{3\,a{x}^{2}} \left ( \sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{x}^{5}a-b\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}xb \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b/x^4)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + \frac{b}{x^{4}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(a + b/x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(a + b/x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.4267, size = 42, normalized size = 0.38 \[ - \frac{x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a+b/x**4)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{a + \frac{b}{x^{4}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(a + b/x^4),x, algorithm="giac")
[Out]