Optimal. Leaf size=131 \[ \frac{5 b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt{a+\frac{b}{x^4}}}+\frac{5 x^3 \sqrt{a+\frac{b}{x^4}}}{6 a^2}-\frac{x^3}{2 a \sqrt{a+\frac{b}{x^4}}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.186721, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{5 b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt{a+\frac{b}{x^4}}}+\frac{5 x^3 \sqrt{a+\frac{b}{x^4}}}{6 a^2}-\frac{x^3}{2 a \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b/x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.8762, size = 117, normalized size = 0.89 \[ - \frac{x^{3}}{2 a \sqrt{a + \frac{b}{x^{4}}}} + \frac{5 x^{3} \sqrt{a + \frac{b}{x^{4}}}}{6 a^{2}} + \frac{5 b^{\frac{3}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{9}{4}} \sqrt{a + \frac{b}{x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b/x**4)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.13519, size = 116, normalized size = 0.89 \[ \frac{x \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (2 a x^4+5 b\right )+5 i b \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )}{6 a^2 x^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a + b/x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.027, size = 133, normalized size = 1. \[{\frac{a{x}^{4}+b}{6\,{a}^{2}{x}^{6}} \left ( 2\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{5}a-5\,b\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ) +5\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}xb \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b/x^4)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a + b/x^4)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a + b/x^4)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.11349, size = 42, normalized size = 0.32 \[ - \frac{x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a+b/x**4)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(a + b/x^4)^(3/2),x, algorithm="giac")
[Out]