3.2092 \(\int \frac{x^2}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx\)

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]

-x^3/(2*a*Sqrt[a + b/x^4]) + (5*Sqrt[a + b/x^4]*x^3)/(6*a^2) + (5*b^(3/4)*Sqrt[(
a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcCot
[(a^(1/4)*x)/b^(1/4)], 1/2])/(12*a^(9/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

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]

-x^3/(2*a*Sqrt[a + b/x^4]) + (5*Sqrt[a + b/x^4]*x^3)/(6*a^2) + (5*b^(3/4)*Sqrt[(
a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcCot
[(a^(1/4)*x)/b^(1/4)], 1/2])/(12*a^(9/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

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]

-x**3/(2*a*sqrt(a + b/x**4)) + 5*x**3*sqrt(a + b/x**4)/(6*a**2) + 5*b**(3/4)*sqr
t((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + sqrt(b)/x**2)*elliptic_f(
2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(12*a**(9/4)*sqrt(a + b/x**4))

_______________________________________________________________________________________

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]

(Sqrt[(I*Sqrt[a])/Sqrt[b]]*x*(5*b + 2*a*x^4) + (5*I)*b*Sqrt[1 + (a*x^4)/b]*Ellip
ticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1])/(6*a^2*Sqrt[(I*Sqrt[a])/Sqrt[b
]]*Sqrt[a + b/x^4]*x^2)

_______________________________________________________________________________________

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]

1/6*(a*x^4+b)*(2*(I*a^(1/2)/b^(1/2))^(1/2)*x^5*a-5*b*(-(I*a^(1/2)*x^2-b^(1/2))/b
^(1/2))^(1/2)*((I*a^(1/2)*x^2+b^(1/2))/b^(1/2))^(1/2)*EllipticF(x*(I*a^(1/2)/b^(
1/2))^(1/2),I)+5*(I*a^(1/2)/b^(1/2))^(1/2)*x*b)/((a*x^4+b)/x^4)^(3/2)/x^6/a^2/(I
*a^(1/2)/b^(1/2))^(1/2)

_______________________________________________________________________________________

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]

integrate(x^2/(a + b/x^4)^(3/2), x)

_______________________________________________________________________________________

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]

integral(x^6/((a*x^4 + b)*sqrt((a*x^4 + b)/x^4)), x)

_______________________________________________________________________________________

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]

-x**3*gamma(-3/4)*hyper((-3/4, 3/2), (1/4,), b*exp_polar(I*pi)/(a*x**4))/(4*a**(
3/2)*gamma(1/4))

_______________________________________________________________________________________

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]

integrate(x^2/(a + b/x^4)^(3/2), x)