3.2102 \(\int \frac{1}{\left (a+\frac{b}{x^4}\right )^{5/2} x^2} \, dx\)

Optimal. Leaf size=131 \[ -\frac{5 \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{24 a^{9/4} \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{5}{12 a^2 x \sqrt{a+\frac{b}{x^4}}}-\frac{1}{6 a x \left (a+\frac{b}{x^4}\right )^{3/2}} \]

[Out]

-1/(6*a*(a + b/x^4)^(3/2)*x) - 5/(12*a^2*Sqrt[a + b/x^4]*x) - (5*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])/(24*a^(9/4)*b^(1/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.162898, antiderivative size = 131, 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 \[ -\frac{5 \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{24 a^{9/4} \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{5}{12 a^2 x \sqrt{a+\frac{b}{x^4}}}-\frac{1}{6 a x \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^4)^(5/2)*x^2),x]

[Out]

-1/(6*a*(a + b/x^4)^(3/2)*x) - 5/(12*a^2*Sqrt[a + b/x^4]*x) - (5*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])/(24*a^(9/4)*b^(1/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.0075, size = 117, normalized size = 0.89 \[ - \frac{1}{6 a x \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}} - \frac{5}{12 a^{2} x \sqrt{a + \frac{b}{x^{4}}}} - \frac{5 \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{24 a^{\frac{9}{4}} \sqrt [4]{b} \sqrt{a + \frac{b}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**4)**(5/2)/x**2,x)

[Out]

-1/(6*a*x*(a + b/x**4)**(3/2)) - 5/(12*a**2*x*sqrt(a + b/x**4)) - 5*sqrt((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)/(24*a**(9/4)*b**(1/4)*sqrt(a + b/x**4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.307096, size = 107, normalized size = 0.82 \[ \frac{-\frac{7 a x^5+5 b x}{a x^4+b}-\frac{5 i \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}}{12 a^2 x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^4)^(5/2)*x^2),x]

[Out]

(-((5*b*x + 7*a*x^5)/(b + a*x^4)) - ((5*I)*Sqrt[1 + (a*x^4)/b]*EllipticF[I*ArcSi
nh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1])/Sqrt[(I*Sqrt[a])/Sqrt[b]])/(12*a^2*Sqrt[a
+ b/x^4]*x^2)

_______________________________________________________________________________________

Maple [C]  time = 0.031, size = 279, normalized size = 2.1 \[ -{\frac{1}{12\,{a}^{2}{x}^{10}} \left ( -5\,\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{8}{a}^{2}+7\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{9}{a}^{2}-10\,\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{4}ab+12\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{5}ab-5\,\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){b}^{2}+5\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}x{b}^{2} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4)^(5/2)/x^2,x)

[Out]

-1/12*(-5*(-(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)*x^8*a^2+7*(I*a^(1/2)/b^(1/2)
)^(1/2)*x^9*a^2-10*(-(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)*x^4*a*b+12*(I*a^(1/
2)/b^(1/2))^(1/2)*x^5*a*b-5*(-(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)*b^2+5*(I*a
^(1/2)/b^(1/2))^(1/2)*x*b^2)/a^2/((a*x^4+b)/x^4)^(5/2)/x^10/(I*a^(1/2)/b^(1/2))^
(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(5/2)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((a + b/x^4)^(5/2)*x^2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(5/2)*x^2),x, algorithm="fricas")

[Out]

integral(x^6/((a^2*x^8 + 2*a*b*x^4 + b^2)*sqrt((a*x^4 + b)/x^4)), x)

_______________________________________________________________________________________

Sympy [A]  time = 10.299, size = 37, normalized size = 0.28 \[ - \frac{\Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{5}{2}} x \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**4)**(5/2)/x**2,x)

[Out]

-gamma(1/4)*hyper((1/4, 5/2), (5/4,), b*exp_polar(I*pi)/(a*x**4))/(4*a**(5/2)*x*
gamma(5/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(5/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^4)^(5/2)*x^2), x)