3.2070 \(\int \frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=126 \[ -\frac{2 a^{7/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{7 x}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{7 x} \]

[Out]

(-2*a*Sqrt[a + b/x^4])/(7*x) - (a + b/x^4)^(3/2)/(7*x) - (2*a^(7/4)*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])/(7*b^(1/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.152631, antiderivative size = 126, 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 \[ -\frac{2 a^{7/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{7 x}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{7 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(3/2)/x^2,x]

[Out]

(-2*a*Sqrt[a + b/x^4])/(7*x) - (a + b/x^4)^(3/2)/(7*x) - (2*a^(7/4)*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])/(7*b^(1/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.15675, size = 112, normalized size = 0.89 \[ - \frac{2 a^{\frac{7}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{7 \sqrt [4]{b} \sqrt{a + \frac{b}{x^{4}}}} - \frac{2 a \sqrt{a + \frac{b}{x^{4}}}}{7 x} - \frac{\left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{7 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**4)**(3/2)/x**2,x)

[Out]

-2*a**(7/4)*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)/(7*b**(1/4)*sqrt(a + b/x**4))
- 2*a*sqrt(a + b/x**4)/(7*x) - (a + b/x**4)**(3/2)/(7*x)

_______________________________________________________________________________________

Mathematica [C]  time = 0.193086, size = 135, normalized size = 1.07 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (3 a^2 x^8+4 a b x^4+b^2\right )+4 i a^2 x^7 \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{7 x^5 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(3/2)/x^2,x]

[Out]

-(Sqrt[a + b/x^4]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(b^2 + 4*a*b*x^4 + 3*a^2*x^8) + (4*
I)*a^2*x^7*Sqrt[1 + (a*x^4)/b]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x],
-1]))/(7*Sqrt[(I*Sqrt[a])/Sqrt[b]]*x^5*(b + a*x^4))

_______________________________________________________________________________________

Maple [C]  time = 0.028, size = 157, normalized size = 1.3 \[ -{\frac{1}{7\,x \left ( a{x}^{4}+b \right ) ^{2}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( -4\,{a}^{2}\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{7}+3\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{8}{a}^{2}+4\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{4}ab+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{b}^{2} \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(3/2)/x^2,x)

[Out]

-1/7*((a*x^4+b)/x^4)^(3/2)*(-4*a^2*(-(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^7
+3*(I*a^(1/2)/b^(1/2))^(1/2)*x^8*a^2+4*(I*a^(1/2)/b^(1/2))^(1/2)*x^4*a*b+(I*a^(1
/2)/b^(1/2))^(1/2)*b^2)/x/(a*x^4+b)^2/(I*a^(1/2)/b^(1/2))^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(3/2)/x^2,x, algorithm="maxima")

[Out]

integrate((a + b/x^4)^(3/2)/x^2, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(3/2)/x^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 5.59632, size = 39, normalized size = 0.31 \[ - \frac{a^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 x \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**4)**(3/2)/x**2,x)

[Out]

-a**(3/2)*gamma(1/4)*hyper((-3/2, 1/4), (5/4,), b*exp_polar(I*pi)/(a*x**4))/(4*x
*gamma(5/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(3/2)/x^2,x, algorithm="giac")

[Out]

integrate((a + b/x^4)^(3/2)/x^2, x)