3.2095 \(\int \frac{1}{\left (a+\frac{b}{x^4}\right )^{3/2} x^4} \, dx\)

Optimal. Leaf size=241 \[ \frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{3/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{1}{2 a x^3 \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}}}{2 a \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x^3) + Sqrt[a + b/x^4]/(2*a*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x
^2)*x) - (Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*El
lipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(2*a^(3/4)*b^(3/4)*Sqrt[a + b/x^4])
+ (Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Elliptic
F[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(4*a^(3/4)*b^(3/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.320025, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{3/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{3/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{1}{2 a x^3 \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}}}{2 a \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^4)^(3/2)*x^4),x]

[Out]

-1/(2*a*Sqrt[a + b/x^4]*x^3) + Sqrt[a + b/x^4]/(2*a*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x
^2)*x) - (Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*El
lipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(2*a^(3/4)*b^(3/4)*Sqrt[a + b/x^4])
+ (Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Elliptic
F[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(4*a^(3/4)*b^(3/4)*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 25.2176, size = 211, normalized size = 0.88 \[ - \frac{1}{2 a x^{3} \sqrt{a + \frac{b}{x^{4}}}} + \frac{\sqrt{a + \frac{b}{x^{4}}}}{2 a \sqrt{b} x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} - \frac{\sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} + \frac{\sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{3}{4}} b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**4)**(3/2)/x**4,x)

[Out]

-1/(2*a*x**3*sqrt(a + b/x**4)) + sqrt(a + b/x**4)/(2*a*sqrt(b)*x*(sqrt(a) + sqrt
(b)/x**2)) - sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + sqrt(b)/x
**2)*elliptic_e(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(2*a**(3/4)*b**(3/4)*sqrt(a
+ b/x**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)/(4*a**(3/4)*b**(3/4)*sqrt(a
+ b/x**4))

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Mathematica [C]  time = 0.177907, size = 166, normalized size = 0.69 \[ \frac{i \left (\sqrt{b} \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-\sqrt{b} \sqrt{\frac{a x^4}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )+\sqrt{a} x^3 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}\right )}{2 b^{3/2} x^2 \left (\frac{i \sqrt{a}}{\sqrt{b}}\right )^{3/2} \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^4)^(3/2)*x^4),x]

[Out]

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

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Maple [C]  time = 0.021, size = 187, normalized size = 0.8 \[{\frac{a{x}^{4}+b}{2\,{x}^{6}} \left ({x}^{3}\sqrt{b}\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}\sqrt{a}-i\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}b{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +i\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}b{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4)^(3/2)/x^4,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((a + b/x^4)^(3/2)*x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (a x^{4} + b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x^4),x, algorithm="fricas")

[Out]

integral(1/((a*x^4 + b)*sqrt((a*x^4 + b)/x^4)), x)

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Sympy [A]  time = 8.59647, size = 39, normalized size = 0.16 \[ - \frac{\Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{3}{2}} x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**4)**(3/2)/x**4,x)

[Out]

-gamma(3/4)*hyper((3/4, 3/2), (7/4,), b*exp_polar(I*pi)/(a*x**4))/(4*a**(3/2)*x*
*3*gamma(7/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(3/2)*x^4),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^4)^(3/2)*x^4), x)