3.2093 \(\int \frac{1}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx\)

Optimal. Leaf size=258 \[ -\frac{3 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{7/4} \sqrt{a+\frac{b}{x^4}}}+\frac{3 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+\frac{b}{x^4}}}+\frac{3 x \sqrt{a+\frac{b}{x^4}}}{2 a^2}-\frac{3 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{2 a^2 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{x}{2 a \sqrt{a+\frac{b}{x^4}}} \]

[Out]

(-3*Sqrt[b]*Sqrt[a + b/x^4])/(2*a^2*(Sqrt[a] + Sqrt[b]/x^2)*x) - x/(2*a*Sqrt[a +
b/x^4]) + (3*Sqrt[a + b/x^4]*x)/(2*a^2) + (3*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a]
+ Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)
], 1/2])/(2*a^(7/4)*Sqrt[a + b/x^4]) - (3*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sq
rt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1
/2])/(4*a^(7/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.366454, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ -\frac{3 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{7/4} \sqrt{a+\frac{b}{x^4}}}+\frac{3 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+\frac{b}{x^4}}}+\frac{3 x \sqrt{a+\frac{b}{x^4}}}{2 a^2}-\frac{3 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{2 a^2 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{x}{2 a \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(-3/2),x]

[Out]

(-3*Sqrt[b]*Sqrt[a + b/x^4])/(2*a^2*(Sqrt[a] + Sqrt[b]/x^2)*x) - x/(2*a*Sqrt[a +
b/x^4]) + (3*Sqrt[a + b/x^4]*x)/(2*a^2) + (3*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a]
+ Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)
], 1/2])/(2*a^(7/4)*Sqrt[a + b/x^4]) - (3*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sq
rt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*EllipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1
/2])/(4*a^(7/4)*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 29.9066, size = 233, normalized size = 0.9 \[ - \frac{x}{2 a \sqrt{a + \frac{b}{x^{4}}}} - \frac{3 \sqrt{b} \sqrt{a + \frac{b}{x^{4}}}}{2 a^{2} x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} + \frac{3 x \sqrt{a + \frac{b}{x^{4}}}}{2 a^{2}} + \frac{3 \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{7}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{3 \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{7}{4}} \sqrt{a + \frac{b}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.171908, size = 166, normalized size = 0.64 \[ \frac{-3 \sqrt{b} \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )+3 \sqrt{b} \sqrt{\frac{a x^4}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-\sqrt{a} x^3 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{2 a^{3/2} x^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(-3/2),x]

[Out]

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

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Maple [C]  time = 0.018, size = 187, normalized size = 0.7 \[ -{\frac{a{x}^{4}+b}{2\,{x}^{6}} \left ({x}^{3}{a}^{{\frac{3}{2}}}\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}-3\,i\sqrt{b}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}a{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +3\,i\sqrt{b}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}a{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.1885, size = 41, normalized size = 0.16 \[ - \frac{x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(-3/2), x)