3.2078 \(\int \frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{x^2} \, dx\)

Optimal. Leaf size=147 \[ -\frac{20 a^{11/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{20 a^2 \sqrt{a+\frac{b}{x^4}}}{77 x}-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{77 x}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{11 x} \]

[Out]

(-20*a^2*Sqrt[a + b/x^4])/(77*x) - (10*a*(a + b/x^4)^(3/2))/(77*x) - (a + b/x^4)
^(5/2)/(11*x) - (20*a^(11/4)*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])/(77*b^(1/4)*Sqrt
[a + b/x^4])

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Rubi [A]  time = 0.18914, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{20 a^{11/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{77 \sqrt [4]{b} \sqrt{a+\frac{b}{x^4}}}-\frac{20 a^2 \sqrt{a+\frac{b}{x^4}}}{77 x}-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{77 x}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{11 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(5/2)/x^2,x]

[Out]

(-20*a^2*Sqrt[a + b/x^4])/(77*x) - (10*a*(a + b/x^4)^(3/2))/(77*x) - (a + b/x^4)
^(5/2)/(11*x) - (20*a^(11/4)*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])/(77*b^(1/4)*Sqrt
[a + b/x^4])

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Rubi in Sympy [A]  time = 11.4276, size = 131, normalized size = 0.89 \[ - \frac{20 a^{\frac{11}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{77 \sqrt [4]{b} \sqrt{a + \frac{b}{x^{4}}}} - \frac{20 a^{2} \sqrt{a + \frac{b}{x^{4}}}}{77 x} - \frac{10 a \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{77 x} - \frac{\left (a + \frac{b}{x^{4}}\right )^{\frac{5}{2}}}{11 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20*a**(11/4)*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)/(77*b**(1/4)*sqrt(a + b/x**
4)) - 20*a**2*sqrt(a + b/x**4)/(77*x) - 10*a*(a + b/x**4)**(3/2)/(77*x) - (a + b
/x**4)**(5/2)/(11*x)

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Mathematica [C]  time = 0.24143, size = 148, normalized size = 1.01 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (40 i a^3 x^{11} \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (37 a^3 x^{12}+61 a^2 b x^8+31 a b^2 x^4+7 b^3\right )\right )}{77 x^9 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b/x^4]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(7*b^3 + 31*a*b^2*x^4 + 61*a^2*b*x^
8 + 37*a^3*x^12) + (40*I)*a^3*x^11*Sqrt[1 + (a*x^4)/b]*EllipticF[I*ArcSinh[Sqrt[
(I*Sqrt[a])/Sqrt[b]]*x], -1]))/(77*Sqrt[(I*Sqrt[a])/Sqrt[b]]*x^9*(b + a*x^4))

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Maple [C]  time = 0.031, size = 180, normalized size = 1.2 \[ -{\frac{1}{77\,x \left ( a{x}^{4}+b \right ) ^{3}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( -40\,{a}^{3}\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{11}+37\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{12}{a}^{3}+61\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{8}{a}^{2}b+31\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{4}a{b}^{2}+7\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{3} \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(5/2)/x^2,x)

[Out]

-1/77*((a*x^4+b)/x^4)^(5/2)*(-40*a^3*(-(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
^11+37*(I*a^(1/2)/b^(1/2))^(1/2)*x^12*a^3+61*(I*a^(1/2)/b^(1/2))^(1/2)*x^8*a^2*b
+31*(I*a^(1/2)/b^(1/2))^(1/2)*x^4*a*b^2+7*(I*a^(1/2)/b^(1/2))^(1/2)*b^3)/x/(a*x^
4+b)^3/(I*a^(1/2)/b^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(5/2)/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{x^{10}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((a^2*x^8 + 2*a*b*x^4 + b^2)*sqrt((a*x^4 + b)/x^4)/x^10, x)

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Sympy [A]  time = 10.9865, size = 39, normalized size = 0.27 \[ - \frac{a^{\frac{5}{2}} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 x \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(5/2)/x^2, x)