3.2079 \(\int \frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=278 \[ -\frac{4 a^{13/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{8 a^{13/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{8 a^3 \sqrt{a+\frac{b}{x^4}}}{39 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{4 a^2 \sqrt{a+\frac{b}{x^4}}}{39 x^3}-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{117 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{13 x^3} \]

[Out]

(-4*a^2*Sqrt[a + b/x^4])/(39*x^3) - (10*a*(a + b/x^4)^(3/2))/(117*x^3) - (a + b/
x^4)^(5/2)/(13*x^3) - (8*a^3*Sqrt[a + b/x^4])/(39*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x^2
)*x) + (8*a^(13/4)*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])/(39*b^(3/4)*Sqrt[a + b/x^4
]) - (4*a^(13/4)*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])/(39*b^(3/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.445405, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{4 a^{13/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{8 a^{13/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{39 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{8 a^3 \sqrt{a+\frac{b}{x^4}}}{39 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{4 a^2 \sqrt{a+\frac{b}{x^4}}}{39 x^3}-\frac{10 a \left (a+\frac{b}{x^4}\right )^{3/2}}{117 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{5/2}}{13 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(5/2)/x^4,x]

[Out]

(-4*a^2*Sqrt[a + b/x^4])/(39*x^3) - (10*a*(a + b/x^4)^(3/2))/(117*x^3) - (a + b/
x^4)^(5/2)/(13*x^3) - (8*a^3*Sqrt[a + b/x^4])/(39*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x^2
)*x) + (8*a^(13/4)*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])/(39*b^(3/4)*Sqrt[a + b/x^4
]) - (4*a^(13/4)*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])/(39*b^(3/4)*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 35.9919, size = 253, normalized size = 0.91 \[ \frac{8 a^{\frac{13}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{39 b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{4 a^{\frac{13}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{39 b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{8 a^{3} \sqrt{a + \frac{b}{x^{4}}}}{39 \sqrt{b} x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} - \frac{4 a^{2} \sqrt{a + \frac{b}{x^{4}}}}{39 x^{3}} - \frac{10 a \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{117 x^{3}} - \frac{\left (a + \frac{b}{x^{4}}\right )^{\frac{5}{2}}}{13 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*a**(13/4)*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)/(39*b**(3/4)*sqrt(a + b/x**4)
) - 4*a**(13/4)*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)/(39*b**(3/4)*sqrt(a + b/x
**4)) - 8*a**3*sqrt(a + b/x**4)/(39*sqrt(b)*x*(sqrt(a) + sqrt(b)/x**2)) - 4*a**2
*sqrt(a + b/x**4)/(39*x**3) - 10*a*(a + b/x**4)**(3/2)/(117*x**3) - (a + b/x**4)
**(5/2)/(13*x**3)

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Mathematica [C]  time = 0.420107, size = 223, normalized size = 0.8 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (24 a^{7/2} \sqrt{b} x^{13} \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-24 a^{7/2} \sqrt{b} x^{13} \sqrt{\frac{a x^4}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (24 a^4 x^{16}+55 a^3 b x^{12}+59 a^2 b^2 x^8+37 a b^3 x^4+9 b^4\right )\right )}{117 b x^{11} \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b/x^4]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(9*b^4 + 37*a*b^3*x^4 + 59*a^2*b^2*
x^8 + 55*a^3*b*x^12 + 24*a^4*x^16) - 24*a^(7/2)*Sqrt[b]*x^13*Sqrt[1 + (a*x^4)/b]
*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1] + 24*a^(7/2)*Sqrt[b]*x^13
*Sqrt[1 + (a*x^4)/b]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1]))/(11
7*Sqrt[(I*Sqrt[a])/Sqrt[b]]*b*x^11*(b + a*x^4))

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Maple [C]  time = 0.034, size = 279, normalized size = 1. \[ -{\frac{1}{117\,{x}^{3} \left ( a{x}^{4}+b \right ) ^{3}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( -24\,i{a}^{{\frac{7}{2}}}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{13}b{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +24\,i{a}^{{\frac{7}{2}}}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{13}b{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +24\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}\sqrt{b}{x}^{16}{a}^{4}+55\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{3/2}{x}^{12}{a}^{3}+59\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{5/2}{x}^{8}{a}^{2}+37\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{7/2}{x}^{4}a+9\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{9/2} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(5/2)/x^4,x)

[Out]

-1/117*((a*x^4+b)/x^4)^(5/2)*(-24*I*a^(7/2)*(-(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)*x^13*b*EllipticF(x*(I*a^(1/2)/b^(1/
2))^(1/2),I)+24*I*a^(7/2)*(-(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)*x^13*b*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)+24*(I
*a^(1/2)/b^(1/2))^(1/2)*b^(1/2)*x^16*a^4+55*(I*a^(1/2)/b^(1/2))^(1/2)*b^(3/2)*x^
12*a^3+59*(I*a^(1/2)/b^(1/2))^(1/2)*b^(5/2)*x^8*a^2+37*(I*a^(1/2)/b^(1/2))^(1/2)
*b^(7/2)*x^4*a+9*(I*a^(1/2)/b^(1/2))^(1/2)*b^(9/2))/x^3/(a*x^4+b)^3/b^(3/2)/(I*a
^(1/2)/b^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(5/2)/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{x^{12}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((a^2*x^8 + 2*a*b*x^4 + b^2)*sqrt((a*x^4 + b)/x^4)/x^12, x)

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Sympy [A]  time = 11.081, size = 41, normalized size = 0.15 \[ - \frac{a^{\frac{5}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(5/2)*gamma(3/4)*hyper((-5/2, 3/4), (7/4,), b*exp_polar(I*pi)/(a*x**4))/(4*x
**3*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(5/2)/x^4, x)