3.2101 \(\int \frac{1}{\left (a+\frac{b}{x^4}\right )^{5/2}} \, dx\)

Optimal. Leaf size=277 \[ -\frac{7 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{8 a^{11/4} \sqrt{a+\frac{b}{x^4}}}+\frac{7 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{11/4} \sqrt{a+\frac{b}{x^4}}}+\frac{7 x \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{7 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{4 a^3 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{7 x}{12 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

[Out]

(-7*Sqrt[b]*Sqrt[a + b/x^4])/(4*a^3*(Sqrt[a] + Sqrt[b]/x^2)*x) - x/(6*a*(a + b/x
^4)^(3/2)) - (7*x)/(12*a^2*Sqrt[a + b/x^4]) + (7*Sqrt[a + b/x^4]*x)/(4*a^3) + (7
*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Ell
ipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(4*a^(11/4)*Sqrt[a + b/x^4]) - (7*b^
(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Ellipt
icF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(8*a^(11/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.433029, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ -\frac{7 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{8 a^{11/4} \sqrt{a+\frac{b}{x^4}}}+\frac{7 \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{11/4} \sqrt{a+\frac{b}{x^4}}}+\frac{7 x \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{7 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{4 a^3 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{7 x}{12 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(-5/2),x]

[Out]

(-7*Sqrt[b]*Sqrt[a + b/x^4])/(4*a^3*(Sqrt[a] + Sqrt[b]/x^2)*x) - x/(6*a*(a + b/x
^4)^(3/2)) - (7*x)/(12*a^2*Sqrt[a + b/x^4]) + (7*Sqrt[a + b/x^4]*x)/(4*a^3) + (7
*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Ell
ipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(4*a^(11/4)*Sqrt[a + b/x^4]) - (7*b^
(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*Ellipt
icF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(8*a^(11/4)*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 36.4579, size = 252, normalized size = 0.91 \[ - \frac{x}{6 a \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}} - \frac{7 x}{12 a^{2} \sqrt{a + \frac{b}{x^{4}}}} - \frac{7 \sqrt{b} \sqrt{a + \frac{b}{x^{4}}}}{4 a^{3} x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} + \frac{7 x \sqrt{a + \frac{b}{x^{4}}}}{4 a^{3}} + \frac{7 \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{11}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{7 \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{8 a^{\frac{11}{4}} \sqrt{a + \frac{b}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**4)**(5/2),x)

[Out]

-x/(6*a*(a + b/x**4)**(3/2)) - 7*x/(12*a**2*sqrt(a + b/x**4)) - 7*sqrt(b)*sqrt(a
+ b/x**4)/(4*a**3*x*(sqrt(a) + sqrt(b)/x**2)) + 7*x*sqrt(a + b/x**4)/(4*a**3) +
7*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)/(4*a**(11/4)*sqrt(a + b/x**4)
) - 7*b**(1/4)*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)/(8*a**(11/4)*sqrt(a + b/x*
*4))

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Mathematica [C]  time = 0.407822, size = 153, normalized size = 0.55 \[ \frac{\left (a x^4+b\right )^2 \left (-\frac{x^3 \left (9 a x^4+7 b\right )}{3 a^2 \left (a x^4+b\right )}+\frac{7 i \sqrt{\frac{a x^4}{b}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{a^2 \left (\frac{i \sqrt{a}}{\sqrt{b}}\right )^{3/2}}\right )}{4 x^{10} \left (a+\frac{b}{x^4}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(-5/2),x]

[Out]

((b + a*x^4)^2*(-(x^3*(7*b + 9*a*x^4))/(3*a^2*(b + a*x^4)) + ((7*I)*Sqrt[1 + (a*
x^4)/b]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1] - EllipticF[I*Arc
Sinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1]))/(a^2*((I*Sqrt[a])/Sqrt[b])^(3/2))))/(4*
(a + b/x^4)^(5/2)*x^10)

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Maple [C]  time = 0.03, size = 503, normalized size = 1.8 \[ -{\frac{1}{12\,{x}^{10}} \left ( 9\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{9/2}{x}^{11}-21\,i\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{b}{x}^{8}{a}^{4}+21\,i\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{b}{x}^{8}{a}^{4}+16\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{7/2}{x}^{7}b-42\,i\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ){b}^{{\frac{3}{2}}}{x}^{4}{a}^{3}+42\,i\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ){b}^{{\frac{3}{2}}}{x}^{4}{a}^{3}+7\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{5/2}{x}^{3}{b}^{2}-21\,i\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ){b}^{{\frac{5}{2}}}{a}^{2}+21\,i\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ){b}^{{\frac{5}{2}}}{a}^{2} \right ){a}^{-{\frac{9}{2}}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4)^(5/2),x)

[Out]

-1/12*(9*(I*a^(1/2)/b^(1/2))^(1/2)*a^(9/2)*x^11-21*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)*EllipticF(x*(I*a^(1/2)/b^(
1/2))^(1/2),I)*b^(1/2)*x^8*a^4+21*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)*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*b^
(1/2)*x^8*a^4+16*(I*a^(1/2)/b^(1/2))^(1/2)*a^(7/2)*x^7*b-42*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)*EllipticF(x*(I*a^
(1/2)/b^(1/2))^(1/2),I)*b^(3/2)*x^4*a^3+42*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)*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1
/2),I)*b^(3/2)*x^4*a^3+7*(I*a^(1/2)/b^(1/2))^(1/2)*a^(5/2)*x^3*b^2-21*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)*Ellipti
cF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*b^(5/2)*a^2+21*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)*EllipticE(x*(I*a^(1/2)/b^(1/
2))^(1/2),I)*b^(5/2)*a^2)/a^(9/2)/((a*x^4+b)/x^4)^(5/2)/x^10/(I*a^(1/2)/b^(1/2))
^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(-5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.19938, size = 41, normalized size = 0.15 \[ - \frac{x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{5}{2}} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**4)**(5/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(-5/2), x)