3.2103 \(\int \frac{1}{\left (a+\frac{b}{x^4}\right )^{5/2} x^4} \, dx\)

Optimal. Leaf size=262 \[ \frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{8 a^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{1}{4 a^2 x^3 \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}}}{4 a^2 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{1}{6 a x^3 \left (a+\frac{b}{x^4}\right )^{3/2}} \]

[Out]

-1/(6*a*(a + b/x^4)^(3/2)*x^3) - 1/(4*a^2*Sqrt[a + b/x^4]*x^3) + Sqrt[a + b/x^4]
/(4*a^2*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)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])
/(4*a^(7/4)*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)*EllipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(8*a^
(7/4)*b^(3/4)*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.390801, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{8 a^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{\sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{1}{4 a^2 x^3 \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}}}{4 a^2 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{1}{6 a x^3 \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(6*a*(a + b/x^4)^(3/2)*x^3) - 1/(4*a^2*Sqrt[a + b/x^4]*x^3) + Sqrt[a + b/x^4]
/(4*a^2*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)*EllipticE[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])
/(4*a^(7/4)*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)*EllipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(8*a^
(7/4)*b^(3/4)*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 31.4734, size = 233, normalized size = 0.89 \[ - \frac{1}{6 a x^{3} \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}} - \frac{1}{4 a^{2} x^{3} \sqrt{a + \frac{b}{x^{4}}}} + \frac{\sqrt{a + \frac{b}{x^{4}}}}{4 a^{2} \sqrt{b} x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} - \frac{\sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{7}{4}} b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} + \frac{\sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{8 a^{\frac{7}{4}} b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.415921, size = 155, normalized size = 0.59 \[ \frac{\left (a x^4+b\right )^2 \left (\frac{3 a x^7+b x^3}{3 a^2 b x^4+3 a b^2}+\frac{i \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{\frac{a x^4}{b}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{a^2}\right )}{4 x^{10} \left (a+\frac{b}{x^4}\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.03, size = 503, normalized size = 1.9 \[ -{\frac{1}{12\,{x}^{10}} \left ( -3\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{7/2}\sqrt{b}{x}^{11}+3\,i{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{8}{a}^{3}b-3\,i{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{8}{a}^{3}b-4\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{a}^{5/2}{b}^{3/2}{x}^{7}+6\,i{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{4}{a}^{2}{b}^{2}-6\,i{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{4}{a}^{2}{b}^{2}-\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{a}^{{\frac{3}{2}}}{b}^{{\frac{5}{2}}}{x}^{3}+3\,i{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}a{b}^{3}-3\,i{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) \sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}a{b}^{3} \right ){a}^{-{\frac{5}{2}}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^4)^(5/2)/x^4,x)

[Out]

-1/12*(-3*(I*a^(1/2)/b^(1/2))^(1/2)*a^(7/2)*b^(1/2)*x^11+3*I*EllipticF(x*(I*a^(1
/2)/b^(1/2))^(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)*x^8*a^3*b-3*I*EllipticE(x*(I*a^(1/2)/b^(1/2))^(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)*x
^8*a^3*b-4*(I*a^(1/2)/b^(1/2))^(1/2)*a^(5/2)*b^(3/2)*x^7+6*I*EllipticF(x*(I*a^(1
/2)/b^(1/2))^(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)*x^4*a^2*b^2-6*I*EllipticE(x*(I*a^(1/2)/b^(1/2))^(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)
*x^4*a^2*b^2-(I*a^(1/2)/b^(1/2))^(1/2)*a^(3/2)*b^(5/2)*x^3+3*I*EllipticF(x*(I*a^
(1/2)/b^(1/2))^(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)*a*b^3-3*I*EllipticE(x*(I*a^(1/2)/b^(1/2))^(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)*a*b
^3)/a^(5/2)/((a*x^4+b)/x^4)^(5/2)/x^10/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. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(5/2)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((a + b/x^4)^(5/2)*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(5/2)*x^4),x, algorithm="fricas")

[Out]

integral(x^4/((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 = 15.4528, size = 39, normalized size = 0.15 \[ - \frac{\Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{5}{2}} x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^4)^(5/2)*x^4),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^4)^(5/2)*x^4), x)