3.2071 \(\int \frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=257 \[ -\frac{2 a^{9/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{4 a^{9/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{4 a^2 \sqrt{a+\frac{b}{x^4}}}{15 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{15 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{9 x^3} \]

[Out]

(-2*a*Sqrt[a + b/x^4])/(15*x^3) - (a + b/x^4)^(3/2)/(9*x^3) - (4*a^2*Sqrt[a + b/
x^4])/(15*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x^2)*x) + (4*a^(9/4)*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])/(15*b^(3/4)*Sqrt[a + b/x^4]) - (2*a^(9/4)*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])/(15*b^(3/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.374442, antiderivative size = 257, 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 \[ -\frac{2 a^{9/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+\frac{b}{x^4}}}+\frac{4 a^{9/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+\frac{b}{x^4}}}-\frac{4 a^2 \sqrt{a+\frac{b}{x^4}}}{15 \sqrt{b} x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}-\frac{2 a \sqrt{a+\frac{b}{x^4}}}{15 x^3}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{9 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(3/2)/x^4,x]

[Out]

(-2*a*Sqrt[a + b/x^4])/(15*x^3) - (a + b/x^4)^(3/2)/(9*x^3) - (4*a^2*Sqrt[a + b/
x^4])/(15*Sqrt[b]*(Sqrt[a] + Sqrt[b]/x^2)*x) + (4*a^(9/4)*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])/(15*b^(3/4)*Sqrt[a + b/x^4]) - (2*a^(9/4)*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])/(15*b^(3/4)*Sqrt[a + b/x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.8456, size = 233, normalized size = 0.91 \[ \frac{4 a^{\frac{9}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{2 a^{\frac{9}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} \sqrt{a + \frac{b}{x^{4}}}} - \frac{4 a^{2} \sqrt{a + \frac{b}{x^{4}}}}{15 \sqrt{b} x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} - \frac{2 a \sqrt{a + \frac{b}{x^{4}}}}{15 x^{3}} - \frac{\left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{9 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(9/4)*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)/(15*b**(3/4)*sqrt(a + b/x**4))
- 2*a**(9/4)*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)/(15*b**(3/4)*sqrt(a + b/x**
4)) - 4*a**2*sqrt(a + b/x**4)/(15*sqrt(b)*x*(sqrt(a) + sqrt(b)/x**2)) - 2*a*sqrt
(a + b/x**4)/(15*x**3) - (a + b/x**4)**(3/2)/(9*x**3)

_______________________________________________________________________________________

Mathematica [C]  time = 0.339974, size = 213, normalized size = 0.83 \[ \frac{x^6 \left (a+\frac{b}{x^4}\right )^{3/2} \left (-\frac{4 a^2}{15 b x}-\frac{11 a}{45 x^5}-\frac{b}{9 x^9}\right )}{a x^4+b}+\frac{4 a^{5/2} x^6 \left (a+\frac{b}{x^4}\right )^{3/2} \sqrt{1-\frac{i \sqrt{a} x^2}{\sqrt{b}}} \sqrt{1+\frac{i \sqrt{a} x^2}{\sqrt{b}}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{15 \sqrt{b} \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(3/2)/x^4,x]

[Out]

((a + b/x^4)^(3/2)*(-b/(9*x^9) - (11*a)/(45*x^5) - (4*a^2)/(15*b*x))*x^6)/(b + a
*x^4) + (4*a^(5/2)*(a + b/x^4)^(3/2)*x^6*Sqrt[1 - (I*Sqrt[a]*x^2)/Sqrt[b]]*Sqrt[
1 + (I*Sqrt[a]*x^2)/Sqrt[b]]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x],
-1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1]))/(15*Sqrt[(I*Sqrt[a
])/Sqrt[b]]*Sqrt[b]*(b + a*x^4)^2)

_______________________________________________________________________________________

Maple [C]  time = 0.031, size = 257, normalized size = 1. \[ -{\frac{1}{45\,{x}^{3} \left ( a{x}^{4}+b \right ) ^{2}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( -12\,i{a}^{{\frac{5}{2}}}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{9}b{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +12\,i{a}^{{\frac{5}{2}}}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{9}b{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +12\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}\sqrt{b}{x}^{12}{a}^{3}+23\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{3/2}{x}^{8}{a}^{2}+16\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{5/2}{x}^{4}a+5\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{7/2} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(3/2)/x^4,x)

[Out]

-1/45*((a*x^4+b)/x^4)^(3/2)*(-12*I*a^(5/2)*(-(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^9*b*EllipticF(x*(I*a^(1/2)/b^(1/2)
)^(1/2),I)+12*I*a^(5/2)*(-(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^9*b*EllipticE(x*(I*a^(1/2)/b^(1/2))^(1/2),I)+12*(I*a^
(1/2)/b^(1/2))^(1/2)*b^(1/2)*x^12*a^3+23*(I*a^(1/2)/b^(1/2))^(1/2)*b^(3/2)*x^8*a
^2+16*(I*a^(1/2)/b^(1/2))^(1/2)*b^(5/2)*x^4*a+5*(I*a^(1/2)/b^(1/2))^(1/2)*b^(7/2
))/x^3/(a*x^4+b)^2/b^(3/2)/(I*a^(1/2)/b^(1/2))^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(3/2)/x^4, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 6.92694, size = 41, normalized size = 0.16 \[ - \frac{a^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(3/2)/x^4, x)