3.2069 \(\int \left (a+\frac{b}{x^4}\right )^{3/2} \, dx\)

Optimal. Leaf size=250 \[ -\frac{6 a^{5/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+\frac{b}{x^4}}}+\frac{12 a^{5/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+\frac{b}{x^4}}}+x \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{6 b \sqrt{a+\frac{b}{x^4}}}{5 x^3}-\frac{12 a \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{5 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

[Out]

(-6*b*Sqrt[a + b/x^4])/(5*x^3) - (12*a*Sqrt[b]*Sqrt[a + b/x^4])/(5*(Sqrt[a] + Sq
rt[b]/x^2)*x) + (a + b/x^4)^(3/2)*x + (12*a^(5/4)*b^(1/4)*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])/(5*Sqrt[a + b/x^4]) - (6*a^(5/4)*b^(1/4)*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])/(5*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.358019, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ -\frac{6 a^{5/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+\frac{b}{x^4}}}+\frac{12 a^{5/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 \sqrt{a+\frac{b}{x^4}}}+x \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{6 b \sqrt{a+\frac{b}{x^4}}}{5 x^3}-\frac{12 a \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{5 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(3/2),x]

[Out]

(-6*b*Sqrt[a + b/x^4])/(5*x^3) - (12*a*Sqrt[b]*Sqrt[a + b/x^4])/(5*(Sqrt[a] + Sq
rt[b]/x^2)*x) + (a + b/x^4)^(3/2)*x + (12*a^(5/4)*b^(1/4)*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])/(5*Sqrt[a + b/x^4]) - (6*a^(5/4)*b^(1/4)*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])/(5*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 28.2384, size = 228, normalized size = 0.91 \[ \frac{12 a^{\frac{5}{4}} \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{5 \sqrt{a + \frac{b}{x^{4}}}} - \frac{6 a^{\frac{5}{4}} \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{5 \sqrt{a + \frac{b}{x^{4}}}} - \frac{12 a \sqrt{b} \sqrt{a + \frac{b}{x^{4}}}}{5 x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} - \frac{6 b \sqrt{a + \frac{b}{x^{4}}}}{5 x^{3}} + x \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*a**(5/4)*b**(1/4)*sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + s
qrt(b)/x**2)*elliptic_e(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(5*sqrt(a + b/x**4))
- 6*a**(5/4)*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)/(5*sqrt(a + b/x**4
)) - 12*a*sqrt(b)*sqrt(a + b/x**4)/(5*x*(sqrt(a) + sqrt(b)/x**2)) - 6*b*sqrt(a +
b/x**4)/(5*x**3) + x*(a + b/x**4)**(3/2)

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Mathematica [C]  time = 0.307057, size = 196, normalized size = 0.78 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (12 a^{3/2} \sqrt{b} x^5 \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-12 a^{3/2} \sqrt{b} x^5 \sqrt{\frac{a x^4}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (7 a^2 x^8+8 a b x^4+b^2\right )\right )}{5 x^3 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(3/2),x]

[Out]

-(Sqrt[a + b/x^4]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(b^2 + 8*a*b*x^4 + 7*a^2*x^8) - 12*
a^(3/2)*Sqrt[b]*x^5*Sqrt[1 + (a*x^4)/b]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqr
t[b]]*x], -1] + 12*a^(3/2)*Sqrt[b]*x^5*Sqrt[1 + (a*x^4)/b]*EllipticF[I*ArcSinh[S
qrt[(I*Sqrt[a])/Sqrt[b]]*x], -1]))/(5*Sqrt[(I*Sqrt[a])/Sqrt[b]]*x^3*(b + a*x^4))

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Maple [C]  time = 0.023, size = 228, normalized size = 0.9 \[ -{\frac{x}{5\, \left ( a{x}^{4}+b \right ) ^{2}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( -12\,i{a}^{{\frac{3}{2}}}\sqrt{b}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{5}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +12\,i{a}^{{\frac{3}{2}}}\sqrt{b}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{5}{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +7\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{8}{a}^{2}+8\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{4}ab+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{b}^{2} \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.68254, size = 42, normalized size = 0.17 \[ - \frac{a^{\frac{3}{2}} x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**4)**(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(3/2), x)