3.2077 \(\int \left (a+\frac{b}{x^4}\right )^{5/2} \, dx\)

Optimal. Leaf size=272 \[ -\frac{4 a^{9/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}+\frac{8 a^{9/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}-\frac{8 a^2 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{3 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}+x \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{9 x^3}-\frac{4 a b \sqrt{a+\frac{b}{x^4}}}{3 x^3} \]

[Out]

(-4*a*b*Sqrt[a + b/x^4])/(3*x^3) - (10*b*(a + b/x^4)^(3/2))/(9*x^3) - (8*a^2*Sqr
t[b]*Sqrt[a + b/x^4])/(3*(Sqrt[a] + Sqrt[b]/x^2)*x) + (a + b/x^4)^(5/2)*x + (8*a
^(9/4)*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])/(3*Sqrt[a + b/x^4]) - (4*a^(9/
4)*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*E
llipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(3*Sqrt[a + b/x^4])

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Rubi [A]  time = 0.447423, antiderivative size = 272, 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 \[ -\frac{4 a^{9/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}+\frac{8 a^{9/4} \sqrt [4]{b} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{3 \sqrt{a+\frac{b}{x^4}}}-\frac{8 a^2 \sqrt{b} \sqrt{a+\frac{b}{x^4}}}{3 x \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )}+x \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{9 x^3}-\frac{4 a b \sqrt{a+\frac{b}{x^4}}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(5/2),x]

[Out]

(-4*a*b*Sqrt[a + b/x^4])/(3*x^3) - (10*b*(a + b/x^4)^(3/2))/(9*x^3) - (8*a^2*Sqr
t[b]*Sqrt[a + b/x^4])/(3*(Sqrt[a] + Sqrt[b]/x^2)*x) + (a + b/x^4)^(5/2)*x + (8*a
^(9/4)*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])/(3*Sqrt[a + b/x^4]) - (4*a^(9/
4)*b^(1/4)*Sqrt[(a + b/x^4)/(Sqrt[a] + Sqrt[b]/x^2)^2]*(Sqrt[a] + Sqrt[b]/x^2)*E
llipticF[2*ArcCot[(a^(1/4)*x)/b^(1/4)], 1/2])/(3*Sqrt[a + b/x^4])

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Rubi in Sympy [A]  time = 33.9662, size = 250, normalized size = 0.92 \[ \frac{8 a^{\frac{9}{4}} \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{a + \frac{b}{x^{4}}}} - \frac{4 a^{\frac{9}{4}} \sqrt [4]{b} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{a + \frac{b}{x^{4}}}} - \frac{8 a^{2} \sqrt{b} \sqrt{a + \frac{b}{x^{4}}}}{3 x \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )} - \frac{4 a b \sqrt{a + \frac{b}{x^{4}}}}{3 x^{3}} - \frac{10 b \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{9 x^{3}} + x \left (a + \frac{b}{x^{4}}\right )^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*a**(9/4)*b**(1/4)*sqrt((a + b/x**4)/(sqrt(a) + sqrt(b)/x**2)**2)*(sqrt(a) + sq
rt(b)/x**2)*elliptic_e(2*atan(b**(1/4)/(a**(1/4)*x)), 1/2)/(3*sqrt(a + b/x**4))
- 4*a**(9/4)*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)/(3*sqrt(a + b/x**4)
) - 8*a**2*sqrt(b)*sqrt(a + b/x**4)/(3*x*(sqrt(a) + sqrt(b)/x**2)) - 4*a*b*sqrt(
a + b/x**4)/(3*x**3) - 10*b*(a + b/x**4)**(3/2)/(9*x**3) + x*(a + b/x**4)**(5/2)

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Mathematica [C]  time = 0.333811, size = 207, normalized size = 0.76 \[ -\frac{\sqrt{a+\frac{b}{x^4}} \left (24 a^{5/2} \sqrt{b} x^9 \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )-24 a^{5/2} \sqrt{b} x^9 \sqrt{\frac{a x^4}{b}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (15 a^3 x^{12}+19 a^2 b x^8+5 a b^2 x^4+b^3\right )\right )}{9 x^7 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^4)^(5/2),x]

[Out]

-(Sqrt[a + b/x^4]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(b^3 + 5*a*b^2*x^4 + 19*a^2*b*x^8 +
15*a^3*x^12) - 24*a^(5/2)*Sqrt[b]*x^9*Sqrt[1 + (a*x^4)/b]*EllipticE[I*ArcSinh[S
qrt[(I*Sqrt[a])/Sqrt[b]]*x], -1] + 24*a^(5/2)*Sqrt[b]*x^9*Sqrt[1 + (a*x^4)/b]*El
lipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1]))/(9*Sqrt[(I*Sqrt[a])/Sqrt[b
]]*x^7*(b + a*x^4))

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Maple [C]  time = 0.028, size = 250, normalized size = 0.9 \[ -{\frac{x}{9\, \left ( a{x}^{4}+b \right ) ^{3}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 24\,i{a}^{{\frac{5}{2}}}\sqrt{b}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{9}{\it EllipticE} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) -24\,i{a}^{{\frac{5}{2}}}\sqrt{b}\sqrt{-{1 \left ( i\sqrt{a}{x}^{2}-\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}\sqrt{{1 \left ( i\sqrt{a}{x}^{2}+\sqrt{b} \right ){\frac{1}{\sqrt{b}}}}}{x}^{9}{\it EllipticF} \left ( x\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}},i \right ) +15\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{12}{a}^{3}+19\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{8}{a}^{2}b+5\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{4}a{b}^{2}+\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}{b}^{3} \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(5/2),x)

[Out]

-1/9*((a*x^4+b)/x^4)^(5/2)*x*(24*I*a^(5/2)*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^9*EllipticE(x*(I*a^(1/2)/b
^(1/2))^(1/2),I)-24*I*a^(5/2)*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^9*EllipticF(x*(I*a^(1/2)/b^(1/2))^(1/2)
,I)+15*(I*a^(1/2)/b^(1/2))^(1/2)*x^12*a^3+19*(I*a^(1/2)/b^(1/2))^(1/2)*x^8*a^2*b
+5*(I*a^(1/2)/b^(1/2))^(1/2)*x^4*a*b^2+(I*a^(1/2)/b^(1/2))^(1/2)*b^3)/(a*x^4+b)^
3/(I*a^(1/2)/b^(1/2))^(1/2)

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

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. \[{\rm integral}\left (\frac{{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.8476, size = 42, normalized size = 0.15 \[ - \frac{a^{\frac{5}{2}} x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**4)**(5/2),x)

[Out]

-a**(5/2)*x*gamma(-1/4)*hyper((-5/2, -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. \[ \int{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}}\,{d x} \]

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)