3.2076 \(\int \left (a+\frac{b}{x^4}\right )^{5/2} x^2 \, dx\)

Optimal. Leaf size=146 \[ -\frac{20 a^{7/4} b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 \sqrt{a+\frac{b}{x^4}}}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{21 x}-\frac{20 a b \sqrt{a+\frac{b}{x^4}}}{21 x}+\frac{1}{3} x^3 \left (a+\frac{b}{x^4}\right )^{5/2} \]

[Out]

(-20*a*b*Sqrt[a + b/x^4])/(21*x) - (10*b*(a + b/x^4)^(3/2))/(21*x) + ((a + b/x^4
)^(5/2)*x^3)/3 - (20*a^(7/4)*b^(3/4)*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])/(21*Sqrt
[a + b/x^4])

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Rubi [A]  time = 0.198572, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{20 a^{7/4} b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 \sqrt{a+\frac{b}{x^4}}}-\frac{10 b \left (a+\frac{b}{x^4}\right )^{3/2}}{21 x}-\frac{20 a b \sqrt{a+\frac{b}{x^4}}}{21 x}+\frac{1}{3} x^3 \left (a+\frac{b}{x^4}\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(5/2)*x^2,x]

[Out]

(-20*a*b*Sqrt[a + b/x^4])/(21*x) - (10*b*(a + b/x^4)^(3/2))/(21*x) + ((a + b/x^4
)^(5/2)*x^3)/3 - (20*a^(7/4)*b^(3/4)*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])/(21*Sqrt
[a + b/x^4])

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Rubi in Sympy [A]  time = 12.6947, size = 131, normalized size = 0.9 \[ - \frac{20 a^{\frac{7}{4}} b^{\frac{3}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{21 \sqrt{a + \frac{b}{x^{4}}}} - \frac{20 a b \sqrt{a + \frac{b}{x^{4}}}}{21 x} - \frac{10 b \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{21 x} + \frac{x^{3} \left (a + \frac{b}{x^{4}}\right )^{\frac{5}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20*a**(7/4)*b**(3/4)*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)/(21*sqrt(a + b/x**4
)) - 20*a*b*sqrt(a + b/x**4)/(21*x) - 10*b*(a + b/x**4)**(3/2)/(21*x) + x**3*(a
+ b/x**4)**(5/2)/3

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Mathematica [C]  time = 0.238476, size = 149, normalized size = 1.02 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (7 a^3 x^{12}-9 a^2 b x^8-19 a b^2 x^4-3 b^3\right )-40 i a^2 b x^7 \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )\right )}{21 x^5 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a x^4+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^4]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(-3*b^3 - 19*a*b^2*x^4 - 9*a^2*b*x^8
+ 7*a^3*x^12) - (40*I)*a^2*b*x^7*Sqrt[1 + (a*x^4)/b]*EllipticF[I*ArcSinh[Sqrt[(
I*Sqrt[a])/Sqrt[b]]*x], -1]))/(21*Sqrt[(I*Sqrt[a])/Sqrt[b]]*x^5*(b + a*x^4))

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Maple [C]  time = 0.027, size = 181, normalized size = 1.2 \[{\frac{{x}^{3}}{21\, \left ( a{x}^{4}+b \right ) ^{3}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 7\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{12}{a}^{3}+40\,{a}^{2}b\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{7}-9\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{8}{a}^{2}b-19\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{4}a{b}^{2}-3\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{b}^{3} \right ){\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(5/2)*x^2,x)

[Out]

1/21*((a*x^4+b)/x^4)^(5/2)*x^3*(7*(I*a^(1/2)/b^(1/2))^(1/2)*x^12*a^3+40*a^2*b*(-
(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)*E
llipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*x^7-9*(I*a^(1/2)/b^(1/2))^(1/2)*x^8*a^2*
b-19*(I*a^(1/2)/b^(1/2))^(1/2)*x^4*a*b^2-3*(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. \[ \int{\left (a + \frac{b}{x^{4}}\right )}^{\frac{5}{2}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(5/2)*x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^4)^(5/2)*x^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^6, x)

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Sympy [A]  time = 15.6653, size = 44, normalized size = 0.3 \[ - \frac{a^{\frac{5}{2}} x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a + b/x^4)^(5/2)*x^2, x)