3.2100 \(\int \frac{x^2}{\left (a+\frac{b}{x^4}\right )^{5/2}} \, dx\)

Optimal. Leaf size=152 \[ \frac{5 b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{8 a^{13/4} \sqrt{a+\frac{b}{x^4}}}+\frac{5 x^3 \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{3 x^3}{4 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^3}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

[Out]

-x^3/(6*a*(a + b/x^4)^(3/2)) - (3*x^3)/(4*a^2*Sqrt[a + b/x^4]) + (5*Sqrt[a + b/x
^4]*x^3)/(4*a^3) + (5*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])/(8*a^(13/4)*Sqr
t[a + b/x^4])

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Rubi [A]  time = 0.23594, antiderivative size = 152, 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 \[ \frac{5 b^{3/4} \sqrt{\frac{a+\frac{b}{x^4}}{\left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right )^2}} \left (\sqrt{a}+\frac{\sqrt{b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{8 a^{13/4} \sqrt{a+\frac{b}{x^4}}}+\frac{5 x^3 \sqrt{a+\frac{b}{x^4}}}{4 a^3}-\frac{3 x^3}{4 a^2 \sqrt{a+\frac{b}{x^4}}}-\frac{x^3}{6 a \left (a+\frac{b}{x^4}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b/x^4)^(5/2),x]

[Out]

-x^3/(6*a*(a + b/x^4)^(3/2)) - (3*x^3)/(4*a^2*Sqrt[a + b/x^4]) + (5*Sqrt[a + b/x
^4]*x^3)/(4*a^3) + (5*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])/(8*a^(13/4)*Sqr
t[a + b/x^4])

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Rubi in Sympy [A]  time = 17.0398, size = 138, normalized size = 0.91 \[ - \frac{x^{3}}{6 a \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}} - \frac{3 x^{3}}{4 a^{2} \sqrt{a + \frac{b}{x^{4}}}} + \frac{5 x^{3} \sqrt{a + \frac{b}{x^{4}}}}{4 a^{3}} + \frac{5 b^{\frac{3}{4}} \sqrt{\frac{a + \frac{b}{x^{4}}}{\left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right )^{2}}} \left (\sqrt{a} + \frac{\sqrt{b}}{x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b}}{\sqrt [4]{a} x} \right )}\middle | \frac{1}{2}\right )}{8 a^{\frac{13}{4}} \sqrt{a + \frac{b}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**3/(6*a*(a + b/x**4)**(3/2)) - 3*x**3/(4*a**2*sqrt(a + b/x**4)) + 5*x**3*sqrt
(a + b/x**4)/(4*a**3) + 5*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)/(8*a**
(13/4)*sqrt(a + b/x**4))

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Mathematica [C]  time = 0.378637, size = 118, normalized size = 0.78 \[ \frac{\frac{4 a^2 x^9+21 a b x^5+15 b^2 x}{a x^4+b}+\frac{15 i b \sqrt{\frac{a x^4}{b}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}}{12 a^3 x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

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

[Out]

((15*b^2*x + 21*a*b*x^5 + 4*a^2*x^9)/(b + a*x^4) + ((15*I)*b*Sqrt[1 + (a*x^4)/b]
*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]*x], -1])/Sqrt[(I*Sqrt[a])/Sqrt[b]
])/(12*a^3*Sqrt[a + b/x^4]*x^2)

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Maple [C]  time = 0.031, size = 304, normalized size = 2. \[{\frac{1}{12\,{a}^{3}{x}^{10}} \left ( 4\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{13}{a}^{3}-15\,\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{8}{a}^{2}b+25\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{9}{a}^{2}b-30\,\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){x}^{4}a{b}^{2}+36\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}{x}^{5}a{b}^{2}-15\,\sqrt{-{\frac{i\sqrt{a}{x}^{2}-\sqrt{b}}{\sqrt{b}}}}\sqrt{{\frac{i\sqrt{a}{x}^{2}+\sqrt{b}}{\sqrt{b}}}}{\it EllipticF} \left ( x\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}},i \right ){b}^{3}+15\,\sqrt{{\frac{i\sqrt{a}}{\sqrt{b}}}}x{b}^{3} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{i\sqrt{a}{\frac{1}{\sqrt{b}}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(a+b/x^4)^(5/2),x)

[Out]

1/12*(4*(I*a^(1/2)/b^(1/2))^(1/2)*x^13*a^3-15*(-(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)*EllipticF(x*(I*a^(1/2)/b^(1/2))^(
1/2),I)*x^8*a^2*b+25*(I*a^(1/2)/b^(1/2))^(1/2)*x^9*a^2*b-30*(-(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)*EllipticF(x*(I*a^(1
/2)/b^(1/2))^(1/2),I)*x^4*a*b^2+36*(I*a^(1/2)/b^(1/2))^(1/2)*x^5*a*b^2-15*(-(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)*Ellip
ticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)*b^3+15*(I*a^(1/2)/b^(1/2))^(1/2)*x*b^3)/a^3/
((a*x^4+b)/x^4)^(5/2)/x^10/(I*a^(1/2)/b^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(a + b/x^4)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{10}}{{\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^10/((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 = 7.39366, size = 42, normalized size = 0.28 \[ - \frac{x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{5}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac{5}{2}} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(a + b/x^4)^(5/2), x)