3.863 \(\int \frac{1}{x^4 \left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=131 \[ -\frac{5 b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt{a+b x^4}}-\frac{5 \sqrt{a+b x^4}}{6 a^2 x^3}+\frac{1}{2 a x^3 \sqrt{a+b x^4}} \]

[Out]

1/(2*a*x^3*Sqrt[a + b*x^4]) - (5*Sqrt[a + b*x^4])/(6*a^2*x^3) - (5*b^(3/4)*(Sqrt
[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTa
n[(b^(1/4)*x)/a^(1/4)], 1/2])/(12*a^(9/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.103376, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 b^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{9/4} \sqrt{a+b x^4}}-\frac{5 \sqrt{a+b x^4}}{6 a^2 x^3}+\frac{1}{2 a x^3 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^4)^(3/2)),x]

[Out]

1/(2*a*x^3*Sqrt[a + b*x^4]) - (5*Sqrt[a + b*x^4])/(6*a^2*x^3) - (5*b^(3/4)*(Sqrt
[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTa
n[(b^(1/4)*x)/a^(1/4)], 1/2])/(12*a^(9/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 11.1125, size = 119, normalized size = 0.91 \[ \frac{1}{2 a x^{3} \sqrt{a + b x^{4}}} - \frac{5 \sqrt{a + b x^{4}}}{6 a^{2} x^{3}} - \frac{5 b^{\frac{3}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{9}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**4+a)**(3/2),x)

[Out]

1/(2*a*x**3*sqrt(a + b*x**4)) - 5*sqrt(a + b*x**4)/(6*a**2*x**3) - 5*b**(3/4)*sq
rt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*elliptic_f
(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(12*a**(9/4)*sqrt(a + b*x**4))

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Mathematica [C]  time = 0.217908, size = 93, normalized size = 0.71 \[ \frac{\frac{5 i b \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}}}-\frac{2 a}{x^3}-5 b x}{6 a^2 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^4)^(3/2)),x]

[Out]

((-2*a)/x^3 - 5*b*x + ((5*I)*b*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*S
qrt[b])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[b])/Sqrt[a]])/(6*a^2*Sqrt[a + b*x^4])

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Maple [C]  time = 0.025, size = 113, normalized size = 0.9 \[ -{\frac{bx}{2\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{\frac{1}{3\,{x}^{3}{a}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{5\,b}{6\,{a}^{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^4+a)^(3/2),x)

[Out]

-1/2*b/a^2*x/((x^4+a/b)*b)^(1/2)-1/3*(b*x^4+a)^(1/2)/x^3/a^2-5/6*b/a^2/(I/a^(1/2
)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)
/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(3/2)*x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{8} + a x^{4}\right )} \sqrt{b x^{4} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^4),x, algorithm="fricas")

[Out]

integral(1/((b*x^8 + a*x^4)*sqrt(b*x^4 + a)), x)

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Sympy [A]  time = 3.46837, size = 41, normalized size = 0.31 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**4+a)**(3/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^4),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/2)*x^4), x)