3.828 \(\int \frac{1}{x^6 \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=261 \[ -\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} \sqrt{a+b x^4}}-\frac{3 b^{3/2} x \sqrt{a+b x^4}}{5 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3 b \sqrt{a+b x^4}}{5 a^2 x}-\frac{\sqrt{a+b x^4}}{5 a x^5} \]

[Out]

-Sqrt[a + b*x^4]/(5*a*x^5) + (3*b*Sqrt[a + b*x^4])/(5*a^2*x) - (3*b^(3/2)*x*Sqrt
[a + b*x^4])/(5*a^2*(Sqrt[a] + Sqrt[b]*x^2)) + (3*b^(5/4)*(Sqrt[a] + Sqrt[b]*x^2
)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(
1/4)], 1/2])/(5*a^(7/4)*Sqrt[a + b*x^4]) - (3*b^(5/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sq
rt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)
], 1/2])/(10*a^(7/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.243216, antiderivative size = 261, 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 \[ -\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{7/4} \sqrt{a+b x^4}}+\frac{3 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{7/4} \sqrt{a+b x^4}}-\frac{3 b^{3/2} x \sqrt{a+b x^4}}{5 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{3 b \sqrt{a+b x^4}}{5 a^2 x}-\frac{\sqrt{a+b x^4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*Sqrt[a + b*x^4]),x]

[Out]

-Sqrt[a + b*x^4]/(5*a*x^5) + (3*b*Sqrt[a + b*x^4])/(5*a^2*x) - (3*b^(3/2)*x*Sqrt
[a + b*x^4])/(5*a^2*(Sqrt[a] + Sqrt[b]*x^2)) + (3*b^(5/4)*(Sqrt[a] + Sqrt[b]*x^2
)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(
1/4)], 1/2])/(5*a^(7/4)*Sqrt[a + b*x^4]) - (3*b^(5/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sq
rt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)
], 1/2])/(10*a^(7/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 29.256, size = 236, normalized size = 0.9 \[ - \frac{\sqrt{a + b x^{4}}}{5 a x^{5}} - \frac{3 b^{\frac{3}{2}} x \sqrt{a + b x^{4}}}{5 a^{2} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{3 b \sqrt{a + b x^{4}}}{5 a^{2} x} + \frac{3 b^{\frac{5}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{7}{4}} \sqrt{a + b x^{4}}} - \frac{3 b^{\frac{5}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{10 a^{\frac{7}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**4+a)**(1/2),x)

[Out]

-sqrt(a + b*x**4)/(5*a*x**5) - 3*b**(3/2)*x*sqrt(a + b*x**4)/(5*a**2*(sqrt(a) +
sqrt(b)*x**2)) + 3*b*sqrt(a + b*x**4)/(5*a**2*x) + 3*b**(5/4)*sqrt((a + b*x**4)/
(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x**2)*elliptic_e(2*atan(b**(1/4)
*x/a**(1/4)), 1/2)/(5*a**(7/4)*sqrt(a + b*x**4)) - 3*b**(5/4)*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)
*x/a**(1/4)), 1/2)/(10*a**(7/4)*sqrt(a + b*x**4))

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Mathematica [C]  time = 0.493914, size = 135, normalized size = 0.52 \[ \frac{\frac{\left (a+b x^4\right ) \left (3 b x^4-a\right )}{x^5}+3 i a b \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{\frac{b x^4}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{5 a^2 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*Sqrt[a + b*x^4]),x]

[Out]

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

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Maple [C]  time = 0.02, size = 133, normalized size = 0.5 \[ -{\frac{1}{5\,a{x}^{5}}\sqrt{b{x}^{4}+a}}+{\frac{3\,b}{5\,x{a}^{2}}\sqrt{b{x}^{4}+a}}-{{\frac{3\,i}{5}}{b}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^4+a)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a)*x^6),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{4} + a} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a)*x^6),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x^4 + a)*x^6), x)

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Sympy [A]  time = 3.33616, size = 44, normalized size = 0.17 \[ \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**4+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{4} + a} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^4 + a)*x^6),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^6), x)