3.843 \(\int \frac{1}{x^8 \sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=102 \[ \frac{5 b^{7/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 a^{7/4} \sqrt{a-b x^4}}-\frac{5 b \sqrt{a-b x^4}}{21 a^2 x^3}-\frac{\sqrt{a-b x^4}}{7 a x^7} \]

[Out]

-Sqrt[a - b*x^4]/(7*a*x^7) - (5*b*Sqrt[a - b*x^4])/(21*a^2*x^3) + (5*b^(7/4)*Sqr
t[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(21*a^(7/4)*Sqrt[a
- b*x^4])

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Rubi [A]  time = 0.0914435, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{5 b^{7/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 a^{7/4} \sqrt{a-b x^4}}-\frac{5 b \sqrt{a-b x^4}}{21 a^2 x^3}-\frac{\sqrt{a-b x^4}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*Sqrt[a - b*x^4]),x]

[Out]

-Sqrt[a - b*x^4]/(7*a*x^7) - (5*b*Sqrt[a - b*x^4])/(21*a^2*x^3) + (5*b^(7/4)*Sqr
t[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(21*a^(7/4)*Sqrt[a
- b*x^4])

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Rubi in Sympy [A]  time = 12.5357, size = 90, normalized size = 0.88 \[ - \frac{\sqrt{a - b x^{4}}}{7 a x^{7}} - \frac{5 b \sqrt{a - b x^{4}}}{21 a^{2} x^{3}} + \frac{5 b^{\frac{7}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{21 a^{\frac{7}{4}} \sqrt{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(-b*x**4+a)**(1/2),x)

[Out]

-sqrt(a - b*x**4)/(7*a*x**7) - 5*b*sqrt(a - b*x**4)/(21*a**2*x**3) + 5*b**(7/4)*
sqrt(1 - b*x**4/a)*elliptic_f(asin(b**(1/4)*x/a**(1/4)), -1)/(21*a**(7/4)*sqrt(a
- b*x**4))

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Mathematica [C]  time = 0.248317, size = 104, normalized size = 1.02 \[ \frac{-\frac{3 a^2}{x^7}-\frac{5 i b^2 \sqrt{1-\frac{b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}}}-\frac{2 a b}{x^3}+5 b^2 x}{21 a^2 \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*Sqrt[a - b*x^4]),x]

[Out]

((-3*a^2)/x^7 - (2*a*b)/x^3 + 5*b^2*x - ((5*I)*b^2*Sqrt[1 - (b*x^4)/a]*EllipticF
[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]*x], -1])/Sqrt[-(Sqrt[b]/Sqrt[a])])/(21*a^2*S
qrt[a - b*x^4])

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Maple [A]  time = 0.022, size = 109, normalized size = 1.1 \[ -{\frac{1}{7\,a{x}^{7}}\sqrt{-b{x}^{4}+a}}-{\frac{5\,b}{21\,{x}^{3}{a}^{2}}\sqrt{-b{x}^{4}+a}}+{\frac{5\,{b}^{2}}{21\,{a}^{2}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(-b*x^4+a)^(1/2),x)

[Out]

-1/7*(-b*x^4+a)^(1/2)/a/x^7-5/21*b*(-b*x^4+a)^(1/2)/x^3/a^2+5/21/a^2*b^2/(1/a^(1
/2)*b^(1/2))^(1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(
-b*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b x^{4} + a} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^8),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^8),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-b*x^4 + a)*x^8), x)

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Sympy [A]  time = 4.57097, size = 46, normalized size = 0.45 \[ \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} x^{7} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(-b*x**4+a)**(1/2),x)

[Out]

gamma(-7/4)*hyper((-7/4, 1/2), (-3/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*x
**7*gamma(-3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b x^{4} + a} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x^4 + a)*x^8),x, algorithm="giac")

[Out]

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