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

Optimal. Leaf size=158 \[ \frac{3 b^{5/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}-\frac{3 b^{5/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}-\frac{3 b \sqrt{a-b x^4}}{5 a^2 x}-\frac{\sqrt{a-b x^4}}{5 a x^5} \]

[Out]

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

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Rubi [A]  time = 0.265528, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{3 b^{5/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}-\frac{3 b^{5/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 a^{5/4} \sqrt{a-b x^4}}-\frac{3 b \sqrt{a-b x^4}}{5 a^2 x}-\frac{\sqrt{a-b x^4}}{5 a x^5} \]

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

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Rubi in Sympy [A]  time = 44.818, size = 141, normalized size = 0.89 \[ - \frac{\sqrt{a - b x^{4}}}{5 a x^{5}} - \frac{3 b \sqrt{a - b x^{4}}}{5 a^{2} x} - \frac{3 b^{\frac{5}{4}} \sqrt{1 - \frac{b x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{5 a^{\frac{5}{4}} \sqrt{a - b x^{4}}} + \frac{3 b^{\frac{5}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{5 a^{\frac{5}{4}} \sqrt{a - b x^{4}}} \]

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

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Mathematica [C]  time = 0.558353, size = 131, normalized size = 0.83 \[ \frac{\frac{\left (b x^4-a\right ) \left (a+3 b x^4\right )}{x^5}-3 i a b \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{1-\frac{b x^4}{a}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{5 a^2 \sqrt{a-b x^4}} \]

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

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Maple [A]  time = 0.02, size = 126, normalized size = 0.8 \[ -{\frac{1}{5\,a{x}^{5}}\sqrt{-b{x}^{4}+a}}-{\frac{3\,b}{5\,x{a}^{2}}\sqrt{-b{x}^{4}+a}}+{\frac{3}{5}{b}^{{\frac{3}{2}}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^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/a^(3/2)*b^(3/2)/(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)-EllipticE(x*(1/a^(
1/2)*b^(1/2))^(1/2),I))

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

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 = 3.43552, size = 46, normalized size = 0.29 \[ \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

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(2*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)