3.1215 \(\int \frac{1}{x^7 \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{b^{3/2} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a-b x^4}}-\frac{b \left (a-b x^4\right )^{3/4}}{4 a^2 x^2}-\frac{\left (a-b x^4\right )^{3/4}}{6 a x^6} \]

[Out]

-(a - b*x^4)^(3/4)/(6*a*x^6) - (b*(a - b*x^4)^(3/4))/(4*a^2*x^2) - (b^(3/2)*(1 -
(b*x^4)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*a^(3/2)*(a -
b*x^4)^(1/4))

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Rubi [A]  time = 0.15419, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b^{3/2} \sqrt [4]{1-\frac{b x^4}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a-b x^4}}-\frac{b \left (a-b x^4\right )^{3/4}}{4 a^2 x^2}-\frac{\left (a-b x^4\right )^{3/4}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(6*a*x^6) - (b*(a - b*x^4)^(3/4))/(4*a^2*x^2) - (b^(3/2)*(1 -
(b*x^4)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*a^(3/2)*(a -
b*x^4)^(1/4))

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Rubi in Sympy [A]  time = 17.4422, size = 90, normalized size = 0.83 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{6 a x^{6}} - \frac{b \left (a - b x^{4}\right )^{\frac{3}{4}}}{4 a^{2} x^{2}} - \frac{b^{\frac{3}{2}} \sqrt [4]{1 - \frac{b x^{4}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{4 a^{\frac{3}{2}} \sqrt [4]{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(-b*x**4+a)**(1/4),x)

[Out]

-(a - b*x**4)**(3/4)/(6*a*x**6) - b*(a - b*x**4)**(3/4)/(4*a**2*x**2) - b**(3/2)
*(1 - b*x**4/a)**(1/4)*elliptic_e(asin(sqrt(b)*x**2/sqrt(a))/2, 2)/(4*a**(3/2)*(
a - b*x**4)**(1/4))

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Mathematica [C]  time = 0.06457, size = 84, normalized size = 0.78 \[ \frac{-4 a^2-3 b^2 x^8 \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^4}{a}\right )-2 a b x^4+6 b^2 x^8}{24 a^2 x^6 \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(a - b*x^4)^(1/4)),x]

[Out]

(-4*a^2 - 2*a*b*x^4 + 6*b^2*x^8 - 3*b^2*x^8*(1 - (b*x^4)/a)^(1/4)*Hypergeometric
2F1[1/4, 1/2, 3/2, (b*x^4)/a])/(24*a^2*x^6*(a - b*x^4)^(1/4))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(-b*x^4+a)^(1/4),x)

[Out]

int(1/x^7/(-b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^7),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^7), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^7),x, algorithm="fricas")

[Out]

integral(1/((-b*x^4 + a)^(1/4)*x^7), x)

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Sympy [A]  time = 4.66196, size = 34, normalized size = 0.31 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{6 \sqrt [4]{a} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(-b*x**4+a)**(1/4),x)

[Out]

-hyper((-3/2, 1/4), (-1/2,), b*x**4*exp_polar(2*I*pi)/a)/(6*a**(1/4)*x**6)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^7),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^7), x)