3.1230 \(\int \frac{1}{x^{10} \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{4 b^{5/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 a^{5/2} \sqrt [4]{a-b x^4}}-\frac{2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}-\frac{\left (a-b x^4\right )^{3/4}}{9 a x^9} \]

[Out]

-(a - b*x^4)^(3/4)/(9*a*x^9) - (2*b*(a - b*x^4)^(3/4))/(15*a^2*x^5) - (4*b^(5/2)
*(1 - a/(b*x^4))^(1/4)*x*EllipticE[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(15*a^(5
/2)*(a - b*x^4)^(1/4))

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Rubi [A]  time = 0.154601, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{4 b^{5/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 a^{5/2} \sqrt [4]{a-b x^4}}-\frac{2 b \left (a-b x^4\right )^{3/4}}{15 a^2 x^5}-\frac{\left (a-b x^4\right )^{3/4}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(9*a*x^9) - (2*b*(a - b*x^4)^(3/4))/(15*a^2*x^5) - (4*b^(5/2)
*(1 - a/(b*x^4))^(1/4)*x*EllipticE[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(15*a^(5
/2)*(a - b*x^4)^(1/4))

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Rubi in Sympy [A]  time = 19.957, size = 95, normalized size = 0.87 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{9 a x^{9}} - \frac{2 b \left (a - b x^{4}\right )^{\frac{3}{4}}}{15 a^{2} x^{5}} - \frac{4 b^{\frac{5}{2}} x \sqrt [4]{- \frac{a}{b x^{4}} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{15 a^{\frac{5}{2}} \sqrt [4]{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(-b*x**4+a)**(1/4),x)

[Out]

-(a - b*x**4)**(3/4)/(9*a*x**9) - 2*b*(a - b*x**4)**(3/4)/(15*a**2*x**5) - 4*b**
(5/2)*x*(-a/(b*x**4) + 1)**(1/4)*elliptic_e(asin(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(
15*a**(5/2)*(a - b*x**4)**(1/4))

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Mathematica [C]  time = 0.0680857, size = 95, normalized size = 0.87 \[ \frac{-5 a^3-a^2 b x^4-8 b^3 x^{12} \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )-6 a b^2 x^8+12 b^3 x^{12}}{45 a^3 x^9 \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*(a - b*x^4)^(1/4)),x]

[Out]

(-5*a^3 - a^2*b*x^4 - 6*a*b^2*x^8 + 12*b^3*x^12 - 8*b^3*x^12*(1 - (b*x^4)/a)^(1/
4)*Hypergeometric2F1[1/4, 3/4, 7/4, (b*x^4)/a])/(45*a^3*x^9*(a - b*x^4)^(1/4))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(-b*x^4+a)^(1/4),x)

[Out]

int(1/x^10/(-b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^10),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^10), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{10}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^10),x, algorithm="fricas")

[Out]

integral(1/((-b*x^4 + a)^(1/4)*x^10), x)

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Sympy [A]  time = 7.67815, size = 32, normalized size = 0.29 \[ \frac{i e^{\frac{9 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{10 \sqrt [4]{b} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**10/(-b*x**4+a)**(1/4),x)

[Out]

I*exp(9*I*pi/4)*hyper((1/4, 5/2), (7/2,), a/(b*x**4))/(10*b**(1/4)*x**10)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^10),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^10), x)