3.1246 \(\int \frac{1}{x^{11} \left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=133 \[ \frac{3 b^{5/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 a^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}-\frac{3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac{\sqrt [4]{a-b x^4}}{10 a x^{10}} \]

[Out]

-(a - b*x^4)^(1/4)/(10*a*x^10) - (3*b*(a - b*x^4)^(1/4))/(20*a^2*x^6) - (3*b^2*(
a - b*x^4)^(1/4))/(8*a^3*x^2) + (3*b^(5/2)*(1 - (b*x^4)/a)^(3/4)*EllipticF[ArcSi
n[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(8*a^(5/2)*(a - b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.204491, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{3 b^{5/2} \left (1-\frac{b x^4}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 a^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{3 b^2 \sqrt [4]{a-b x^4}}{8 a^3 x^2}-\frac{3 b \sqrt [4]{a-b x^4}}{20 a^2 x^6}-\frac{\sqrt [4]{a-b x^4}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^11*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(10*a*x^10) - (3*b*(a - b*x^4)^(1/4))/(20*a^2*x^6) - (3*b^2*(
a - b*x^4)^(1/4))/(8*a^3*x^2) + (3*b^(5/2)*(1 - (b*x^4)/a)^(3/4)*EllipticF[ArcSi
n[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(8*a^(5/2)*(a - b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.0554, size = 116, normalized size = 0.87 \[ - \frac{\sqrt [4]{a - b x^{4}}}{10 a x^{10}} - \frac{3 b \sqrt [4]{a - b x^{4}}}{20 a^{2} x^{6}} - \frac{3 b^{2} \sqrt [4]{a - b x^{4}}}{8 a^{3} x^{2}} + \frac{3 b^{\frac{5}{2}} \left (1 - \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{8 a^{\frac{5}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**11/(-b*x**4+a)**(3/4),x)

[Out]

-(a - b*x**4)**(1/4)/(10*a*x**10) - 3*b*(a - b*x**4)**(1/4)/(20*a**2*x**6) - 3*b
**2*(a - b*x**4)**(1/4)/(8*a**3*x**2) + 3*b**(5/2)*(1 - b*x**4/a)**(3/4)*ellipti
c_f(asin(sqrt(b)*x**2/sqrt(a))/2, 2)/(8*a**(5/2)*(a - b*x**4)**(3/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0695509, size = 95, normalized size = 0.71 \[ \frac{-8 a^3-4 a^2 b x^4+15 b^3 x^{12} \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^4}{a}\right )-18 a b^2 x^8+30 b^3 x^{12}}{80 a^3 x^{10} \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^11*(a - b*x^4)^(3/4)),x]

[Out]

(-8*a^3 - 4*a^2*b*x^4 - 18*a*b^2*x^8 + 30*b^3*x^12 + 15*b^3*x^12*(1 - (b*x^4)/a)
^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, (b*x^4)/a])/(80*a^3*x^10*(a - b*x^4)^(3/
4))

_______________________________________________________________________________________

Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{11}} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^11/(-b*x^4+a)^(3/4),x)

[Out]

int(1/x^11/(-b*x^4+a)^(3/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^11),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^11), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{11}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^11),x, algorithm="fricas")

[Out]

integral(1/((-b*x^4 + a)^(3/4)*x^11), x)

_______________________________________________________________________________________

Sympy [A]  time = 12.5245, size = 34, normalized size = 0.26 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{10 a^{\frac{3}{4}} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**11/(-b*x**4+a)**(3/4),x)

[Out]

-hyper((-5/2, 3/4), (-3/2,), b*x**4*exp_polar(2*I*pi)/a)/(10*a**(3/4)*x**10)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^11),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^11), x)