3.1231 \(\int \frac{1}{x^{14} \sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=134 \[ -\frac{8 b^{7/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{7/2} \sqrt [4]{a-b x^4}}-\frac{4 b^2 \left (a-b x^4\right )^{3/4}}{39 a^3 x^5}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}} \]

[Out]

-(a - b*x^4)^(3/4)/(13*a*x^13) - (10*b*(a - b*x^4)^(3/4))/(117*a^2*x^9) - (4*b^2
*(a - b*x^4)^(3/4))/(39*a^3*x^5) - (8*b^(7/2)*(1 - a/(b*x^4))^(1/4)*x*EllipticE[
ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(39*a^(7/2)*(a - b*x^4)^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.187178, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{8 b^{7/2} x \sqrt [4]{1-\frac{a}{b x^4}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{7/2} \sqrt [4]{a-b x^4}}-\frac{4 b^2 \left (a-b x^4\right )^{3/4}}{39 a^3 x^5}-\frac{10 b \left (a-b x^4\right )^{3/4}}{117 a^2 x^9}-\frac{\left (a-b x^4\right )^{3/4}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^14*(a - b*x^4)^(1/4)),x]

[Out]

-(a - b*x^4)^(3/4)/(13*a*x^13) - (10*b*(a - b*x^4)^(3/4))/(117*a^2*x^9) - (4*b^2
*(a - b*x^4)^(3/4))/(39*a^3*x^5) - (8*b^(7/2)*(1 - a/(b*x^4))^(1/4)*x*EllipticE[
ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(39*a^(7/2)*(a - b*x^4)^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.0158, size = 119, normalized size = 0.89 \[ - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{13 a x^{13}} - \frac{10 b \left (a - b x^{4}\right )^{\frac{3}{4}}}{117 a^{2} x^{9}} - \frac{4 b^{2} \left (a - b x^{4}\right )^{\frac{3}{4}}}{39 a^{3} x^{5}} - \frac{8 b^{\frac{7}{2}} x \sqrt [4]{- \frac{a}{b x^{4}} + 1} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{39 a^{\frac{7}{2}} \sqrt [4]{a - b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**14/(-b*x**4+a)**(1/4),x)

[Out]

-(a - b*x**4)**(3/4)/(13*a*x**13) - 10*b*(a - b*x**4)**(3/4)/(117*a**2*x**9) - 4
*b**2*(a - b*x**4)**(3/4)/(39*a**3*x**5) - 8*b**(7/2)*x*(-a/(b*x**4) + 1)**(1/4)
*elliptic_e(asin(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(39*a**(7/2)*(a - b*x**4)**(1/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0777162, size = 106, normalized size = 0.79 \[ \frac{-9 a^4-a^3 b x^4-2 a^2 b^2 x^8-16 b^4 x^{16} \sqrt [4]{1-\frac{b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )-12 a b^3 x^{12}+24 b^4 x^{16}}{117 a^4 x^{13} \sqrt [4]{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^14*(a - b*x^4)^(1/4)),x]

[Out]

(-9*a^4 - a^3*b*x^4 - 2*a^2*b^2*x^8 - 12*a*b^3*x^12 + 24*b^4*x^16 - 16*b^4*x^16*
(1 - (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (b*x^4)/a])/(117*a^4*x^13
*(a - b*x^4)^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{14}}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^14/(-b*x^4+a)^(1/4),x)

[Out]

int(1/x^14/(-b*x^4+a)^(1/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^14),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^14), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{14}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^14),x, algorithm="fricas")

[Out]

integral(1/((-b*x^4 + a)^(1/4)*x^14), x)

_______________________________________________________________________________________

Sympy [A]  time = 15.8023, size = 34, normalized size = 0.25 \[ - \frac{i e^{\frac{13 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{14 \sqrt [4]{b} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**14/(-b*x**4+a)**(1/4),x)

[Out]

-I*exp(13*I*pi/4)*hyper((1/4, 7/2), (9/2,), a/(b*x**4))/(14*b**(1/4)*x**14)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(1/4)*x^14),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(1/4)*x^14), x)