3.1260 \(\int \frac{1}{x^{12} \left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{40 b^{7/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} \left (a-b x^4\right )^{3/4}}-\frac{20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}-\frac{10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac{\sqrt [4]{a-b x^4}}{11 a x^{11}} \]

[Out]

-(a - b*x^4)^(1/4)/(11*a*x^11) - (10*b*(a - b*x^4)^(1/4))/(77*a^2*x^7) - (20*b^2
*(a - b*x^4)^(1/4))/(77*a^3*x^3) - (40*b^(7/2)*(1 - a/(b*x^4))^(3/4)*x^3*Ellipti
cF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(77*a^(7/2)*(a - b*x^4)^(3/4))

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Rubi [A]  time = 0.17492, antiderivative size = 136, 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 \[ -\frac{40 b^{7/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} \left (a-b x^4\right )^{3/4}}-\frac{20 b^2 \sqrt [4]{a-b x^4}}{77 a^3 x^3}-\frac{10 b \sqrt [4]{a-b x^4}}{77 a^2 x^7}-\frac{\sqrt [4]{a-b x^4}}{11 a x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^12*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(11*a*x^11) - (10*b*(a - b*x^4)^(1/4))/(77*a^2*x^7) - (20*b^2
*(a - b*x^4)^(1/4))/(77*a^3*x^3) - (40*b^(7/2)*(1 - a/(b*x^4))^(3/4)*x^3*Ellipti
cF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(77*a^(7/2)*(a - b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 23.2864, size = 121, normalized size = 0.89 \[ - \frac{\sqrt [4]{a - b x^{4}}}{11 a x^{11}} - \frac{10 b \sqrt [4]{a - b x^{4}}}{77 a^{2} x^{7}} - \frac{20 b^{2} \sqrt [4]{a - b x^{4}}}{77 a^{3} x^{3}} - \frac{40 b^{\frac{7}{2}} x^{3} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{77 a^{\frac{7}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**12/(-b*x**4+a)**(3/4),x)

[Out]

-(a - b*x**4)**(1/4)/(11*a*x**11) - 10*b*(a - b*x**4)**(1/4)/(77*a**2*x**7) - 20
*b**2*(a - b*x**4)**(1/4)/(77*a**3*x**3) - 40*b**(7/2)*x**3*(-a/(b*x**4) + 1)**(
3/4)*elliptic_f(asin(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(77*a**(7/2)*(a - b*x**4)**(3
/4))

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Mathematica [C]  time = 0.0670435, size = 95, normalized size = 0.7 \[ \frac{-7 a^3-3 a^2 b x^4+40 b^3 x^{12} \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )-10 a b^2 x^8+20 b^3 x^{12}}{77 a^3 x^{11} \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^12*(a - b*x^4)^(3/4)),x]

[Out]

(-7*a^3 - 3*a^2*b*x^4 - 10*a*b^2*x^8 + 20*b^3*x^12 + 40*b^3*x^12*(1 - (b*x^4)/a)
^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (b*x^4)/a])/(77*a^3*x^11*(a - b*x^4)^(3/
4))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{12}} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^12/(-b*x^4+a)^(3/4),x)

[Out]

int(1/x^12/(-b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^12),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^12), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{12}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^12),x, algorithm="fricas")

[Out]

integral(1/((-b*x^4 + a)^(3/4)*x^12), x)

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Sympy [A]  time = 15.1347, size = 34, normalized size = 0.25 \[ - \frac{i e^{\frac{11 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{14 b^{\frac{3}{4}} x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**12/(-b*x**4+a)**(3/4),x)

[Out]

-I*exp(11*I*pi/4)*hyper((3/4, 7/2), (9/2,), a/(b*x**4))/(14*b**(3/4)*x**14)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x^{12}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^12),x, algorithm="giac")

[Out]

integrate(1/((-b*x^4 + a)^(3/4)*x^12), x)