3.1136 \(\int \frac{1}{x^{12} \left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=131 \[ \frac{40 b^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac{20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}+\frac{10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac{\sqrt [4]{a+b x^4}}{11 a x^{11}} \]

[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[ArcCot[(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 = 0.162316, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{40 b^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac{20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}+\frac{10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac{\sqrt [4]{a+b x^4}}{11 a x^{11}} \]

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[ArcCot[(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 = 18.8474, size = 119, normalized size = 0.91 \[ - \frac{\sqrt [4]{a + b x^{4}}}{11 a x^{11}} + \frac{10 b \sqrt [4]{a + b x^{4}}}{77 a^{2} x^{7}} - \frac{20 b^{2} \sqrt [4]{a + b x^{4}}}{77 a^{3} x^{3}} + \frac{40 b^{\frac{7}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{77 a^{\frac{7}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

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(atan(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 = 0.063467, size = 94, normalized size = 0.72 \[ \frac{-7 a^3+3 a^2 b x^4-40 b^3 x^{12} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-10 a b^2 x^8-20 b^3 x^{12}}{77 a^3 x^{11} \left (a+b x^4\right )^{3/4}} \]

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 = 0.048, 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 = 14.7161, size = 44, normalized size = 0.34 \[ \frac{\Gamma \left (- \frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{11}{4}, \frac{3}{4} \\ - \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} x^{11} \Gamma \left (- \frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-11/4)*hyper((-11/4, 3/4), (-7/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/4)*
x**11*gamma(-7/4))

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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)