3.1135 \(\int \frac{1}{x^8 \left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=107 \[ -\frac{4 b^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{7 a^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{2 b \sqrt [4]{a+b x^4}}{7 a^2 x^3}-\frac{\sqrt [4]{a+b x^4}}{7 a x^7} \]

[Out]

-(a + b*x^4)^(1/4)/(7*a*x^7) + (2*b*(a + b*x^4)^(1/4))/(7*a^2*x^3) - (4*b^(5/2)*
(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(7*a^(5
/2)*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.130348, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{4 b^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{7 a^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{2 b \sqrt [4]{a+b x^4}}{7 a^2 x^3}-\frac{\sqrt [4]{a+b x^4}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(a + b*x^4)^(3/4)),x]

[Out]

-(a + b*x^4)^(1/4)/(7*a*x^7) + (2*b*(a + b*x^4)^(1/4))/(7*a^2*x^3) - (4*b^(5/2)*
(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(7*a^(5
/2)*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 15.1045, size = 95, normalized size = 0.89 \[ - \frac{\sqrt [4]{a + b x^{4}}}{7 a x^{7}} + \frac{2 b \sqrt [4]{a + b x^{4}}}{7 a^{2} x^{3}} - \frac{4 b^{\frac{5}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{7 a^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(b*x**4+a)**(3/4),x)

[Out]

-(a + b*x**4)**(1/4)/(7*a*x**7) + 2*b*(a + b*x**4)**(1/4)/(7*a**2*x**3) - 4*b**(
5/2)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2, 2)/
(7*a**(5/2)*(a + b*x**4)**(3/4))

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Mathematica [C]  time = 0.0557465, size = 82, normalized size = 0.77 \[ \frac{-a^2+4 b^2 x^8 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+a b x^4+2 b^2 x^8}{7 a^2 x^7 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(a + b*x^4)^(3/4)),x]

[Out]

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

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{8}} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(b*x^4+a)^(3/4),x)

[Out]

int(1/x^8/(b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^8),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(3/4)*x^8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^8),x, algorithm="fricas")

[Out]

integral(1/((b*x^4 + a)^(3/4)*x^8), x)

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Sympy [A]  time = 7.00327, size = 44, normalized size = 0.41 \[ \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{3}{4} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} x^{7} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(b*x**4+a)**(3/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^8),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/4)*x^8), x)