3.1133 \(\int \frac{1}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=61 \[ -\frac{\sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \left (a+b x^4\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0727929, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{\sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(-3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.59202, size = 54, normalized size = 0.89 \[ - \frac{\sqrt{b} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{\sqrt{a} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0205531, size = 47, normalized size = 0.77 \[ \frac{x \left (\frac{a+b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{\left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(-3/4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(3/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(-3/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(-3/4), x)

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Sympy [A]  time = 2.25052, size = 36, normalized size = 0.59 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*gamma(1/4)*hyper((1/4, 3/4), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/4)*gam
ma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(-3/4), x)