3.1134 \(\int \frac{1}{x^4 \left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=85 \[ \frac{2 b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.100221, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{2 b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.7028, size = 73, normalized size = 0.86 \[ - \frac{\sqrt [4]{a + b x^{4}}}{3 a x^{3}} + \frac{2 b^{\frac{3}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{3 a^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0454827, size = 70, normalized size = 0.82 \[ \frac{-2 b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-a-b x^4}{3 a x^3 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^4+a)^(3/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.47346, size = 41, normalized size = 0.48 \[ \frac{\Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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