3.1119 \(\int \frac{x}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=57 \[ \frac{\sqrt{a} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \left (a+b x^4\right )^{3/4}} \]

[Out]

(Sqrt[a]*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(S
qrt[b]*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.0681445, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{a} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^4)^(3/4),x]

[Out]

(Sqrt[a]*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(S
qrt[b]*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 6.70818, size = 49, normalized size = 0.86 \[ \frac{\sqrt{a} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{\sqrt{b} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0285281, size = 52, normalized size = 0.91 \[ \frac{x^2 \left (\frac{a+b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )}{2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{x \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^4+a)^(3/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^4 + a)^(3/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/(b*x^4 + a)^(3/4), x)

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Sympy [A]  time = 2.25065, size = 27, normalized size = 0.47 \[ \frac{x^{2}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*hyper((1/2, 3/4), (3/2,), b*x**4*exp_polar(I*pi)/a)/(2*a**(3/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^4 + a)^(3/4), x)