3.1089 \(\int \frac{x}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=74 \[ \frac{x^2}{\sqrt [4]{a+b x^4}}-\frac{\sqrt{a} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a+b x^4}} \]

[Out]

x^2/(a + b*x^4)^(1/4) - (Sqrt[a]*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]
*x^2)/Sqrt[a]]/2, 2])/(Sqrt[b]*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.0851433, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{x^2}{\sqrt [4]{a+b x^4}}-\frac{\sqrt{a} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^4)^(1/4),x]

[Out]

x^2/(a + b*x^4)^(1/4) - (Sqrt[a]*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]
*x^2)/Sqrt[a]]/2, 2])/(Sqrt[b]*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x**2)**(-1/4), (x, x**2))/2

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Mathematica [C]  time = 0.0303462, size = 52, normalized size = 0.7 \[ \frac{x^2 \sqrt [4]{\frac{a+b x^4}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )}{2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int{x{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^4+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^4 + a)^(1/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/(b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 2.20951, size = 27, normalized size = 0.36 \[ \frac{x^{2}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^4 + a)^(1/4), x)