3.1148 \(\int \frac{x}{\left (a+b x^4\right )^{5/4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0667427, 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 \[ \frac{\sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^4)^(5/4),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(b*x**4+a)**(5/4),x)

[Out]

Integral((a + b*x**2)**(-5/4), (x, x**2))/2

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Mathematica [C]  time = 0.0404977, size = 57, normalized size = 1. \[ -\frac{x^2 \left (\sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-2\right )}{2 a \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^4)^(5/4),x]

[Out]

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

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{x \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^4+a)^(5/4),x)

[Out]

int(x/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^4 + a)^(5/4),x, algorithm="maxima")

[Out]

integrate(x/(b*x^4 + a)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

integral(x/(b*x^4 + a)^(5/4), x)

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Sympy [A]  time = 2.61752, size = 27, normalized size = 0.47 \[ \frac{x^{2}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**4+a)**(5/4),x)

[Out]

x**2*hyper((1/2, 5/4), (3/2,), b*x**4*exp_polar(I*pi)/a)/(2*a**(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

integrate(x/(b*x^4 + a)^(5/4), x)