3.1088 \(\int \frac{x^5}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=104 \[ \frac{2 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 b^{3/2} \sqrt [4]{a+b x^4}}+\frac{x^2 \left (a+b x^4\right )^{3/4}}{5 b}-\frac{2 a x^2}{5 b \sqrt [4]{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.130058, antiderivative size = 104, 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 \[ \frac{2 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 b^{3/2} \sqrt [4]{a+b x^4}}+\frac{x^2 \left (a+b x^4\right )^{3/4}}{5 b}-\frac{2 a x^2}{5 b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^4)^(1/4),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{5 b} - \frac{2 a x^{2}}{5 b \sqrt [4]{a + b x^{4}}} + \frac{x^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**4+a)**(1/4),x)

[Out]

a**2*Integral((a + b*x**2)**(-5/4), (x, x**2))/(5*b) - 2*a*x**2/(5*b*(a + b*x**4
)**(1/4)) + x**2*(a + b*x**4)**(3/4)/(5*b)

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Mathematica [C]  time = 0.0533053, size = 64, normalized size = 0.62 \[ \frac{x^2 \left (-a \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+a+b x^4\right )}{5 b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^4)^(1/4),x]

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{{x}^{5}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^4+a)^(1/4),x)

[Out]

int(x^5/(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^4 + a)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^5/(b*x^4 + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^4 + a)^(1/4),x, algorithm="fricas")

[Out]

integral(x^5/(b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 2.67598, size = 27, normalized size = 0.26 \[ \frac{x^{6}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 \sqrt [4]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**4+a)**(1/4),x)

[Out]

x**6*hyper((1/4, 3/2), (5/2,), b*x**4*exp_polar(I*pi)/a)/(6*a**(1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^4 + a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^5/(b*x^4 + a)^(1/4), x)