3.1147 \(\int \frac{x^5}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=77 \[ \frac{x^2}{b \sqrt [4]{a+b x^4}}-\frac{2 \sqrt{a} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.111006, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x^2}{b \sqrt [4]{a+b x^4}}-\frac{2 \sqrt{a} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^4)^(5/4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0510082, size = 54, normalized size = 0.7 \[ \frac{x^2 \left (\sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-1\right )}{b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{5} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^4+a)^(5/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^5/(b*x^4 + a)^(5/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^5/(b*x^4 + a)^(5/4), x)

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Sympy [A]  time = 2.63554, size = 27, normalized size = 0.35 \[ \frac{x^{6}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^5/(b*x^4 + a)^(5/4), x)