3.1163 \(\int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0866284, antiderivative size = 59, 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 \[ -\frac{x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^4)^(5/4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**4+a)**(5/4),x)

[Out]

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

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Mathematica [C]  time = 0.0428857, size = 58, normalized size = 0.98 \[ -\frac{x^3 \left (2 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-3\right )}{3 a \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x^4)^(5/4),x]

[Out]

-(x^3*(-3 + 2*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^4)/a
)]))/(3*a*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^4+a)^(5/4),x)

[Out]

int(x^2/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^4 + a)^(5/4),x, algorithm="maxima")

[Out]

integrate(x^2/(b*x^4 + a)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

integral(x^2/(b*x^4 + a)^(5/4), x)

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Sympy [A]  time = 2.64641, size = 37, normalized size = 0.63 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**4+a)**(5/4),x)

[Out]

x**3*gamma(3/4)*hyper((3/4, 5/4), (7/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(5/4)*
gamma(7/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

integrate(x^2/(b*x^4 + a)^(5/4), x)