3.1090 \(\int \frac{1}{x^3 \sqrt [4]{a+b x^4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.128741, 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 \[ \frac{b x^2}{2 a \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{2 a x^2}-\frac{\sqrt{b} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^4)^(1/4)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**4+a)**(1/4),x)

[Out]

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

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Mathematica [C]  time = 0.0491708, size = 69, normalized size = 0.66 \[ \frac{b x^4 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-2 \left (a+b x^4\right )}{4 a x^2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^4)^(1/4)),x]

[Out]

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

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^4+a)^(1/4),x)

[Out]

int(1/x^3/(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^3),x, algorithm="fricas")

[Out]

integral(1/((b*x^4 + a)^(1/4)*x^3), x)

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Sympy [A]  time = 2.68967, size = 31, normalized size = 0.3 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**4+a)**(1/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^3),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^3), x)