3.1105 \(\int \frac{1}{x^6 \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{2 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^4}}+\frac{2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 a x^5} \]

[Out]

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

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Rubi [A]  time = 0.142548, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{2 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^4}}+\frac{2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^4)^(1/4)),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**4+a)**(1/4),x)

[Out]

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

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Mathematica [C]  time = 0.0565743, size = 83, normalized size = 0.79 \[ \frac{-3 a^2-4 b^2 x^8 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+3 a b x^4+6 b^2 x^8}{15 a^2 x^5 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^4)^(1/4)),x]

[Out]

(-3*a^2 + 3*a*b*x^4 + 6*b^2*x^8 - 4*b^2*x^8*(1 + (b*x^4)/a)^(1/4)*Hypergeometric
2F1[1/4, 3/4, 7/4, -((b*x^4)/a)])/(15*a^2*x^5*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^4+a)^(1/4),x)

[Out]

int(1/x^6/(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^6),x, algorithm="fricas")

[Out]

integral(1/((b*x^4 + a)^(1/4)*x^6), x)

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Sympy [A]  time = 3.85368, size = 29, normalized size = 0.28 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{6 \sqrt [4]{b} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**4+a)**(1/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^6),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^6), x)