3.1106 \(\int \frac{1}{x^{10} \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=129 \[ \frac{4 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 a^{5/2} \sqrt [4]{a+b x^4}}-\frac{4 b^2}{15 a^2 x \sqrt [4]{a+b x^4}}+\frac{2 b \left (a+b x^4\right )^{3/4}}{15 a^2 x^5}-\frac{\left (a+b x^4\right )^{3/4}}{9 a x^9} \]

[Out]

(-4*b^2)/(15*a^2*x*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(9*a*x^9) + (2*b*(a +
b*x^4)^(3/4))/(15*a^2*x^5) + (4*b^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot
[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(15*a^(5/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.176704, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{4 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 a^{5/2} \sqrt [4]{a+b x^4}}-\frac{4 b^2}{15 a^2 x \sqrt [4]{a+b x^4}}+\frac{2 b \left (a+b x^4\right )^{3/4}}{15 a^2 x^5}-\frac{\left (a+b x^4\right )^{3/4}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*(a + b*x^4)^(1/4)),x]

[Out]

(-4*b^2)/(15*a^2*x*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(9*a*x^9) + (2*b*(a +
b*x^4)^(3/4))/(15*a^2*x^5) + (4*b^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot
[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(15*a^(5/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{9 a x^{9}} - \frac{2 b^{2} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{15 a^{2} \sqrt [4]{a + b x^{4}}} + \frac{2 b \left (a + b x^{4}\right )^{\frac{3}{4}}}{15 a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(b*x**4+a)**(1/4),x)

[Out]

-(a + b*x**4)**(3/4)/(9*a*x**9) - 2*b**2*x*(a/(b*x**4) + 1)**(1/4)*Integral((a*x
**2/b + 1)**(-1/4), (x, x**(-2)))/(15*a**2*(a + b*x**4)**(1/4)) + 2*b*(a + b*x**
4)**(3/4)/(15*a**2*x**5)

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Mathematica [C]  time = 0.068613, size = 93, normalized size = 0.72 \[ \frac{-5 a^3+a^2 b x^4+8 b^3 x^{12} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-6 a b^2 x^8-12 b^3 x^{12}}{45 a^3 x^9 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*(a + b*x^4)^(1/4)),x]

[Out]

(-5*a^3 + a^2*b*x^4 - 6*a*b^2*x^8 - 12*b^3*x^12 + 8*b^3*x^12*(1 + (b*x^4)/a)^(1/
4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^4)/a)])/(45*a^3*x^9*(a + b*x^4)^(1/4)
)

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(b*x^4+a)^(1/4),x)

[Out]

int(1/x^10/(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^10),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^10), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{10}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^10),x, algorithm="fricas")

[Out]

integral(1/((b*x^4 + a)^(1/4)*x^10), x)

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Sympy [A]  time = 7.35426, size = 44, normalized size = 0.34 \[ \frac{\Gamma \left (- \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{9}{4}, \frac{1}{4} \\ - \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} x^{9} \Gamma \left (- \frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**10/(b*x**4+a)**(1/4),x)

[Out]

gamma(-9/4)*hyper((-9/4, 1/4), (-5/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(1/4)*x*
*9*gamma(-5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^10),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^10), x)