3.1165 \(\int \frac{1}{x^6 \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{12 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{5/2} \sqrt [4]{a+b x^4}}+\frac{6 b}{5 a^2 x \sqrt [4]{a+b x^4}}-\frac{1}{5 a x^5 \sqrt [4]{a+b x^4}} \]

[Out]

-1/(5*a*x^5*(a + b*x^4)^(1/4)) + (6*b)/(5*a^2*x*(a + b*x^4)^(1/4)) - (12*b^(3/2)
*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*a^(5/
2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.144276, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{12 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 a^{5/2} \sqrt [4]{a+b x^4}}+\frac{6 b}{5 a^2 x \sqrt [4]{a+b x^4}}-\frac{1}{5 a x^5 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^4)^(5/4)),x]

[Out]

-1/(5*a*x^5*(a + b*x^4)^(1/4)) + (6*b)/(5*a^2*x*(a + b*x^4)^(1/4)) - (12*b^(3/2)
*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(5*a^(5/
2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{1}{5 a x^{5} \sqrt [4]{a + b x^{4}}} - \frac{6 b x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{5 a^{2} \sqrt [4]{a + b x^{4}}} + \frac{6 b}{5 a^{2} x \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**4+a)**(5/4),x)

[Out]

-1/(5*a*x**5*(a + b*x**4)**(1/4)) - 6*b*x*(a/(b*x**4) + 1)**(1/4)*Integral((a*x*
*2/b + 1)**(-5/4), (x, x**(-2)))/(5*a**2*(a + b*x**4)**(1/4)) + 6*b/(5*a**2*x*(a
+ b*x**4)**(1/4))

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Mathematica [C]  time = 0.0644839, size = 83, normalized size = 0.79 \[ \frac{-a^2-8 b^2 x^8 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+6 a b x^4+12 b^2 x^8}{5 a^3 x^5 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^4)^(5/4)),x]

[Out]

(-a^2 + 6*a*b*x^4 + 12*b^2*x^8 - 8*b^2*x^8*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2
F1[1/4, 3/4, 7/4, -((b*x^4)/a)])/(5*a^3*x^5*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^4+a)^(5/4),x)

[Out]

int(1/x^6/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{10} + a x^{6}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^6),x, algorithm="fricas")

[Out]

integral(1/((b*x^10 + a*x^6)*(b*x^4 + a)^(1/4)), x)

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Sympy [A]  time = 7.03511, size = 44, normalized size = 0.42 \[ \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{5}{4} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**4+a)**(5/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^6),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^6), x)