3.1329 \(\int \frac{x^8}{\left (a+b x^6\right )^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{6 \sqrt{a} b^{3/2}}-\frac{x^3}{6 b \left (a+b x^6\right )} \]

[Out]

-x^3/(6*b*(a + b*x^6)) + ArcTan[(Sqrt[b]*x^3)/Sqrt[a]]/(6*Sqrt[a]*b^(3/2))

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Rubi [A]  time = 0.0636449, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{6 \sqrt{a} b^{3/2}}-\frac{x^3}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^6)^2,x]

[Out]

-x^3/(6*b*(a + b*x^6)) + ArcTan[(Sqrt[b]*x^3)/Sqrt[a]]/(6*Sqrt[a]*b^(3/2))

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Rubi in Sympy [A]  time = 8.93265, size = 39, normalized size = 0.8 \[ - \frac{x^{3}}{6 b \left (a + b x^{6}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{3}}{\sqrt{a}} \right )}}{6 \sqrt{a} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**6+a)**2,x)

[Out]

-x**3/(6*b*(a + b*x**6)) + atan(sqrt(b)*x**3/sqrt(a))/(6*sqrt(a)*b**(3/2))

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Mathematica [A]  time = 0.0473194, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{6 \sqrt{a} b^{3/2}}-\frac{x^3}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x^6)^2,x]

[Out]

-x^3/(6*b*(a + b*x^6)) + ArcTan[(Sqrt[b]*x^3)/Sqrt[a]]/(6*Sqrt[a]*b^(3/2))

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Maple [A]  time = 0.007, size = 40, normalized size = 0.8 \[ -{\frac{{x}^{3}}{6\,b \left ( b{x}^{6}+a \right ) }}+{\frac{1}{6\,b}\arctan \left ({b{x}^{3}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^6+a)^2,x)

[Out]

-1/6*x^3/b/(b*x^6+a)+1/6/b/(a*b)^(1/2)*arctan(x^3*b/(a*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^6 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222615, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{-a b} x^{3} -{\left (b x^{6} + a\right )} \log \left (\frac{2 \, a b x^{3} +{\left (b x^{6} - a\right )} \sqrt{-a b}}{b x^{6} + a}\right )}{12 \,{\left (b^{2} x^{6} + a b\right )} \sqrt{-a b}}, -\frac{\sqrt{a b} x^{3} -{\left (b x^{6} + a\right )} \arctan \left (\frac{\sqrt{a b} x^{3}}{a}\right )}{6 \,{\left (b^{2} x^{6} + a b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^6 + a)^2,x, algorithm="fricas")

[Out]

[-1/12*(2*sqrt(-a*b)*x^3 - (b*x^6 + a)*log((2*a*b*x^3 + (b*x^6 - a)*sqrt(-a*b))/
(b*x^6 + a)))/((b^2*x^6 + a*b)*sqrt(-a*b)), -1/6*(sqrt(a*b)*x^3 - (b*x^6 + a)*ar
ctan(sqrt(a*b)*x^3/a))/((b^2*x^6 + a*b)*sqrt(a*b))]

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Sympy [A]  time = 4.2502, size = 83, normalized size = 1.69 \[ - \frac{x^{3}}{6 a b + 6 b^{2} x^{6}} - \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (- a b \sqrt{- \frac{1}{a b^{3}}} + x^{3} \right )}}{12} + \frac{\sqrt{- \frac{1}{a b^{3}}} \log{\left (a b \sqrt{- \frac{1}{a b^{3}}} + x^{3} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**6+a)**2,x)

[Out]

-x**3/(6*a*b + 6*b**2*x**6) - sqrt(-1/(a*b**3))*log(-a*b*sqrt(-1/(a*b**3)) + x**
3)/12 + sqrt(-1/(a*b**3))*log(a*b*sqrt(-1/(a*b**3)) + x**3)/12

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GIAC/XCAS [A]  time = 0.227198, size = 53, normalized size = 1.08 \[ -\frac{x^{3}}{6 \,{\left (b x^{6} + a\right )} b} + \frac{\arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{6 \, \sqrt{a b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^6 + a)^2,x, algorithm="giac")

[Out]

-1/6*x^3/((b*x^6 + a)*b) + 1/6*arctan(b*x^3/sqrt(a*b))/(sqrt(a*b)*b)