3.1316 \(\int \frac{x^8}{a+b x^6} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^3}{3 b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0543462, antiderivative size = 40, 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 \[ \frac{x^3}{3 b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.38121, size = 32, normalized size = 0.8 \[ - \frac{\sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} x^{3}}{\sqrt{a}} \right )}}{3 b^{\frac{3}{2}}} + \frac{x^{3}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0270648, size = 40, normalized size = 1. \[ \frac{x^3}{3 b}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x^6),x]

[Out]

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

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \[{\frac{{x}^{3}}{3\,b}}-{\frac{a}{3\,b}\arctan \left ({b{x}^{3}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^6+a),x)

[Out]

1/3*x^3/b-1/3*a/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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21969, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, x^{3} + \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{6} - 2 \, b x^{3} \sqrt{-\frac{a}{b}} - a}{b x^{6} + a}\right )}{6 \, b}, \frac{x^{3} - \sqrt{\frac{a}{b}} \arctan \left (\frac{x^{3}}{\sqrt{\frac{a}{b}}}\right )}{3 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.49669, size = 63, normalized size = 1.58 \[ \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (- b \sqrt{- \frac{a}{b^{3}}} + x^{3} \right )}}{6} - \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (b \sqrt{- \frac{a}{b^{3}}} + x^{3} \right )}}{6} + \frac{x^{3}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**6+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223972, size = 42, normalized size = 1.05 \[ \frac{x^{3}}{3 \, b} - \frac{a \arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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