3.1328 \(\int \frac{1}{x^4 \left (a+b x^6\right )} \, dx\)

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

[Out]

-1/(3*a*x^3) - (Sqrt[b]*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]])/(3*a^(3/2))

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Rubi [A]  time = 0.0540656, 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 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^6)),x]

[Out]

-1/(3*a*x^3) - (Sqrt[b]*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]])/(3*a^(3/2))

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Rubi in Sympy [A]  time = 8.8729, size = 36, normalized size = 0.9 \[ - \frac{1}{3 a x^{3}} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} x^{3}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*a*x**3) - sqrt(b)*atan(sqrt(b)*x**3/sqrt(a))/(3*a**(3/2))

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Mathematica [B]  time = 0.0453729, size = 101, normalized size = 2.52 \[ \frac{\sqrt{b} x^3 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+\sqrt{b} x^3 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-\sqrt{b} x^3 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-\sqrt{a}}{3 a^{3/2} x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-Sqrt[a] + Sqrt[b]*x^3*ArcTan[(b^(1/6)*x)/a^(1/6)] + Sqrt[b]*x^3*ArcTan[Sqrt[3]
- (2*b^(1/6)*x)/a^(1/6)] - Sqrt[b]*x^3*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])
/(3*a^(3/2)*x^3)

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \[ -{\frac{b}{3\,a}\arctan \left ({b{x}^{3}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{3\,a{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^6+a),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22561, size = 1, normalized size = 0.02 \[ \left [\frac{x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{6} - 2 \, a x^{3} \sqrt{-\frac{b}{a}} - a}{b x^{6} + a}\right ) - 2}{6 \, a x^{3}}, -\frac{x^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x^{3}}{a \sqrt{\frac{b}{a}}}\right ) + 1}{3 \, a x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.0193, size = 71, normalized size = 1.78 \[ \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x^{3} \right )}}{6} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x^{3} \right )}}{6} - \frac{1}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**6+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220301, size = 42, normalized size = 1.05 \[ -\frac{b \arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} a} - \frac{1}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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