3.1341 \(\int \frac{1}{x^4 \left (a+b x^6\right )^2} \, dx\)

Optimal. Leaf size=59 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{1}{2 a^2 x^3}+\frac{1}{6 a x^3 \left (a+b x^6\right )} \]

[Out]

-1/(2*a^2*x^3) + 1/(6*a*x^3*(a + b*x^6)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]
])/(2*a^(5/2))

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Rubi [A]  time = 0.0776455, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{1}{2 a^2 x^3}+\frac{1}{6 a x^3 \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^6)^2),x]

[Out]

-1/(2*a^2*x^3) + 1/(6*a*x^3*(a + b*x^6)) - (Sqrt[b]*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]
])/(2*a^(5/2))

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Rubi in Sympy [A]  time = 12.5341, size = 51, normalized size = 0.86 \[ \frac{1}{6 a x^{3} \left (a + b x^{6}\right )} - \frac{1}{2 a^{2} x^{3}} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} x^{3}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**6+a)**2,x)

[Out]

1/(6*a*x**3*(a + b*x**6)) - 1/(2*a**2*x**3) - sqrt(b)*atan(sqrt(b)*x**3/sqrt(a))
/(2*a**(5/2))

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Mathematica [A]  time = 0.157234, size = 114, normalized size = 1.93 \[ \frac{-\frac{\sqrt{a} b x^3}{a+b x^6}+3 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+3 \sqrt{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-3 \sqrt{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-\frac{2 \sqrt{a}}{x^3}}{6 a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^6)^2),x]

[Out]

((-2*Sqrt[a])/x^3 - (Sqrt[a]*b*x^3)/(a + b*x^6) + 3*Sqrt[b]*ArcTan[(b^(1/6)*x)/a
^(1/6)] + 3*Sqrt[b]*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] - 3*Sqrt[b]*ArcTan[S
qrt[3] + (2*b^(1/6)*x)/a^(1/6)])/(6*a^(5/2))

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Maple [A]  time = 0.017, size = 50, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{3}{a}^{2}}}-{\frac{b{x}^{3}}{6\,{a}^{2} \left ( b{x}^{6}+a \right ) }}-{\frac{b}{2\,{a}^{2}}\arctan \left ({b{x}^{3}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^6+a)^2,x)

[Out]

-1/3/x^3/a^2-1/6*b/a^2*x^3/(b*x^6+a)-1/2*b/a^2/(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(1/((b*x^6 + a)^2*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230666, size = 1, normalized size = 0.02 \[ \left [-\frac{6 \, b x^{6} - 3 \,{\left (b x^{9} + a x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{6} - 2 \, a x^{3} \sqrt{-\frac{b}{a}} - a}{b x^{6} + a}\right ) + 4 \, a}{12 \,{\left (a^{2} b x^{9} + a^{3} x^{3}\right )}}, -\frac{3 \, b x^{6} + 3 \,{\left (b x^{9} + a x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x^{3}}{a \sqrt{\frac{b}{a}}}\right ) + 2 \, a}{6 \,{\left (a^{2} b x^{9} + a^{3} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*x^4),x, algorithm="fricas")

[Out]

[-1/12*(6*b*x^6 - 3*(b*x^9 + a*x^3)*sqrt(-b/a)*log((b*x^6 - 2*a*x^3*sqrt(-b/a) -
a)/(b*x^6 + a)) + 4*a)/(a^2*b*x^9 + a^3*x^3), -1/6*(3*b*x^6 + 3*(b*x^9 + a*x^3)
*sqrt(b/a)*arctan(b*x^3/(a*sqrt(b/a))) + 2*a)/(a^2*b*x^9 + a^3*x^3)]

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Sympy [A]  time = 41.0366, size = 92, normalized size = 1.56 \[ \frac{\sqrt{- \frac{b}{a^{5}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b}{a^{5}}}}{b} + x^{3} \right )}}{4} - \frac{\sqrt{- \frac{b}{a^{5}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b}{a^{5}}}}{b} + x^{3} \right )}}{4} - \frac{2 a + 3 b x^{6}}{6 a^{3} x^{3} + 6 a^{2} b x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**6+a)**2,x)

[Out]

sqrt(-b/a**5)*log(-a**3*sqrt(-b/a**5)/b + x**3)/4 - sqrt(-b/a**5)*log(a**3*sqrt(
-b/a**5)/b + x**3)/4 - (2*a + 3*b*x**6)/(6*a**3*x**3 + 6*a**2*b*x**9)

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GIAC/XCAS [A]  time = 0.226304, size = 69, normalized size = 1.17 \[ -\frac{b \arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} - \frac{3 \, b x^{6} + 2 \, a}{6 \,{\left (b x^{9} + a x^{3}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*x^4),x, algorithm="giac")

[Out]

-1/2*b*arctan(b*x^3/sqrt(a*b))/(sqrt(a*b)*a^2) - 1/6*(3*b*x^6 + 2*a)/((b*x^9 + a
*x^3)*a^2)