3.1340 \(\int \frac{1}{x^3 \left (a+b x^6\right )^2} \, dx\)

Optimal. Leaf size=152 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}+\frac{2 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{2}{3 a^2 x^2}+\frac{1}{6 a x^2 \left (a+b x^6\right )} \]

[Out]

-2/(3*a^2*x^2) + 1/(6*a*x^2*(a + b*x^6)) + (2*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3
)*x^2)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)) + (2*b^(1/3)*Log[a^(1/3) + b^(1/3
)*x^2])/(9*a^(7/3)) - (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4])
/(9*a^(7/3))

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Rubi [A]  time = 0.261644, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{9 a^{7/3}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{9 a^{7/3}}+\frac{2 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{2}{3 a^2 x^2}+\frac{1}{6 a x^2 \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^6)^2),x]

[Out]

-2/(3*a^2*x^2) + 1/(6*a*x^2*(a + b*x^6)) + (2*b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3
)*x^2)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)) + (2*b^(1/3)*Log[a^(1/3) + b^(1/3
)*x^2])/(9*a^(7/3)) - (b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4])
/(9*a^(7/3))

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Rubi in Sympy [A]  time = 38.5649, size = 143, normalized size = 0.94 \[ \frac{1}{6 a x^{2} \left (a + b x^{6}\right )} - \frac{2}{3 a^{2} x^{2}} + \frac{2 \sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x^{2} \right )}}{9 a^{\frac{7}{3}}} - \frac{\sqrt [3]{b} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{2} + b^{\frac{2}{3}} x^{4} \right )}}{9 a^{\frac{7}{3}}} + \frac{2 \sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x^{2}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**6+a)**2,x)

[Out]

1/(6*a*x**2*(a + b*x**6)) - 2/(3*a**2*x**2) + 2*b**(1/3)*log(a**(1/3) + b**(1/3)
*x**2)/(9*a**(7/3)) - b**(1/3)*log(a**(2/3) - a**(1/3)*b**(1/3)*x**2 + b**(2/3)*
x**4)/(9*a**(7/3)) + 2*sqrt(3)*b**(1/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x*
*2/3)/a**(1/3))/(9*a**(7/3))

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Mathematica [A]  time = 0.33005, size = 208, normalized size = 1.37 \[ \frac{4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-2 \sqrt [3]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-2 \sqrt [3]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\frac{3 \sqrt [3]{a} b x^4}{a+b x^6}+4 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+4 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-\frac{9 \sqrt [3]{a}}{x^2}}{18 a^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^6)^2),x]

[Out]

((-9*a^(1/3))/x^2 - (3*a^(1/3)*b*x^4)/(a + b*x^6) + 4*Sqrt[3]*b^(1/3)*ArcTan[Sqr
t[3] - (2*b^(1/6)*x)/a^(1/6)] + 4*Sqrt[3]*b^(1/3)*ArcTan[Sqrt[3] + (2*b^(1/6)*x)
/a^(1/6)] + 4*b^(1/3)*Log[a^(1/3) + b^(1/3)*x^2] - 2*b^(1/3)*Log[a^(1/3) - Sqrt[
3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] - 2*b^(1/3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^
(1/6)*x + b^(1/3)*x^2])/(18*a^(7/3))

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Maple [A]  time = 0.019, size = 123, normalized size = 0.8 \[ -{\frac{1}{2\,{a}^{2}{x}^{2}}}-{\frac{b{x}^{4}}{6\,{a}^{2} \left ( b{x}^{6}+a \right ) }}+{\frac{2}{9\,{a}^{2}}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{9\,{a}^{2}}\ln \left ({x}^{4}-{x}^{2}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^6+a)^2,x)

[Out]

-1/2/a^2/x^2-1/6*b/a^2*x^4/(b*x^6+a)+2/9/a^2/(a/b)^(1/3)*ln(x^2+(a/b)^(1/3))-1/9
/a^2/(a/b)^(1/3)*ln(x^4-x^2*(a/b)^(1/3)+(a/b)^(2/3))-2/9/a^2*3^(1/2)/(a/b)^(1/3)
*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^2-1))

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232861, size = 251, normalized size = 1.65 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b x^{8} + a x^{2}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{4} - a x^{2} \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 4 \, \sqrt{3}{\left (b x^{8} + a x^{2}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 12 \,{\left (b x^{8} + a x^{2}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x^{2} - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (4 \, b x^{6} + 3 \, a\right )}\right )}}{54 \,{\left (a^{2} b x^{8} + a^{3} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*x^3),x, algorithm="fricas")

[Out]

-1/54*sqrt(3)*(2*sqrt(3)*(b*x^8 + a*x^2)*(b/a)^(1/3)*log(b*x^4 - a*x^2*(b/a)^(2/
3) + a*(b/a)^(1/3)) - 4*sqrt(3)*(b*x^8 + a*x^2)*(b/a)^(1/3)*log(b*x^2 + a*(b/a)^
(2/3)) - 12*(b*x^8 + a*x^2)*(b/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x^2 - sqrt(3)*a
*(b/a)^(2/3))/(a*(b/a)^(2/3))) + 3*sqrt(3)*(4*b*x^6 + 3*a))/(a^2*b*x^8 + a^3*x^2
)

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Sympy [A]  time = 21.5891, size = 58, normalized size = 0.38 \[ - \frac{3 a + 4 b x^{6}}{6 a^{3} x^{2} + 6 a^{2} b x^{8}} + \operatorname{RootSum}{\left (729 t^{3} a^{7} - 8 b, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{5}}{4 b} + x^{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**6+a)**2,x)

[Out]

-(3*a + 4*b*x**6)/(6*a**3*x**2 + 6*a**2*b*x**8) + RootSum(729*_t**3*a**7 - 8*b,
Lambda(_t, _t*log(81*_t**2*a**5/(4*b) + x**2)))

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GIAC/XCAS [A]  time = 0.228231, size = 198, normalized size = 1.3 \[ \frac{2 \, b \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b} - \frac{4 \, b x^{6} + 3 \, a}{6 \,{\left (b x^{8} + a x^{2}\right )} a^{2}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^6 + a)^2*x^3),x, algorithm="giac")

[Out]

2/9*b*(-a/b)^(2/3)*ln(abs(x^2 - (-a/b)^(1/3)))/a^3 + 2/9*sqrt(3)*(-a*b^2)^(2/3)*
arctan(1/3*sqrt(3)*(2*x^2 + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b) - 1/6*(4*b*x^6 +
3*a)/((b*x^8 + a*x^2)*a^2) - 1/9*(-a*b^2)^(2/3)*ln(x^4 + x^2*(-a/b)^(1/3) + (-a
/b)^(2/3))/(a^3*b)