3.1327 \(\int \frac{1}{x^3 \left (a+b x^6\right )} \, dx\)

Optimal. Leaf size=133 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{4/3}}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} a^{4/3}}-\frac{1}{2 a x^2} \]

[Out]

-1/(2*a*x^2) + (b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))])/(2*
Sqrt[3]*a^(4/3)) + (b^(1/3)*Log[a^(1/3) + b^(1/3)*x^2])/(6*a^(4/3)) - (b^(1/3)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4])/(12*a^(4/3))

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Rubi [A]  time = 0.22321, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{4/3}}+\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}+\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} a^{4/3}}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^6)),x]

[Out]

-1/(2*a*x^2) + (b^(1/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x^2)/(Sqrt[3]*a^(1/3))])/(2*
Sqrt[3]*a^(4/3)) + (b^(1/3)*Log[a^(1/3) + b^(1/3)*x^2])/(6*a^(4/3)) - (b^(1/3)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4])/(12*a^(4/3))

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Rubi in Sympy [A]  time = 34.0088, size = 122, normalized size = 0.92 \[ - \frac{1}{2 a x^{2}} + \frac{\sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x^{2} \right )}}{6 a^{\frac{4}{3}}} - \frac{\sqrt [3]{b} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x^{2} + b^{\frac{2}{3}} x^{4} \right )}}{12 a^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x^{2}}{3}\right )}{\sqrt [3]{a}} \right )}}{6 a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a*x**2) + b**(1/3)*log(a**(1/3) + b**(1/3)*x**2)/(6*a**(4/3)) - b**(1/3)*l
og(a**(2/3) - a**(1/3)*b**(1/3)*x**2 + b**(2/3)*x**4)/(12*a**(4/3)) + sqrt(3)*b*
*(1/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x**2/3)/a**(1/3))/(6*a**(4/3))

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Mathematica [A]  time = 0.0640942, size = 203, normalized size = 1.53 \[ \frac{2 \sqrt [3]{b} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt [3]{b} x^2 \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt [3]{b} x^2 \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+2 \sqrt{3} \sqrt [3]{b} x^2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt{3} \sqrt [3]{b} x^2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-6 \sqrt [3]{a}}{12 a^{4/3} x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-6*a^(1/3) + 2*Sqrt[3]*b^(1/3)*x^2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*
Sqrt[3]*b^(1/3)*x^2*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] + 2*b^(1/3)*x^2*Log[
a^(1/3) + b^(1/3)*x^2] - b^(1/3)*x^2*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b
^(1/3)*x^2] - b^(1/3)*x^2*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]
)/(12*a^(4/3)*x^2)

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Maple [A]  time = 0.007, size = 105, normalized size = 0.8 \[{\frac{1}{6\,a}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{12\,a}\ln \left ({x}^{4}-{x}^{2}\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}}{6\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{2\,a{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^6+a),x)

[Out]

1/6/a/(a/b)^(1/3)*ln(x^2+(a/b)^(1/3))-1/12/a/(a/b)^(1/3)*ln(x^4-x^2*(a/b)^(1/3)+
(a/b)^(2/3))-1/6/a*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^2-1))
-1/2/a/x^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)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221367, size = 188, normalized size = 1.41 \[ -\frac{\sqrt{3}{\left (\sqrt{3} x^{2} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{4} - a x^{2} \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} x^{2} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 6 \, x^{2} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x^{2} - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 6 \, \sqrt{3}\right )}}{36 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*sqrt(3)*(sqrt(3)*x^2*(b/a)^(1/3)*log(b*x^4 - a*x^2*(b/a)^(2/3) + a*(b/a)^(
1/3)) - 2*sqrt(3)*x^2*(b/a)^(1/3)*log(b*x^2 + a*(b/a)^(2/3)) - 6*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))) + 6*sqrt
(3))/(a*x^2)

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Sympy [A]  time = 1.72441, size = 34, normalized size = 0.26 \[ \operatorname{RootSum}{\left (216 t^{3} a^{4} - b, \left ( t \mapsto t \log{\left (\frac{36 t^{2} a^{3}}{b} + x^{2} \right )} \right )\right )} - \frac{1}{2 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**6+a),x)

[Out]

RootSum(216*_t**3*a**4 - b, Lambda(_t, _t*log(36*_t**2*a**3/b + x**2))) - 1/(2*a
*x**2)

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GIAC/XCAS [A]  time = 0.225764, size = 171, normalized size = 1.29 \[ \frac{b \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x^{2} - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{6 \, a^{2}} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{6 \, a^{2} b} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{4} + x^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{12 \, a^{2} b} - \frac{1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b*(-a/b)^(2/3)*ln(abs(x^2 - (-a/b)^(1/3)))/a^2 + 1/6*sqrt(3)*(-a*b^2)^(2/3)*
arctan(1/3*sqrt(3)*(2*x^2 + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b) - 1/12*(-a*b^2)^
(2/3)*ln(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b) - 1/2/(a*x^2)