3.1974 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right ) x^6} \, dx\)

Optimal. Leaf size=124 \[ \frac{a^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{5/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{5/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{5/3}}-\frac{1}{2 b x^2} \]

[Out]

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

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Rubi [A]  time = 0.158557, antiderivative size = 124, 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 \[ \frac{a^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{5/3}}-\frac{a^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{5/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{5/3}}-\frac{1}{2 b x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)*x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 30.8434, size = 117, normalized size = 0.94 \[ - \frac{a^{\frac{2}{3}} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{3 b^{\frac{5}{3}}} + \frac{a^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{6 b^{\frac{5}{3}}} + \frac{\sqrt{3} a^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{3 b^{\frac{5}{3}}} - \frac{1}{2 b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0380149, size = 119, normalized size = 0.96 \[ \frac{a^{2/3} x^2 \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-2 a^{2/3} x^2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt{3} a^{2/3} x^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )-3 b^{2/3}}{6 b^{5/3} x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 99, normalized size = 0.8 \[ -{\frac{1}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{1}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{2\,b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)/x^6,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224277, size = 217, normalized size = 1.75 \[ -\frac{\sqrt{3}{\left (\sqrt{3} x^{2} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a^{2} x^{2} + a b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} + b^{2} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} x^{2} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x - b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) + 6 \, x^{2} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x + \sqrt{3} b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}}{3 \, b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}\right )}}{18 \, b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*x^2*(-a^2/b^2)^(1/3)*log(a^2*x^2 + a*b*x*(-a^2/b^2)^(1/3)
+ b^2*(-a^2/b^2)^(2/3)) - 2*sqrt(3)*x^2*(-a^2/b^2)^(1/3)*log(a*x - b*(-a^2/b^2)
^(1/3)) + 6*x^2*(-a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x + sqrt(3)*b*(-a^2/b^2
)^(1/3))/(b*(-a^2/b^2)^(1/3))) + 3*sqrt(3))/(b*x^2)

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Sympy [A]  time = 1.45458, size = 32, normalized size = 0.26 \[ \operatorname{RootSum}{\left (27 t^{3} b^{5} + a^{2}, \left ( t \mapsto t \log{\left (- \frac{3 t b^{2}}{a} + x \right )} \right )\right )} - \frac{1}{2 b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)/x**6,x)

[Out]

RootSum(27*_t**3*b**5 + a**2, Lambda(_t, _t*log(-3*_t*b**2/a + x))) - 1/(2*b*x**
2)

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GIAC/XCAS [A]  time = 0.244405, size = 155, normalized size = 1.25 \[ \frac{a \left (-\frac{b}{a}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b^{2}} - \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, b^{2}} - \frac{\left (-a^{2} b\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, b^{2}} - \frac{1}{2 \, b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a*(-b/a)^(1/3)*ln(abs(x - (-b/a)^(1/3)))/b^2 - 1/3*sqrt(3)*(-a^2*b)^(1/3)*ar
ctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/b^2 - 1/6*(-a^2*b)^(1/3)*ln(
x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/b^2 - 1/2/(b*x^2)