3.1970 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right ) x^2} \, dx\)

Optimal. Leaf size=115 \[ \frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]  time = 0.131077, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{2/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.1888, size = 109, normalized size = 0.95 \[ - \frac{\log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b}} + \frac{\log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{6 a^{\frac{2}{3}} \sqrt [3]{b}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0222158, size = 89, normalized size = 0.77 \[ \frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{6 a^{2/3} \sqrt [3]{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 91, normalized size = 0.8 \[ -{\frac{1}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{1}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)/x^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23409, size = 134, normalized size = 1.17 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} a x^{2} - a b + \left (-a^{2} b\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3} \log \left (a b + \left (-a^{2} b\right )^{\frac{2}{3}} x\right ) + 6 \, \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} x}{3 \, a b}\right )\right )}}{18 \, \left (-a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.338662, size = 24, normalized size = 0.21 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b + 1, \left ( t \mapsto t \log{\left (9 t^{2} a b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)/x**2,x)

[Out]

RootSum(27*_t**3*a**2*b + 1, Lambda(_t, _t*log(9*_t**2*a*b + x)))

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GIAC/XCAS [A]  time = 0.242769, size = 151, normalized size = 1.31 \[ -\frac{\left (-\frac{b}{a}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b} - \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b} + \frac{\left (-a^{2} b\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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