3.1983 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^2 x^2} \, dx\)

Optimal. Leaf size=136 \[ \frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}-\frac{x^2}{3 a \left (a x^3+b\right )} \]

[Out]

-x^2/(3*a*(b + a*x^3)) - (2*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(
3*Sqrt[3]*a^(5/3)*b^(1/3)) - (2*Log[b^(1/3) + a^(1/3)*x])/(9*a^(5/3)*b^(1/3)) +
Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/(9*a^(5/3)*b^(1/3))

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Rubi [A]  time = 0.166996, antiderivative size = 136, 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 \[ \frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b}}-\frac{x^2}{3 a \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)^2*x^2),x]

[Out]

-x^2/(3*a*(b + a*x^3)) - (2*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(
3*Sqrt[3]*a^(5/3)*b^(1/3)) - (2*Log[b^(1/3) + a^(1/3)*x])/(9*a^(5/3)*b^(1/3)) +
Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/(9*a^(5/3)*b^(1/3))

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Rubi in Sympy [A]  time = 30.2787, size = 126, normalized size = 0.93 \[ - \frac{x^{2}}{3 a \left (a x^{3} + b\right )} - \frac{2 \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} + \frac{\log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2/(3*a*(a*x**3 + b)) - 2*log(a**(1/3)*x + b**(1/3))/(9*a**(5/3)*b**(1/3)) +
log(a**(2/3)*x**2 - a**(1/3)*b**(1/3)*x + b**(2/3))/(9*a**(5/3)*b**(1/3)) - 2*sq
rt(3)*atan(sqrt(3)*(-2*a**(1/3)*x/3 + b**(1/3)/3)/b**(1/3))/(9*a**(5/3)*b**(1/3)
)

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Mathematica [A]  time = 0.152733, size = 119, normalized size = 0.88 \[ \frac{\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{\sqrt [3]{b}}-\frac{3 a^{2/3} x^2}{a x^3+b}-\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{\sqrt [3]{b}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{9 a^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-3*a^(2/3)*x^2)/(b + a*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sq
rt[3]])/b^(1/3) - (2*Log[b^(1/3) + a^(1/3)*x])/b^(1/3) + Log[b^(2/3) - a^(1/3)*b
^(1/3)*x + a^(2/3)*x^2]/b^(1/3))/(9*a^(5/3))

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Maple [A]  time = 0.01, size = 108, normalized size = 0.8 \[ -{\frac{{x}^{2}}{3\,a \left ( a{x}^{3}+b \right ) }}-{\frac{2}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{1}{9\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{2\,\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)^2/x^2,x)

[Out]

-1/3*x^2/a/(a*x^3+b)-2/9/a^2/(b/a)^(1/3)*ln(x+(b/a)^(1/3))+1/9/a^2/(b/a)^(1/3)*l
n(x^2-x*(b/a)^(1/3)+(b/a)^(2/3))+2/9/a^2*3^(1/2)/(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)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236721, size = 201, normalized size = 1.48 \[ -\frac{\sqrt{3}{\left (3 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x^{2} + \sqrt{3}{\left (a x^{3} + b\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} a x^{2} - a b + \left (-a^{2} b\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left (a x^{3} + b\right )} \log \left (a b + \left (-a^{2} b\right )^{\frac{2}{3}} x\right ) + 6 \,{\left (a x^{3} + b\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} x}{3 \, a b}\right )\right )}}{27 \,{\left (a^{2} x^{3} + a b\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*sqrt(3)*(3*sqrt(3)*(-a^2*b)^(1/3)*x^2 + sqrt(3)*(a*x^3 + b)*log((-a^2*b)^(
1/3)*a*x^2 - a*b + (-a^2*b)^(2/3)*x) - 2*sqrt(3)*(a*x^3 + b)*log(a*b + (-a^2*b)^
(2/3)*x) + 6*(a*x^3 + b)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a^2*b)^(2/3)*x)/
(a*b)))/((a^2*x^3 + a*b)*(-a^2*b)^(1/3))

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Sympy [A]  time = 1.58137, size = 44, normalized size = 0.32 \[ - \frac{x^{2}}{3 a^{2} x^{3} + 3 a b} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b + 8, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{3} b}{4} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.233271, size = 178, normalized size = 1.31 \[ -\frac{x^{2}}{3 \,{\left (a x^{3} + b\right )} a} - \frac{2 \, \left (-\frac{b}{a}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b} - \frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b} + \frac{\left (-a^{2} b\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{9 \, a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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