3.1980 \(\int \frac{x}{\left (a+\frac{b}{x^3}\right )^2} \, dx\)

Optimal. Leaf size=146 \[ -\frac{5 b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{8/3}}+\frac{5 b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{8/3}}+\frac{5 x^2}{6 a^2}-\frac{x^5}{3 a \left (a x^3+b\right )} \]

[Out]

(5*x^2)/(6*a^2) - x^5/(3*a*(b + a*x^3)) + (5*b^(2/3)*ArcTan[(b^(1/3) - 2*a^(1/3)
*x)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*a^(8/3)) + (5*b^(2/3)*Log[b^(1/3) + a^(1/3)*x
])/(9*a^(8/3)) - (5*b^(2/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(18*
a^(8/3))

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Rubi [A]  time = 0.195709, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818 \[ -\frac{5 b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{8/3}}+\frac{5 b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{8/3}}+\frac{5 x^2}{6 a^2}-\frac{x^5}{3 a \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^3)^2,x]

[Out]

(5*x^2)/(6*a^2) - x^5/(3*a*(b + a*x^3)) + (5*b^(2/3)*ArcTan[(b^(1/3) - 2*a^(1/3)
*x)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*a^(8/3)) + (5*b^(2/3)*Log[b^(1/3) + a^(1/3)*x
])/(9*a^(8/3)) - (5*b^(2/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(18*
a^(8/3))

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Rubi in Sympy [A]  time = 35.2461, size = 138, normalized size = 0.95 \[ - \frac{x^{5}}{3 a \left (a x^{3} + b\right )} + \frac{5 x^{2}}{6 a^{2}} + \frac{5 b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{9 a^{\frac{8}{3}}} - \frac{5 b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{18 a^{\frac{8}{3}}} + \frac{5 \sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{9 a^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**5/(3*a*(a*x**3 + b)) + 5*x**2/(6*a**2) + 5*b**(2/3)*log(a**(1/3)*x + b**(1/3
))/(9*a**(8/3)) - 5*b**(2/3)*log(a**(2/3)*x**2 - a**(1/3)*b**(1/3)*x + b**(2/3))
/(18*a**(8/3)) + 5*sqrt(3)*b**(2/3)*atan(sqrt(3)*(-2*a**(1/3)*x/3 + b**(1/3)/3)/
b**(1/3))/(9*a**(8/3))

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Mathematica [A]  time = 0.150089, size = 131, normalized size = 0.9 \[ \frac{-5 b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+\frac{6 a^{2/3} b x^2}{a x^3+b}+9 a^{2/3} x^2+10 b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+10 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{18 a^{8/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x^3)^2,x]

[Out]

(9*a^(2/3)*x^2 + (6*a^(2/3)*b*x^2)/(b + a*x^3) + 10*Sqrt[3]*b^(2/3)*ArcTan[(1 -
(2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] + 10*b^(2/3)*Log[b^(1/3) + a^(1/3)*x] - 5*b^(2/3
)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(18*a^(8/3))

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Maple [A]  time = 0.012, size = 120, normalized size = 0.8 \[{\frac{{x}^{2}}{2\,{a}^{2}}}+{\frac{b{x}^{2}}{3\,{a}^{2} \left ( a{x}^{3}+b \right ) }}+{\frac{5\,b}{9\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{5\,b}{18\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{5\,b\sqrt{3}}{9\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^3)^2,x)

[Out]

1/2*x^2/a^2+1/3*b/a^2*x^2/(a*x^3+b)+5/9*b/a^3/(b/a)^(1/3)*ln(x+(b/a)^(1/3))-5/18
*b/a^3/(b/a)^(1/3)*ln(x^2-x*(b/a)^(1/3)+(b/a)^(2/3))-5/9*b/a^3*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(x/(a + b/x^3)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233698, size = 248, normalized size = 1.7 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (a x^{3} + b\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} + b \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3}{\left (a x^{3} + b\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 30 \,{\left (a x^{3} + b\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (3 \, a x^{5} + 5 \, b x^{2}\right )}\right )}}{54 \,{\left (a^{3} x^{3} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(5*sqrt(3)*(a*x^3 + b)*(b^2/a^2)^(1/3)*log(b*x^2 - a*x*(b^2/a^2)^(
2/3) + b*(b^2/a^2)^(1/3)) - 10*sqrt(3)*(a*x^3 + b)*(b^2/a^2)^(1/3)*log(b*x + a*(
b^2/a^2)^(2/3)) - 30*(a*x^3 + b)*(b^2/a^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - sq
rt(3)*a*(b^2/a^2)^(2/3))/(a*(b^2/a^2)^(2/3))) - 3*sqrt(3)*(3*a*x^5 + 5*b*x^2))/(
a^3*x^3 + a^2*b)

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Sympy [A]  time = 1.82811, size = 58, normalized size = 0.4 \[ \frac{b x^{2}}{3 a^{3} x^{3} + 3 a^{2} b} + \operatorname{RootSum}{\left (729 t^{3} a^{8} - 125 b^{2}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{5}}{25 b} + x \right )} \right )\right )} + \frac{x^{2}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x**3)**2,x)

[Out]

b*x**2/(3*a**3*x**3 + 3*a**2*b) + RootSum(729*_t**3*a**8 - 125*b**2, Lambda(_t,
_t*log(81*_t**2*a**5/(25*b) + x))) + x**2/(2*a**2)

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GIAC/XCAS [A]  time = 0.229015, size = 178, normalized size = 1.22 \[ \frac{x^{2}}{2 \, a^{2}} + \frac{b x^{2}}{3 \,{\left (a x^{3} + b\right )} a^{2}} + \frac{5 \, \left (-\frac{b}{a}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{5 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4}} - \frac{5 \, \left (-a^{2} b\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{18 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2/a^2 + 1/3*b*x^2/((a*x^3 + b)*a^2) + 5/9*(-b/a)^(2/3)*ln(abs(x - (-b/a)^(
1/3)))/a^2 + 5/9*sqrt(3)*(-a^2*b)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/
(-b/a)^(1/3))/a^4 - 5/18*(-a^2*b)^(2/3)*ln(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/
a^4