3.1981 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^2} \, dx\)

Optimal. Leaf size=144 \[ \frac{2 \sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 a^{7/3}}-\frac{4 \sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{7/3}}+\frac{4 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{7/3}}+\frac{4 x}{3 a^2}-\frac{x^4}{3 a \left (a x^3+b\right )} \]

[Out]

(4*x)/(3*a^2) - x^4/(3*a*(b + a*x^3)) + (4*b^(1/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x
)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*a^(7/3)) - (4*b^(1/3)*Log[b^(1/3) + a^(1/3)*x])
/(9*a^(7/3)) + (2*b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(9*a^(
7/3))

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Rubi [A]  time = 0.185682, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1. \[ \frac{2 \sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 a^{7/3}}-\frac{4 \sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{7/3}}+\frac{4 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{7/3}}+\frac{4 x}{3 a^2}-\frac{x^4}{3 a \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^3)^(-2),x]

[Out]

(4*x)/(3*a^2) - x^4/(3*a*(b + a*x^3)) + (4*b^(1/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x
)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*a^(7/3)) - (4*b^(1/3)*Log[b^(1/3) + a^(1/3)*x])
/(9*a^(7/3)) + (2*b^(1/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(9*a^(
7/3))

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Rubi in Sympy [A]  time = 35.0174, size = 136, normalized size = 0.94 \[ - \frac{x^{4}}{3 a \left (a x^{3} + b\right )} + \frac{4 x}{3 a^{2}} - \frac{4 \sqrt [3]{b} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{9 a^{\frac{7}{3}}} + \frac{2 \sqrt [3]{b} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{9 a^{\frac{7}{3}}} + \frac{4 \sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{9 a^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**4/(3*a*(a*x**3 + b)) + 4*x/(3*a**2) - 4*b**(1/3)*log(a**(1/3)*x + b**(1/3))/
(9*a**(7/3)) + 2*b**(1/3)*log(a**(2/3)*x**2 - a**(1/3)*b**(1/3)*x + b**(2/3))/(9
*a**(7/3)) + 4*sqrt(3)*b**(1/3)*atan(sqrt(3)*(-2*a**(1/3)*x/3 + b**(1/3)/3)/b**(
1/3))/(9*a**(7/3))

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Mathematica [A]  time = 0.147449, size = 127, normalized size = 0.88 \[ \frac{2 \sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+\frac{3 \sqrt [3]{a} b x}{a x^3+b}-4 \sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+4 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )+9 \sqrt [3]{a} x}{9 a^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^3)^(-2),x]

[Out]

(9*a^(1/3)*x + (3*a^(1/3)*b*x)/(b + a*x^3) + 4*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*a^
(1/3)*x)/b^(1/3))/Sqrt[3]] - 4*b^(1/3)*Log[b^(1/3) + a^(1/3)*x] + 2*b^(1/3)*Log[
b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(9*a^(7/3))

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Maple [A]  time = 0.013, size = 115, normalized size = 0.8 \[{\frac{x}{{a}^{2}}}+{\frac{bx}{3\,{a}^{2} \left ( a{x}^{3}+b \right ) }}-{\frac{4\,b}{9\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,b}{9\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,b\sqrt{3}}{9\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)^2,x)

[Out]

x/a^2+1/3*b/a^2*x/(a*x^3+b)-4/9*b/a^3/(b/a)^(2/3)*ln(x+(b/a)^(1/3))+2/9*b/a^3/(b
/a)^(2/3)*ln(x^2-x*(b/a)^(1/3)+(b/a)^(2/3))-4/9*b/a^3/(b/a)^(2/3)*3^(1/2)*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((a + b/x^3)^(-2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231813, size = 216, normalized size = 1.5 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (a x^{3} + b\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 4 \, \sqrt{3}{\left (a x^{3} + b\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 12 \,{\left (a x^{3} + b\right )} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{b}{a}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (3 \, a x^{4} + 4 \, b x\right )}\right )}}{27 \,{\left (a^{3} x^{3} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*sqrt(3)*(2*sqrt(3)*(a*x^3 + b)*(-b/a)^(1/3)*log(x^2 + x*(-b/a)^(1/3) + (-b
/a)^(2/3)) - 4*sqrt(3)*(a*x^3 + b)*(-b/a)^(1/3)*log(x - (-b/a)^(1/3)) + 12*(a*x^
3 + b)*(-b/a)^(1/3)*arctan(1/3*(2*sqrt(3)*x + sqrt(3)*(-b/a)^(1/3))/(-b/a)^(1/3)
) - 3*sqrt(3)*(3*a*x^4 + 4*b*x))/(a^3*x^3 + a^2*b)

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Sympy [A]  time = 1.77671, size = 48, normalized size = 0.33 \[ \frac{b x}{3 a^{3} x^{3} + 3 a^{2} b} + \operatorname{RootSum}{\left (729 t^{3} a^{7} + 64 b, \left ( t \mapsto t \log{\left (- \frac{9 t a^{2}}{4} + x \right )} \right )\right )} + \frac{x}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)**2,x)

[Out]

b*x/(3*a**3*x**3 + 3*a**2*b) + RootSum(729*_t**3*a**7 + 64*b, Lambda(_t, _t*log(
-9*_t*a**2/4 + x))) + x/a**2

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GIAC/XCAS [A]  time = 0.227399, size = 171, normalized size = 1.19 \[ \frac{4 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2}} + \frac{x}{a^{2}} + \frac{b x}{3 \,{\left (a x^{3} + b\right )} a^{2}} - \frac{4 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3}} - \frac{2 \, \left (-a^{2} b\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{9 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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