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

Optimal. Leaf size=146 \[ -\frac{2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{7/3}}+\frac{4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3}}+\frac{4 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{4}{3 a^2 x}+\frac{1}{3 a x \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.171969, antiderivative size = 146, 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 \[ -\frac{2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{7/3}}+\frac{4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3}}+\frac{4 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3}}-\frac{4}{3 a^2 x}+\frac{1}{3 a x \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^3)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 34.9992, size = 136, normalized size = 0.93 \[ \frac{1}{3 a x \left (a + b x^{3}\right )} - \frac{4}{3 a^{2} x} + \frac{4 \sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{7}{3}}} - \frac{2 \sqrt [3]{b} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{7}{3}}} + \frac{4 \sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.21496, size = 131, normalized size = 0.9 \[ \frac{-2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{3 \sqrt [3]{a} b x^2}{a+b x^3}+4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+4 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-\frac{9 \sqrt [3]{a}}{x}}{9 a^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 117, normalized size = 0.8 \[ -{\frac{1}{x{a}^{2}}}-{\frac{b{x}^{2}}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{4}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2}{9\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{4\,\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^3+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242484, size = 232, normalized size = 1.59 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b x^{4} + a x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 4 \, \sqrt{3}{\left (b x^{4} + a x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 12 \,{\left (b x^{4} + a x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (4 \, b x^{3} + 3 \, a\right )}\right )}}{27 \,{\left (a^{2} b x^{4} + a^{3} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.04035, size = 54, normalized size = 0.37 \[ - \frac{3 a + 4 b x^{3}}{3 a^{3} x + 3 a^{2} b x^{4}} + \operatorname{RootSum}{\left (729 t^{3} a^{7} - 64 b, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{5}}{16 b} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*a + 4*b*x**3)/(3*a**3*x + 3*a**2*b*x**4) + RootSum(729*_t**3*a**7 - 64*b, La
mbda(_t, _t*log(81*_t**2*a**5/(16*b) + x)))

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GIAC/XCAS [A]  time = 0.224913, size = 188, normalized size = 1.29 \[ \frac{4 \, b \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} + \frac{4 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b} - \frac{4 \, b x^{3} + 3 \, a}{3 \,{\left (b x^{4} + a x\right )} a^{2}} - \frac{2 \, \left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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