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

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

[Out]

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

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Rubi [A]  time = 0.171395, 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 \[ \frac{5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac{5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac{5 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3}}-\frac{5}{6 a^2 x^2}+\frac{1}{3 a x^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^3)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 38.7411, size = 139, normalized size = 0.95 \[ \frac{1}{3 a x^{2} \left (a + b x^{3}\right )} - \frac{5}{6 a^{2} x^{2}} - \frac{5 b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{8}{3}}} + \frac{5 b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{8}{3}}} + \frac{5 \sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.174567, size = 129, normalized size = 0.88 \[ \frac{5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{6 a^{2/3} b x}{a+b x^3}-\frac{9 a^{2/3}}{x^2}-10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+10 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{18 a^{8/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 115, normalized size = 0.8 \[ -{\frac{1}{2\,{a}^{2}{x}^{2}}}-{\frac{bx}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{5}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5}{18\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^3+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239256, size = 281, normalized size = 1.92 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left (b x^{5} + a x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 10 \, \sqrt{3}{\left (b x^{5} + a x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 30 \,{\left (b x^{5} + a x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (5 \, b x^{3} + 3 \, a\right )}\right )}}{54 \,{\left (a^{2} b x^{5} + a^{3} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.38208, size = 56, normalized size = 0.38 \[ - \frac{3 a + 5 b x^{3}}{6 a^{3} x^{2} + 6 a^{2} b x^{5}} + \operatorname{RootSum}{\left (729 t^{3} a^{8} + 125 b^{2}, \left ( t \mapsto t \log{\left (- \frac{9 t a^{3}}{5 b} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3*a + 5*b*x**3)/(6*a**3*x**2 + 6*a**2*b*x**5) + RootSum(729*_t**3*a**8 + 125*b
**2, Lambda(_t, _t*log(-9*_t*a**3/(5*b) + x)))

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GIAC/XCAS [A]  time = 0.220951, size = 177, normalized size = 1.21 \[ \frac{5 \, b \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} - \frac{b x}{3 \,{\left (b x^{3} + a\right )} a^{2}} - \frac{5 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3}} - \frac{5 \, \left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3}} - \frac{1}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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