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

Optimal. Leaf size=134 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3}}-\frac{x}{3 b \left (a+b x^3\right )} \]

[Out]

-x/(3*b*(a + b*x^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]/(3*Sqrt
[3]*a^(2/3)*b^(4/3)) + Log[a^(1/3) + b^(1/3)*x]/(9*a^(2/3)*b^(4/3)) - Log[a^(2/3
) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(18*a^(2/3)*b^(4/3))

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Rubi [A]  time = 0.141962, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3}}-\frac{x}{3 b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^3)^2,x]

[Out]

-x/(3*b*(a + b*x^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]/(3*Sqrt
[3]*a^(2/3)*b^(4/3)) + Log[a^(1/3) + b^(1/3)*x]/(9*a^(2/3)*b^(4/3)) - Log[a^(2/3
) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(18*a^(2/3)*b^(4/3))

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Rubi in Sympy [A]  time = 30.4444, size = 121, normalized size = 0.9 \[ - \frac{x}{3 b \left (a + b x^{3}\right )} + \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{2}{3}} b^{\frac{4}{3}}} - \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{2}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(3*b*(a + b*x**3)) + log(a**(1/3) + b**(1/3)*x)/(9*a**(2/3)*b**(4/3)) - log(a
**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(2/3)*b**(4/3)) - sqrt(3)*
atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(2/3)*b**(4/3))

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Mathematica [A]  time = 0.129832, size = 118, normalized size = 0.88 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{6 \sqrt [3]{b} x}{a+b x^3}}{18 b^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x^3)^2,x]

[Out]

((-6*b^(1/3)*x)/(a + b*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt
[3]])/a^(2/3) + (2*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) - Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2]/a^(2/3))/(18*b^(4/3))

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Maple [A]  time = 0.01, size = 106, normalized size = 0.8 \[ -{\frac{x}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{1}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{18\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^3+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232871, size = 184, normalized size = 1.37 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b x^{3} + a\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left (b x^{3} + a\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \,{\left (b x^{3} + a\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) + 6 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x\right )}}{54 \,{\left (b^{2} x^{3} + a b\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(sqrt(3)*(b*x^3 + a)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a
^2) - 2*sqrt(3)*(b*x^3 + a)*log((a^2*b)^(1/3)*x + a) - 6*(b*x^3 + a)*arctan(1/3*
(2*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) + 6*sqrt(3)*(a^2*b)^(1/3)*x)/((b^2*x^
3 + a*b)*(a^2*b)^(1/3))

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Sympy [A]  time = 1.58304, size = 39, normalized size = 0.29 \[ - \frac{x}{3 a b + 3 b^{2} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{4} - 1, \left ( t \mapsto t \log{\left (9 t a b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**3+a)**2,x)

[Out]

-x/(3*a*b + 3*b**2*x**3) + RootSum(729*_t**3*a**2*b**4 - 1, Lambda(_t, _t*log(9*
_t*a*b + x)))

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GIAC/XCAS [A]  time = 0.223201, size = 176, normalized size = 1.31 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b} - \frac{x}{3 \,{\left (b x^{3} + a\right )} b} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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