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

Optimal. Leaf size=154 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3}}+\frac{x}{18 a b \left (a+b x^3\right )}-\frac{x}{6 b \left (a+b x^3\right )^2} \]

[Out]

-x/(6*b*(a + b*x^3)^2) + x/(18*a*b*(a + b*x^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x)
/(Sqrt[3]*a^(1/3))]/(9*Sqrt[3]*a^(5/3)*b^(4/3)) + Log[a^(1/3) + b^(1/3)*x]/(27*a
^(5/3)*b^(4/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(54*a^(5/3)*b^(
4/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.163812, antiderivative size = 154, 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 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{4/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{5/3} b^{4/3}}+\frac{x}{18 a b \left (a+b x^3\right )}-\frac{x}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^3)^3,x]

[Out]

-x/(6*b*(a + b*x^3)^2) + x/(18*a*b*(a + b*x^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)*x)
/(Sqrt[3]*a^(1/3))]/(9*Sqrt[3]*a^(5/3)*b^(4/3)) + Log[a^(1/3) + b^(1/3)*x]/(27*a
^(5/3)*b^(4/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(54*a^(5/3)*b^(
4/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.1272, size = 136, normalized size = 0.88 \[ - \frac{x}{6 b \left (a + b x^{3}\right )^{2}} + \frac{x}{18 a b \left (a + b x^{3}\right )} + \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{5}{3}} b^{\frac{4}{3}}} - \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{5}{3}} b^{\frac{4}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{5}{3}} b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(6*b*(a + b*x**3)**2) + x/(18*a*b*(a + b*x**3)) + log(a**(1/3) + b**(1/3)*x)/
(27*a**(5/3)*b**(4/3)) - log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(54
*a**(5/3)*b**(4/3)) - sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3
))/(27*a**(5/3)*b**(4/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.164854, size = 138, normalized size = 0.9 \[ \frac{-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{5/3}}+\frac{3 \sqrt [3]{b} x}{a^2+a b x^3}-\frac{9 \sqrt [3]{b} x}{\left (a+b x^3\right )^2}}{54 b^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x^3)^3,x]

[Out]

((-9*b^(1/3)*x)/(a + b*x^3)^2 + (3*b^(1/3)*x)/(a^2 + a*b*x^3) - (2*Sqrt[3]*ArcTa
n[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*Log[a^(1/3) + b^(1/3)*x])/a
^(5/3) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/a^(5/3))/(54*b^(4/3))

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 125, normalized size = 0.8 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{{x}^{4}}{18\,a}}-{\frac{x}{9\,b}} \right ) }+{\frac{1}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^3+a)^3,x)

[Out]

(1/18*x^4/a-1/9*x/b)/(b*x^3+a)^2+1/27/b^2/a/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/54/b
^2/a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/27/b^2/a/(a/b)^(2/3)*3^(1/2
)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

_______________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.233108, size = 259, normalized size = 1.68 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (b x^{4} - 2 \, a x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log((a^2*b)^(2/3)*x^2 - (a^2
*b)^(1/3)*a*x + a^2) - 2*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log((a^2*b)^(1/3)*x
+ a) - 6*(b^2*x^6 + 2*a*b*x^3 + a^2)*arctan(1/3*(2*sqrt(3)*(a^2*b)^(1/3)*x - sq
rt(3)*a)/a) - 3*sqrt(3)*(b*x^4 - 2*a*x)*(a^2*b)^(1/3))/((a*b^3*x^6 + 2*a^2*b^2*x
^3 + a^3*b)*(a^2*b)^(1/3))

_______________________________________________________________________________________

Sympy [A]  time = 2.39234, size = 65, normalized size = 0.42 \[ \frac{- 2 a x + b x^{4}}{18 a^{3} b + 36 a^{2} b^{2} x^{3} + 18 a b^{3} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{5} b^{4} - 1, \left ( t \mapsto t \log{\left (27 t a^{2} b + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**3+a)**3,x)

[Out]

(-2*a*x + b*x**4)/(18*a**3*b + 36*a**2*b**2*x**3 + 18*a*b**3*x**6) + RootSum(196
83*_t**3*a**5*b**4 - 1, Lambda(_t, _t*log(27*_t*a**2*b + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.250812, size = 192, normalized size = 1.25 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{2}} + \frac{b x^{4} - 2 \, a x}{18 \,{\left (b x^{3} + a\right )}^{2} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^2*b) + 1/27*sqrt(3)*(-a*b^2)^(1/
3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^2) + 1/54*(-a*b^
2)^(1/3)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^2) + 1/18*(b*x^4 - 2*a*x
)/((b*x^3 + a)^2*a*b)