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

Optimal. Leaf size=155 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{7/3} b^{2/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3}}+\frac{2 x^2}{9 a^2 \left (a+b x^3\right )}+\frac{x^2}{6 a \left (a+b x^3\right )^2} \]

[Out]

x^2/(6*a*(a + b*x^3)^2) + (2*x^2)/(9*a^2*(a + b*x^3)) - (2*ArcTan[(a^(1/3) - 2*b
^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(2/3)) - (2*Log[a^(1/3) + b^(
1/3)*x])/(27*a^(7/3)*b^(2/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(
27*a^(7/3)*b^(2/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.167625, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{7/3} b^{2/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3}}+\frac{2 x^2}{9 a^2 \left (a+b x^3\right )}+\frac{x^2}{6 a \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^3)^3,x]

[Out]

x^2/(6*a*(a + b*x^3)^2) + (2*x^2)/(9*a^2*(a + b*x^3)) - (2*ArcTan[(a^(1/3) - 2*b
^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(2/3)) - (2*Log[a^(1/3) + b^(
1/3)*x])/(27*a^(7/3)*b^(2/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(
27*a^(7/3)*b^(2/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.4453, size = 144, normalized size = 0.93 \[ \frac{x^{2}}{6 a \left (a + b x^{3}\right )^{2}} + \frac{2 x^{2}}{9 a^{2} \left (a + b x^{3}\right )} - \frac{2 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{7}{3}} b^{\frac{2}{3}}} + \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{27 a^{\frac{7}{3}} b^{\frac{2}{3}}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{7}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2/(6*a*(a + b*x**3)**2) + 2*x**2/(9*a**2*(a + b*x**3)) - 2*log(a**(1/3) + b**
(1/3)*x)/(27*a**(7/3)*b**(2/3)) + log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*
x**2)/(27*a**(7/3)*b**(2/3)) - 2*sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x
/3)/a**(1/3))/(27*a**(7/3)*b**(2/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.125709, size = 139, normalized size = 0.9 \[ \frac{\frac{2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{9 a^{4/3} x^2}{\left (a+b x^3\right )^2}-\frac{4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{12 \sqrt [3]{a} x^2}{a+b x^3}}{54 a^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^3)^3,x]

[Out]

((9*a^(4/3)*x^2)/(a + b*x^3)^2 + (12*a^(1/3)*x^2)/(a + b*x^3) - (4*Sqrt[3]*ArcTa
n[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - (4*Log[a^(1/3) + b^(1/3)*x])/b
^(2/3) + (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3))/(54*a^(7/3)
)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 134, normalized size = 0.9 \[{\frac{{x}^{2}}{6\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{2\,{x}^{2}}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{2}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{27\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,\sqrt{3}}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^3+a)^3,x)

[Out]

1/6*x^2/a/(b*x^3+a)^2+2/9*x^2/a^2/(b*x^3+a)-2/27/a^2/b/(a/b)^(1/3)*ln(x+(a/b)^(1
/3))+1/27/a^2/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/27/a^2*3^(1/2)/b
/(a/b)^(1/3)*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/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.225905, size = 277, normalized size = 1.79 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 4 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 12 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (4 \, b x^{5} + 7 \, a x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(2*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log((-a*b^2)^(1/3)*b*x^2 -
a*b + (-a*b^2)^(2/3)*x) - 4*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log(a*b + (-a*b
^2)^(2/3)*x) + 12*(b^2*x^6 + 2*a*b*x^3 + a^2)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(
3)*(-a*b^2)^(2/3)*x)/(a*b)) - 3*sqrt(3)*(4*b*x^5 + 7*a*x^2)*(-a*b^2)^(1/3))/((a^
2*b^2*x^6 + 2*a^3*b*x^3 + a^4)*(-a*b^2)^(1/3))

_______________________________________________________________________________________

Sympy [A]  time = 2.38758, size = 70, normalized size = 0.45 \[ \frac{7 a x^{2} + 4 b x^{5}}{18 a^{4} + 36 a^{3} b x^{3} + 18 a^{2} b^{2} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{7} b^{2} + 8, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a^{5} b}{4} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**3+a)**3,x)

[Out]

(7*a*x**2 + 4*b*x**5)/(18*a**4 + 36*a**3*b*x**3 + 18*a**2*b**2*x**6) + RootSum(1
9683*_t**3*a**7*b**2 + 8, Lambda(_t, _t*log(729*_t**2*a**5*b/4 + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.252784, size = 188, normalized size = 1.21 \[ -\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{3}} - \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{2}} + \frac{4 \, b x^{5} + 7 \, a x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{27 \, a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/27*(-a/b)^(2/3)*ln(abs(x - (-a/b)^(1/3)))/a^3 - 2/27*sqrt(3)*(-a*b^2)^(2/3)*a
rctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^2) + 1/18*(4*b*x^5 +
7*a*x^2)/((b*x^3 + a)^2*a^2) + 1/27*(-a*b^2)^(2/3)*ln(x^2 + x*(-a/b)^(1/3) + (-
a/b)^(2/3))/(a^3*b^2)