3.333 \(\int \frac{1}{x^4 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 b \log \left (a+b x^3\right )}{3 a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{3 a^2 \left (a+b x^3\right )}-\frac{1}{3 a^2 x^3} \]

[Out]

-1/(3*a^2*x^3) - b/(3*a^2*(a + b*x^3)) - (2*b*Log[x])/a^3 + (2*b*Log[a + b*x^3])
/(3*a^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0788349, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 b \log \left (a+b x^3\right )}{3 a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{3 a^2 \left (a+b x^3\right )}-\frac{1}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^3)^2),x]

[Out]

-1/(3*a^2*x^3) - b/(3*a^2*(a + b*x^3)) - (2*b*Log[x])/a^3 + (2*b*Log[a + b*x^3])
/(3*a^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.5762, size = 53, normalized size = 1.02 \[ - \frac{b}{3 a^{2} \left (a + b x^{3}\right )} - \frac{1}{3 a^{2} x^{3}} - \frac{2 b \log{\left (x^{3} \right )}}{3 a^{3}} + \frac{2 b \log{\left (a + b x^{3} \right )}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**3+a)**2,x)

[Out]

-b/(3*a**2*(a + b*x**3)) - 1/(3*a**2*x**3) - 2*b*log(x**3)/(3*a**3) + 2*b*log(a
+ b*x**3)/(3*a**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0847641, size = 41, normalized size = 0.79 \[ -\frac{a \left (\frac{b}{a+b x^3}+\frac{1}{x^3}\right )-2 b \log \left (a+b x^3\right )+6 b \log (x)}{3 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^3)^2),x]

[Out]

-(a*(x^(-3) + b/(a + b*x^3)) + 6*b*Log[x] - 2*b*Log[a + b*x^3])/(3*a^3)

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 47, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{3}{a}^{2}}}-{\frac{b}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}+{\frac{2\,b\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^3+a)^2,x)

[Out]

-1/3/x^3/a^2-1/3*b/a^2/(b*x^3+a)-2*b*ln(x)/a^3+2/3*b*ln(b*x^3+a)/a^3

_______________________________________________________________________________________

Maxima [A]  time = 1.45801, size = 72, normalized size = 1.38 \[ -\frac{2 \, b x^{3} + a}{3 \,{\left (a^{2} b x^{6} + a^{3} x^{3}\right )}} + \frac{2 \, b \log \left (b x^{3} + a\right )}{3 \, a^{3}} - \frac{2 \, b \log \left (x^{3}\right )}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^2*x^4),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.238807, size = 99, normalized size = 1.9 \[ -\frac{2 \, a b x^{3} + a^{2} - 2 \,{\left (b^{2} x^{6} + a b x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \,{\left (b^{2} x^{6} + a b x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{3} b x^{6} + a^{4} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^2*x^4),x, algorithm="fricas")

[Out]

-1/3*(2*a*b*x^3 + a^2 - 2*(b^2*x^6 + a*b*x^3)*log(b*x^3 + a) + 6*(b^2*x^6 + a*b*
x^3)*log(x))/(a^3*b*x^6 + a^4*x^3)

_______________________________________________________________________________________

Sympy [A]  time = 3.13246, size = 53, normalized size = 1.02 \[ - \frac{a + 2 b x^{3}}{3 a^{3} x^{3} + 3 a^{2} b x^{6}} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{2 b \log{\left (\frac{a}{b} + x^{3} \right )}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**3+a)**2,x)

[Out]

-(a + 2*b*x**3)/(3*a**3*x**3 + 3*a**2*b*x**6) - 2*b*log(x)/a**3 + 2*b*log(a/b +
x**3)/(3*a**3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220423, size = 69, normalized size = 1.33 \[ \frac{2 \, b{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac{2 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{2 \, b x^{3} + a}{3 \,{\left (b x^{6} + a x^{3}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^2*x^4),x, algorithm="giac")

[Out]

2/3*b*ln(abs(b*x^3 + a))/a^3 - 2*b*ln(abs(x))/a^3 - 1/3*(2*b*x^3 + a)/((b*x^6 +
a*x^3)*a^2)