3.1988 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^2 x^7} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\log \left (a x^3+b\right )}{3 b^2}+\frac{1}{3 b \left (a x^3+b\right )}+\frac{\log (x)}{b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0764279, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\log \left (a x^3+b\right )}{3 b^2}+\frac{1}{3 b \left (a x^3+b\right )}+\frac{\log (x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)^2*x^7),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.6625, size = 34, normalized size = 0.89 \[ \frac{1}{3 b \left (a x^{3} + b\right )} + \frac{\log{\left (x^{3} \right )}}{3 b^{2}} - \frac{\log{\left (a x^{3} + b \right )}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**3)**2/x**7,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0216033, size = 33, normalized size = 0.87 \[ \frac{\frac{b}{a x^3+b}-\log \left (a x^3+b\right )+3 \log (x)}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^3)^2*x^7),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 35, normalized size = 0.9 \[{\frac{1}{3\,b \left ( a{x}^{3}+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{2}}}-{\frac{\ln \left ( a{x}^{3}+b \right ) }{3\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)^2/x^7,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.4412, size = 50, normalized size = 1.32 \[ \frac{1}{3 \,{\left (a b x^{3} + b^{2}\right )}} - \frac{\log \left (a x^{3} + b\right )}{3 \, b^{2}} + \frac{\log \left (x^{3}\right )}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^2*x^7),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.235612, size = 63, normalized size = 1.66 \[ -\frac{{\left (a x^{3} + b\right )} \log \left (a x^{3} + b\right ) - 3 \,{\left (a x^{3} + b\right )} \log \left (x\right ) - b}{3 \,{\left (a b^{2} x^{3} + b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^2*x^7),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.9393, size = 34, normalized size = 0.89 \[ \frac{1}{3 a b x^{3} + 3 b^{2}} + \frac{\log{\left (x \right )}}{b^{2}} - \frac{\log{\left (x^{3} + \frac{b}{a} \right )}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)**2/x**7,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.226547, size = 61, normalized size = 1.61 \[ -\frac{{\rm ln}\left ({\left | a x^{3} + b \right |}\right )}{3 \, b^{2}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b^{2}} + \frac{a x^{3} + 2 \, b}{3 \,{\left (a x^{3} + b\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)^2*x^7),x, algorithm="giac")

[Out]

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