Optimal. Leaf size=34 \[ -\frac{a^2 x^{-2 n}}{2 n}-\frac{2 a b x^{-n}}{n}+b^2 \log (x) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0477683, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^2 x^{-2 n}}{2 n}-\frac{2 a b x^{-n}}{n}+b^2 \log (x) \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 2*n)*(a + b*x^n)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.65974, size = 31, normalized size = 0.91 \[ - \frac{a^{2} x^{- 2 n}}{2 n} - \frac{2 a b x^{- n}}{n} + \frac{b^{2} \log{\left (x^{n} \right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-2*n)*(a+b*x**n)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0587054, size = 28, normalized size = 0.82 \[ b^2 \log (x)-\frac{a x^{-2 n} \left (a+4 b x^n\right )}{2 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 2*n)*(a + b*x^n)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.021, size = 43, normalized size = 1.3 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-{\frac{{a}^{2}}{2\,n}}-2\,{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}b}{n}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-2*n)*(a+b*x^n)^2,x)
[Out]
_______________________________________________________________________________________
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((b*x^n + a)^2*x^(-2*n - 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.224981, size = 51, normalized size = 1.5 \[ \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) - 4 \, a b x^{n} - a^{2}}{2 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^(-2*n - 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1-2*n)*(a+b*x**n)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.219392, size = 54, normalized size = 1.59 \[ \frac{{\left (2 \, b^{2} n e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) - 4 \, a b e^{\left (n{\rm ln}\left (x\right )\right )} - a^{2}\right )} e^{\left (-2 \, n{\rm ln}\left (x\right )\right )}}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^(-2*n - 1),x, algorithm="giac")
[Out]