Optimal. Leaf size=23 \[ \frac{\log (x)}{b}-\frac{\log \left (b+c x^n\right )}{b n} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0357283, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{\log (x)}{b}-\frac{\log \left (b+c x^n\right )}{b n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + n)/(b*x^n + c*x^(2*n)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.81461, size = 19, normalized size = 0.83 \[ \frac{\log{\left (x^{n} \right )}}{b n} - \frac{\log{\left (b + c x^{n} \right )}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+n)/(b*x**n+c*x**(2*n)),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0144911, size = 22, normalized size = 0.96 \[ \frac{n \log (x)-\log \left (b+c x^n\right )}{b n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + n)/(b*x^n + c*x^(2*n)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.029, size = 26, normalized size = 1.1 \[{\frac{\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{bn}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+n)/(b*x^n+c*x^(2*n)),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.749337, size = 36, normalized size = 1.57 \[ \frac{\log \left (x\right )}{b} - \frac{\log \left (\frac{c x^{n} + b}{c}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.296322, size = 30, normalized size = 1.3 \[ \frac{n \log \left (x\right ) - \log \left (c x^{n} + b\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 95.7024, size = 66, normalized size = 2.87 \[ \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: b = 0 \wedge c = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{b + c} & \text{for}\: n = 0 \\- \frac{x^{- n}}{c n} & \text{for}\: b = 0 \\\frac{\frac{n^{2} \log{\left (x \right )}}{n^{2} - n} - \frac{n \log{\left (x \right )}}{n^{2} - n}}{b} & \text{for}\: c = 0 \\\frac{2 \log{\left (x \right )}}{b} - \frac{\log{\left (\frac{b x^{n}}{c} + x^{2 n} \right )}}{b n} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1+n)/(b*x**n+c*x**(2*n)),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.271302, size = 34, normalized size = 1.48 \[ \frac{{\rm ln}\left ({\left | x \right |}\right )}{b} - \frac{{\rm ln}\left ({\left | c x^{n} + b \right |}\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="giac")
[Out]