3.496 \(\int \frac{x^{-1+n}}{b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=23 \[ \frac{\log (x)}{b}-\frac{\log \left (b+c x^n\right )}{b n} \]

[Out]

Log[x]/b - Log[b + c*x^n]/(b*n)

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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]

Log[x]/b - Log[b + c*x^n]/(b*n)

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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]

log(x**n)/(b*n) - log(b + c*x**n)/(b*n)

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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]

(n*Log[x] - Log[b + c*x^n])/(b*n)

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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]

ln(x)/b-1/b/n*ln(c*exp(n*ln(x))+b)

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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]

log(x)/b - log((c*x^n + b)/c)/(b*n)

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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]

(n*log(x) - log(c*x^n + b))/(b*n)

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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]

Piecewise((zoo*log(x), Eq(b, 0) & Eq(c, 0) & Eq(n, 0)), (log(x)/(b + c), Eq(n, 0
)), (-x**(-n)/(c*n), Eq(b, 0)), ((n**2*log(x)/(n**2 - n) - n*log(x)/(n**2 - n))/
b, Eq(c, 0)), (2*log(x)/b - log(b*x**n/c + x**(2*n))/(b*n), True))

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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]

ln(abs(x))/b - ln(abs(c*x^n + b))/(b*n)