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 = 0.0185387, 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, Rules used = {1584, 266, 36, 29, 31} \[ \frac{\log (x)}{b}-\frac{\log \left (b+c x^n\right )}{b n} \]

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)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x^{-1+n}}{b x^n+c x^{2 n}} \, dx &=\int \frac{1}{x \left (b+c x^n\right )} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^n\right )}{b n}-\frac{c \operatorname{Subst}\left (\int \frac{1}{b+c x} \, dx,x,x^n\right )}{b n}\\ &=\frac{\log (x)}{b}-\frac{\log \left (b+c x^n\right )}{b n}\\ \end{align*}

Mathematica [A]  time = 0.006655, 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 = 0.02, size = 26, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{bn}} \end{align*}

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 = 0.989826, size = 36, normalized size = 1.57 \begin{align*} \frac{\log \left (x\right )}{b} - \frac{\log \left (\frac{c x^{n} + b}{c}\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)/(b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

log(x)/b - log((c*x^n + b)/c)/(b*n)

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Fricas [A]  time = 1.85103, size = 47, normalized size = 2.04 \begin{align*} \frac{n \log \left (x\right ) - \log \left (c x^{n} + b\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)/(b*x^n+c*x^(2*n)),x, algorithm="fricas")

[Out]

(n*log(x) - log(c*x^n + b))/(b*n)

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Sympy [A]  time = 14.6952, size = 42, normalized size = 1.83 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{c} & \text{for}\: b = 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{2 \log{\left (x \right )}}{b} - \frac{\log{\left (x^{n} + \frac{c x^{2 n}}{b} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}

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

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Giac [A]  time = 1.10034, size = 34, normalized size = 1.48 \begin{align*} \frac{\log \left ({\left | x \right |}\right )}{b} - \frac{\log \left ({\left | c x^{n} + b \right |}\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)/(b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

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