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

Optimal. Leaf size=15 \[ \frac{\log \left (b+c x^n\right )}{c n} \]

[Out]

Log[b + c*x^n]/(c*n)

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Rubi [A]  time = 0.0124989, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1584, 260} \[ \frac{\log \left (b+c x^n\right )}{c n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)/(b*x^n + c*x^(2*n)),x]

[Out]

Log[b + c*x^n]/(c*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 260

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

Rubi steps

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

Mathematica [A]  time = 0.0037878, size = 15, normalized size = 1. \[ \frac{\log \left (b+c x^n\right )}{c n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 2*n)/(b*x^n + c*x^(2*n)),x]

[Out]

Log[b + c*x^n]/(c*n)

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Maple [A]  time = 0.017, size = 18, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{cn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+2*n)/(b*x^n+c*x^(2*n)),x)

[Out]

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

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Maxima [A]  time = 0.987509, size = 26, normalized size = 1.73 \begin{align*} \frac{\log \left (\frac{c x^{n} + b}{c}\right )}{c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((c*x^n + b)/c)/(c*n)

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Fricas [A]  time = 1.90315, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (c x^{n} + b\right )}{c n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*x^n + b)/(c*n)

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Sympy [A]  time = 14.0103, size = 37, normalized size = 2.47 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{b} & \text{for}\: c = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{b + c} & \text{for}\: n = 0 \\\frac{x^{n}}{b n} & \text{for}\: c = 0 \\- \frac{\log{\left (x \right )}}{c} + \frac{\log{\left (\frac{b x^{n}}{c} + x^{2 n} \right )}}{c n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+2*n)/(b*x**n+c*x**(2*n)),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2 \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(2*n - 1)/(c*x^(2*n) + b*x^n), x)