3.355 \(\int \frac{1}{a x+b x^n} \, dx\)

Optimal. Leaf size=23 \[ \frac{\log \left (a x^{1-n}+b\right )}{a (1-n)} \]

[Out]

Log[b + a*x^(1 - n)]/(a*(1 - n))

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Rubi [A]  time = 0.0108914, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 260} \[ \frac{\log \left (a x^{1-n}+b\right )}{a (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^n)^(-1),x]

[Out]

Log[b + a*x^(1 - n)]/(a*(1 - n))

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, 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{1}{a x+b x^n} \, dx &=\int \frac{x^{-n}}{b+a x^{1-n}} \, dx\\ &=\frac{\log \left (b+a x^{1-n}\right )}{a (1-n)}\\ \end{align*}

Mathematica [A]  time = 0.0081442, size = 23, normalized size = 1. \[ \frac{\log \left (a x^{1-n}+b\right )}{a (1-n)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^n)^(-1),x]

[Out]

Log[b + a*x^(1 - n)]/(a*(1 - n))

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Maple [A]  time = 0.01, size = 36, normalized size = 1.6 \begin{align*}{\frac{n\ln \left ( x \right ) }{a \left ( -1+n \right ) }}-{\frac{\ln \left ( ax+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{a \left ( -1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.00649, size = 50, normalized size = 2.17 \begin{align*} \frac{n \log \left (x\right )}{a{\left (n - 1\right )}} - \frac{\log \left (\frac{a x + b x^{n}}{b}\right )}{a{\left (n - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

n*log(x)/(a*(n - 1)) - log((a*x + b*x^n)/b)/(a*(n - 1))

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Fricas [A]  time = 0.930557, size = 55, normalized size = 2.39 \begin{align*} \frac{n \log \left (x\right ) - \log \left (a x + b x^{n}\right )}{a n - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(n*log(x) - log(a*x + b*x^n))/(a*n - a)

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Sympy [A]  time = 0.693469, size = 48, normalized size = 2.09 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{b} & \text{for}\: a = 0 \wedge n = 1 \\- \frac{x}{b \left (n x^{n} - x^{n}\right )} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 1 \\\frac{n \log{\left (x \right )}}{a n - a} - \frac{\log{\left (x + \frac{b x^{n}}{a} \right )}}{a n - a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x**n),x)

[Out]

Piecewise((log(x)/b, Eq(a, 0) & Eq(n, 1)), (-x/(b*(n*x**n - x**n)), Eq(a, 0)), (log(x)/(a + b), Eq(n, 1)), (n*
log(x)/(a*n - a) - log(x + b*x**n/a)/(a*n - a), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(a*x + b*x^n), x)