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

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

[Out]

Log[b + a*x^n]/(a*n)

________________________________________________________________________________________

Rubi [A]  time = 0.0076386, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1593, 260} \[ \frac{\log \left (a x^n+b\right )}{a n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^(1 - n))^(-1),x]

[Out]

Log[b + a*x^n]/(a*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^{1-n}} \, dx &=\int \frac{x^{-1+n}}{b+a x^n} \, dx\\ &=\frac{\log \left (b+a x^n\right )}{a n}\\ \end{align*}

Mathematica [A]  time = 0.0044175, size = 15, normalized size = 1. \[ \frac{\log \left (a x^n+b\right )}{a n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^(1 - n))^(-1),x]

[Out]

Log[b + a*x^n]/(a*n)

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 41, normalized size = 2.7 \begin{align*} -{\frac{\ln \left ( x \right ) }{an}}+{\frac{\ln \left ( x \right ) }{a}}+{\frac{\ln \left ( ax+b{{\rm e}^{ \left ( 1-n \right ) \ln \left ( x \right ) }} \right ) }{an}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02379, size = 26, normalized size = 1.73 \begin{align*} \frac{\log \left (\frac{a x^{n} + b}{a}\right )}{a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((a*x^n + b)/a)/(a*n)

________________________________________________________________________________________

Fricas [A]  time = 0.90674, size = 68, normalized size = 4.53 \begin{align*} \frac{{\left (n - 1\right )} \log \left (x\right ) + \log \left (a x + b x^{-n + 1}\right )}{a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.99376, size = 39, normalized size = 2.6 \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{x^{n}}{b n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} + \frac{\log{\left (\frac{a}{b} + x^{- n} \right )}}{a n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(a*x + b*x^(-n + 1)), x)