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

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

[Out]

Log[x]/a - Log[a + b*x^n]/(a*n)

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Rubi [A]  time = 0.0131801, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {1593, 266, 36, 29, 31} \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^n\right )}{a n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^(1 + n))^(-1),x]

[Out]

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

Mathematica [A]  time = 0.0062417, size = 22, normalized size = 0.96 \[ \frac{n \log (x)-\log \left (a+b x^n\right )}{a n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^(1 + n))^(-1),x]

[Out]

(n*Log[x] - Log[a + b*x^n])/(a*n)

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Maple [A]  time = 0.008, size = 39, normalized size = 1.7 \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)))

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Maxima [A]  time = 1.03796, size = 36, normalized size = 1.57 \begin{align*} \frac{\log \left (x\right )}{a} - \frac{\log \left (\frac{b x^{n} + a}{b}\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(x)/a - log((b*x^n + a)/b)/(a*n)

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Fricas [A]  time = 0.877325, size = 66, normalized size = 2.87 \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)

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Sympy [A]  time = 1.71174, size = 41, normalized size = 1.78 \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))

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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)