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]

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

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.41, size = 28, normalized size = 1.22 \[ \frac {{\left (n + 1\right )} \log \relax (x) - \log \left (a x + b x^{n + 1}\right )}{a n} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a x + b x^{n + 1}}\,{d x} \]

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)

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maple [A]  time = 0.05, size = 39, normalized size = 1.70 \[ \frac {\ln \relax (x )}{a}+\frac {\ln \relax (x )}{a n}-\frac {\ln \left (a x +b \,{\mathrm e}^{\left (n +1\right ) \ln \relax (x )}\right )}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 27, normalized size = 1.17 \[ \frac {\log \relax (x)}{a} - \frac {\log \left (\frac {b x^{n} + a}{b}\right )}{a n} \]

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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mupad [B]  time = 5.23, size = 31, normalized size = 1.35 \[ \frac {\ln \relax (x)\,\left (n+1\right )}{a\,n}-\frac {\ln \left (x\,\left (a+b\,x^n\right )\right )}{a\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x + b*x^(n + 1)),x)

[Out]

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

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sympy [A]  time = 1.81, size = 41, normalized size = 1.78 \[ \begin {cases} \tilde {\infty } \log {\relax (x )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b} & \text {for}\: n = 0 \\- \frac {x^{- n}}{b n} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{n} \right )}}{a n} & \text {otherwise} \end {cases} \]

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

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