3.358 \(\int \frac{1}{2 x+3 x^{1+n}} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{2}-\frac{\log \left (3 x^n+2\right )}{2 n} \]

[Out]

Log[x]/2 - Log[2 + 3*x^n]/(2*n)

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Rubi [A]  time = 0.0097024, antiderivative size = 22, 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} \[ \frac{\log (x)}{2}-\frac{\log \left (3 x^n+2\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[(2*x + 3*x^(1 + n))^(-1),x]

[Out]

Log[x]/2 - Log[2 + 3*x^n]/(2*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}{2 x+3 x^{1+n}} \, dx &=\int \frac{1}{x \left (2+3 x^n\right )} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (2+3 x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^n\right )}{2 n}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{2+3 x} \, dx,x,x^n\right )}{2 n}\\ &=\frac{\log (x)}{2}-\frac{\log \left (2+3 x^n\right )}{2 n}\\ \end{align*}

Mathematica [A]  time = 0.0059449, size = 22, normalized size = 1. \[ \frac{n \log (x)-\log \left (3 x^n+2\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x + 3*x^(1 + n))^(-1),x]

[Out]

(n*Log[x] - Log[2 + 3*x^n])/(2*n)

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Maple [A]  time = 0.028, size = 32, normalized size = 1.5 \begin{align*}{\frac{\ln \left ( x \right ) }{2\,n}}+{\frac{\ln \left ( x \right ) }{2}}-{\frac{\ln \left ( 2\,x+3\,{{\rm e}^{ \left ( 1+n \right ) \ln \left ( x \right ) }} \right ) }{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x+3*x^(1+n)),x)

[Out]

1/2/n*ln(x)+1/2*ln(x)-1/2/n*ln(2*x+3*exp((1+n)*ln(x)))

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Maxima [A]  time = 1.11082, size = 22, normalized size = 1. \begin{align*} -\frac{\log \left (x^{n} + \frac{2}{3}\right )}{2 \, n} + \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x+3*x^(1+n)),x, algorithm="maxima")

[Out]

-1/2*log(x^n + 2/3)/n + 1/2*log(x)

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Fricas [A]  time = 0.831635, size = 66, normalized size = 3. \begin{align*} \frac{{\left (n + 1\right )} \log \left (x\right ) - \log \left (3 \, x^{n + 1} + 2 \, x\right )}{2 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x+3*x^(1+n)),x, algorithm="fricas")

[Out]

1/2*((n + 1)*log(x) - log(3*x^(n + 1) + 2*x))/n

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Sympy [A]  time = 1.989, size = 20, normalized size = 0.91 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} - \frac{\log{\left (x^{n} + \frac{2}{3} \right )}}{2 n} & \text{for}\: n \neq 0 \\\frac{\log{\left (x \right )}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x+3*x**(1+n)),x)

[Out]

Piecewise((log(x)/2 - log(x**n + 2/3)/(2*n), Ne(n, 0)), (log(x)/5, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{3 \, x^{n + 1} + 2 \, x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x+3*x^(1+n)),x, algorithm="giac")

[Out]

integrate(1/(3*x^(n + 1) + 2*x), x)