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

Optimal. Leaf size=15 \[ \frac {\log \left (2 x^n+3\right )}{2 n} \]

[Out]

1/2*ln(3+2*x^n)/n

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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} \[ \frac {\log \left (2 x^n+3\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[(2*x + 3*x^(1 - n))^(-1),x]

[Out]

Log[3 + 2*x^n]/(2*n)

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]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\log \left (2 x^n+3\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x + 3*x^(1 - n))^(-1),x]

[Out]

Log[3 + 2*x^n]/(2*n)

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 26, normalized size = 1.73 \[ \frac {{\left (n - 1\right )} \log \relax (x) + \log \left (3 \, x^{-n + 1} + 2 \, x\right )}{2 \, n} \]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{3 \, x^{-n + 1} + 2 \, x}\,{d x} \]

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)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 34, normalized size = 2.27 \[ \frac {\ln \relax (x )}{2}-\frac {\ln \relax (x )}{2 n}+\frac {\ln \left (2 x +3 \,{\mathrm e}^{\left (-n +1\right ) \ln \relax (x )}\right )}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 11, normalized size = 0.73 \[ \frac {\log \left (x^{n} + \frac {3}{2}\right )}{2 \, n} \]

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 + 3/2)/n

________________________________________________________________________________________

mupad [B]  time = 5.20, size = 28, normalized size = 1.87 \[ \frac {\ln \left (\frac {2\,x}{3}+x^{1-n}\right )}{2\,n}+\frac {\ln \relax (x)\,\left (n-1\right )}{2\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x + 3*x^(1 - n)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.70, size = 22, normalized size = 1.47 \[ \begin {cases} \frac {\log {\relax (x )}}{2} + \frac {\log {\left (\frac {2}{3} + x^{- n} \right )}}{2 n} & \text {for}\: n \neq 0 \\\frac {\log {\relax (x )}}{5} & \text {otherwise} \end {cases} \]

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

________________________________________________________________________________________