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]

Log[3 + 2*x^n]/(2*n)

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Rubi [A]  time = 0.0063023, 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} \[ \frac{\log \left (2 x^n+3\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[(2*x + 3*x^(1 - n))^(-1),x]

[Out]

Log[3 + 2*x^n]/(2*n)

Rule 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}{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 = 0.0033049, size = 15, normalized size = 1. \[ \frac{\log \left (2 x^n+3\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x + 3*x^(1 - n))^(-1),x]

[Out]

Log[3 + 2*x^n]/(2*n)

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Maple [B]  time = 0.009, size = 34, normalized size = 2.3 \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 = 1.04812, size = 15, normalized size = 1. \begin{align*} \frac{\log \left (x^{n} + \frac{3}{2}\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="maxima")

[Out]

1/2*log(x^n + 3/2)/n

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Fricas [A]  time = 0.722041, size = 68, normalized size = 4.53 \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 = 1.64235, size = 22, normalized size = 1.47 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} + \frac{\log{\left (\frac{2}{3} + x^{- n} \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(2/3 + x**(-n))/(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)