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 = 0.0153048, 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 \[ \frac{\log \left (2 x^n+3\right )}{2 n} \]

Antiderivative was successfully verified.

[In]  Int[(2*x + 3*x^(1 - n))^(-1),x]

[Out]

Log[3 + 2*x^n]/(2*n)

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Rubi in Sympy [A]  time = 2.72339, size = 10, normalized size = 0.67 \[ \frac{\log{\left (2 x^{n} + 3 \right )}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*x+3*x**(1-n)),x)

[Out]

log(2*x**n + 3)/(2*n)

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Mathematica [A]  time = 0.00513285, 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 = 0.016, size = 34, normalized size = 2.3 \[ -{\frac{\ln \left ( x \right ) }{2\,n}}+{\frac{\ln \left ( x \right ) }{2}}+{\frac{\ln \left ( 2\,x+3\,{{\rm e}^{ \left ( 1-n \right ) \ln \left ( x \right ) }} \right ) }{2\,n}} \]

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 = 1.37878, size = 15, normalized size = 1. \[ \frac{\log \left (x^{n} + \frac{3}{2}\right )}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(x^n + 3/2)/n

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Fricas [A]  time = 0.23396, size = 35, normalized size = 2.33 \[ \frac{{\left (n - 1\right )} \log \left (x\right ) + \log \left (3 \, x^{-n + 1} + 2 \, x\right )}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x^(-n + 1) + 2*x),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 = 5.03032, size = 22, normalized size = 1.47 \[ \begin{cases} \frac{\log{\left (x \right )}}{2} + \frac{\log{\left (\frac{2}{3} + x^{- n} \right )}}{2 n} & \text{for}\: n \neq 0 \\\frac{\log{\left (x \right )}}{5} & \text{otherwise} \end{cases} \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(3*x^(-n + 1) + 2*x), x)