3.82 \(\int \frac{1}{2+5 x-3 x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{7} \log (3 x+1)-\frac{1}{7} \log (2-x) \]

[Out]

-Log[2 - x]/7 + Log[1 + 3*x]/7

_______________________________________________________________________________________

Rubi [A]  time = 0.0144738, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{7} \log (3 x+1)-\frac{1}{7} \log (2-x) \]

Antiderivative was successfully verified.

[In]  Int[(2 + 5*x - 3*x^2)^(-1),x]

[Out]

-Log[2 - x]/7 + Log[1 + 3*x]/7

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.65507, size = 14, normalized size = 0.67 \[ - \frac{\log{\left (- x + 2 \right )}}{7} + \frac{\log{\left (3 x + 1 \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-3*x**2+5*x+2),x)

[Out]

-log(-x + 2)/7 + log(3*x + 1)/7

_______________________________________________________________________________________

Mathematica [A]  time = 0.0052762, size = 21, normalized size = 1. \[ \frac{1}{7} \log (3 x+1)-\frac{1}{7} \log (2-x) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 5*x - 3*x^2)^(-1),x]

[Out]

-Log[2 - x]/7 + Log[1 + 3*x]/7

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 16, normalized size = 0.8 \[{\frac{\ln \left ( 3\,x+1 \right ) }{7}}-{\frac{\ln \left ( x-2 \right ) }{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^2+5*x+2),x)

[Out]

1/7*ln(3*x+1)-1/7*ln(x-2)

_______________________________________________________________________________________

Maxima [A]  time = 0.728245, size = 20, normalized size = 0.95 \[ \frac{1}{7} \, \log \left (3 \, x + 1\right ) - \frac{1}{7} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*x^2 - 5*x - 2),x, algorithm="maxima")

[Out]

1/7*log(3*x + 1) - 1/7*log(x - 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.213459, size = 20, normalized size = 0.95 \[ \frac{1}{7} \, \log \left (3 \, x + 1\right ) - \frac{1}{7} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*x^2 - 5*x - 2),x, algorithm="fricas")

[Out]

1/7*log(3*x + 1) - 1/7*log(x - 2)

_______________________________________________________________________________________

Sympy [A]  time = 0.211821, size = 14, normalized size = 0.67 \[ - \frac{\log{\left (x - 2 \right )}}{7} + \frac{\log{\left (x + \frac{1}{3} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-3*x**2+5*x+2),x)

[Out]

-log(x - 2)/7 + log(x + 1/3)/7

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210156, size = 23, normalized size = 1.1 \[ \frac{1}{7} \,{\rm ln}\left ({\left | 3 \, x + 1 \right |}\right ) - \frac{1}{7} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*x^2 - 5*x - 2),x, algorithm="giac")

[Out]

1/7*ln(abs(3*x + 1)) - 1/7*ln(abs(x - 2))