3.299 \(\int \frac {2+x^2}{(-5+x) (-3+x) (4+x)} \, dx\)

Optimal. Leaf size=29 \[ -\frac {11}{14} \log (3-x)+\frac {3}{2} \log (5-x)+\frac {2}{7} \log (x+4) \]

[Out]

-11/14*ln(3-x)+3/2*ln(5-x)+2/7*ln(4+x)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1612} \[ -\frac {11}{14} \log (3-x)+\frac {3}{2} \log (5-x)+\frac {2}{7} \log (x+4) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x^2)/((-5 + x)*(-3 + x)*(4 + x)),x]

[Out]

(-11*Log[3 - x])/14 + (3*Log[5 - x])/2 + (2*Log[4 + x])/7

Rule 1612

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[E
xpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && Poly
Q[Px, x] && IntegersQ[m, n]

Rubi steps

\begin {align*} \int \frac {2+x^2}{(-5+x) (-3+x) (4+x)} \, dx &=\int \left (\frac {3}{2 (-5+x)}-\frac {11}{14 (-3+x)}+\frac {2}{7 (4+x)}\right ) \, dx\\ &=-\frac {11}{14} \log (3-x)+\frac {3}{2} \log (5-x)+\frac {2}{7} \log (4+x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \[ -\frac {11}{14} \log (3-x)+\frac {3}{2} \log (5-x)+\frac {2}{7} \log (x+4) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x^2)/((-5 + x)*(-3 + x)*(4 + x)),x]

[Out]

(-11*Log[3 - x])/14 + (3*Log[5 - x])/2 + (2*Log[4 + x])/7

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 19, normalized size = 0.66 \[ \frac {2}{7} \, \log \left (x + 4\right ) - \frac {11}{14} \, \log \left (x - 3\right ) + \frac {3}{2} \, \log \left (x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2)/(-5+x)/(-3+x)/(4+x),x, algorithm="fricas")

[Out]

2/7*log(x + 4) - 11/14*log(x - 3) + 3/2*log(x - 5)

________________________________________________________________________________________

giac [A]  time = 0.27, size = 22, normalized size = 0.76 \[ \frac {2}{7} \, \log \left ({\left | x + 4 \right |}\right ) - \frac {11}{14} \, \log \left ({\left | x - 3 \right |}\right ) + \frac {3}{2} \, \log \left ({\left | x - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2)/(-5+x)/(-3+x)/(4+x),x, algorithm="giac")

[Out]

2/7*log(abs(x + 4)) - 11/14*log(abs(x - 3)) + 3/2*log(abs(x - 5))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 20, normalized size = 0.69 \[ \frac {3 \ln \left (x -5\right )}{2}-\frac {11 \ln \left (x -3\right )}{14}+\frac {2 \ln \left (x +4\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+2)/(x-5)/(x-3)/(x+4),x)

[Out]

2/7*ln(x+4)-11/14*ln(x-3)+3/2*ln(x-5)

________________________________________________________________________________________

maxima [A]  time = 1.17, size = 19, normalized size = 0.66 \[ \frac {2}{7} \, \log \left (x + 4\right ) - \frac {11}{14} \, \log \left (x - 3\right ) + \frac {3}{2} \, \log \left (x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2)/(-5+x)/(-3+x)/(4+x),x, algorithm="maxima")

[Out]

2/7*log(x + 4) - 11/14*log(x - 3) + 3/2*log(x - 5)

________________________________________________________________________________________

mupad [B]  time = 2.18, size = 19, normalized size = 0.66 \[ \frac {2\,\ln \left (x+4\right )}{7}-\frac {11\,\ln \left (x-3\right )}{14}+\frac {3\,\ln \left (x-5\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 2)/((x - 3)*(x + 4)*(x - 5)),x)

[Out]

(2*log(x + 4))/7 - (11*log(x - 3))/14 + (3*log(x - 5))/2

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 24, normalized size = 0.83 \[ \frac {3 \log {\left (x - 5 \right )}}{2} - \frac {11 \log {\left (x - 3 \right )}}{14} + \frac {2 \log {\left (x + 4 \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+2)/(-5+x)/(-3+x)/(4+x),x)

[Out]

3*log(x - 5)/2 - 11*log(x - 3)/14 + 2*log(x + 4)/7

________________________________________________________________________________________