\(\int \frac {-70-20 x-x^2}{300+60 x+3 x^2} \, dx\) [8544]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {-70-20 x-x^2}{300+60 x+3 x^2} \, dx=\frac {1}{3} \left (-x+3 \left (2+\frac {x}{10+x}\right )-\log (5)\right ) \]

[Out]

-1/3*x-1/3*ln(5)+x/(x+10)+2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {27, 12, 697} \[ \int \frac {-70-20 x-x^2}{300+60 x+3 x^2} \, dx=-\frac {x}{3}-\frac {10}{x+10} \]

[In]

Int[(-70 - 20*x - x^2)/(300 + 60*x + 3*x^2),x]

[Out]

-1/3*x - 10/(10 + x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 697

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \frac {-70-20 x-x^2}{3 (10+x)^2} \, dx \\ & = \frac {1}{3} \int \frac {-70-20 x-x^2}{(10+x)^2} \, dx \\ & = \frac {1}{3} \int \left (-1+\frac {30}{(10+x)^2}\right ) \, dx \\ & = -\frac {x}{3}-\frac {10}{10+x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {-70-20 x-x^2}{300+60 x+3 x^2} \, dx=\frac {1}{3} \left (-x-\frac {30}{10+x}\right ) \]

[In]

Integrate[(-70 - 20*x - x^2)/(300 + 60*x + 3*x^2),x]

[Out]

(-x - 30/(10 + x))/3

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52

method	result	size
default	\(-\frac {x}{3}-\frac {10}{x +10}\)	\(12\)
risch	\(-\frac {x}{3}-\frac {10}{x +10}\)	\(12\)
gosper	\(-\frac {x^{2}-70}{3 \left (x +10\right )}\)	\(13\)
parallelrisch	\(-\frac {x^{2}-70}{3 \left (x +10\right )}\)	\(13\)
norman	\(\frac {-\frac {x^{2}}{3}+\frac {70}{3}}{x +10}\)	\(14\)
meijerg	\(\frac {13 x}{30 \left (1+\frac {x}{10}\right )}-\frac {x \left (6+\frac {3 x}{10}\right )}{9 \left (1+\frac {x}{10}\right )}\)	\(27\)

[In]

int((-x^2-20*x-70)/(3*x^2+60*x+300),x,method=_RETURNVERBOSE)

[Out]

-1/3*x-10/(x+10)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {-70-20 x-x^2}{300+60 x+3 x^2} \, dx=-\frac {x^{2} + 10 \, x + 30}{3 \, {\left (x + 10\right )}} \]

[In]

integrate((-x^2-20*x-70)/(3*x^2+60*x+300),x, algorithm="fricas")

[Out]

-1/3*(x^2 + 10*x + 30)/(x + 10)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.35 \[ \int \frac {-70-20 x-x^2}{300+60 x+3 x^2} \, dx=- \frac {x}{3} - \frac {10}{x + 10} \]

[In]

integrate((-x**2-20*x-70)/(3*x**2+60*x+300),x)

[Out]

-x/3 - 10/(x + 10)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int \frac {-70-20 x-x^2}{300+60 x+3 x^2} \, dx=-\frac {1}{3} \, x - \frac {10}{x + 10} \]

[In]

integrate((-x^2-20*x-70)/(3*x^2+60*x+300),x, algorithm="maxima")

[Out]

-1/3*x - 10/(x + 10)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int \frac {-70-20 x-x^2}{300+60 x+3 x^2} \, dx=-\frac {1}{3} \, x - \frac {10}{x + 10} \]

[In]

integrate((-x^2-20*x-70)/(3*x^2+60*x+300),x, algorithm="giac")

[Out]

-1/3*x - 10/(x + 10)

Mupad [B] (verification not implemented)

Time = 13.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int \frac {-70-20 x-x^2}{300+60 x+3 x^2} \, dx=-\frac {x}{3}-\frac {10}{x+10} \]

[In]

int(-(20*x + x^2 + 70)/(60*x + 3*x^2 + 300),x)

[Out]

- x/3 - 10/(x + 10)