\(\int \frac {-432+216 x-3 x^2+36 \log (5)}{1296-72 x+x^2} \, dx\) [9864]

Optimal result

Integrand size = 25, antiderivative size = 19 \[ \int \frac {-432+216 x-3 x^2+36 \log (5)}{1296-72 x+x^2} \, dx=\frac {x (3 (4-x)-\log (5))}{-36+x} \]

[Out]

(-3*x+12-ln(5))/(x-36)*x

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 697} \[ \int \frac {-432+216 x-3 x^2+36 \log (5)}{1296-72 x+x^2} \, dx=\frac {36 (96+\log (5))}{36-x}-3 x \]

[In]

Int[(-432 + 216*x - 3*x^2 + 36*Log[5])/(1296 - 72*x + x^2),x]

[Out]

-3*x + (36*(96 + Log[5]))/(36 - 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 {-432+216 x-3 x^2+36 \log (5)}{(-36+x)^2} \, dx \\ & = \int \left (-3+\frac {36 (96+\log (5))}{(-36+x)^2}\right ) \, dx \\ & = -3 x+\frac {36 (96+\log (5))}{36-x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-432+216 x-3 x^2+36 \log (5)}{1296-72 x+x^2} \, dx=-3 \left (x+\frac {12 (96+\log (5))}{-36+x}\right ) \]

[In]

Integrate[(-432 + 216*x - 3*x^2 + 36*Log[5])/(1296 - 72*x + x^2),x]

[Out]

-3*(x + (12*(96 + Log[5]))/(-36 + x))

Maple [A] (verified)

Time = 1.35 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

method	result	size
gosper	\(-\frac {3 \left (x^{2}+12 \ln \left (5\right )-144\right )}{x -36}\)	\(17\)
default	\(-3 x -\frac {3 \left (1152+12 \ln \left (5\right )\right )}{x -36}\)	\(18\)
norman	\(\frac {-3 x^{2}+432-36 \ln \left (5\right )}{x -36}\)	\(18\)
parallelrisch	\(-\frac {3 x^{2}-432+36 \ln \left (5\right )}{x -36}\)	\(19\)
risch	\(-3 x -\frac {3456}{x -36}-\frac {36 \ln \left (5\right )}{x -36}\)	\(21\)
meijerg	\(\frac {17 x}{3 \left (1-\frac {x}{36}\right )}+\frac {x \ln \left (5\right )}{-x +36}-\frac {x \left (-\frac {x}{12}+6\right )}{1-\frac {x}{36}}\)	\(39\)

[In]

int((36*ln(5)-3*x^2+216*x-432)/(x^2-72*x+1296),x,method=_RETURNVERBOSE)

[Out]

-3*(x^2+12*ln(5)-144)/(x-36)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-432+216 x-3 x^2+36 \log (5)}{1296-72 x+x^2} \, dx=-\frac {3 \, {\left (x^{2} - 36 \, x + 12 \, \log \left (5\right ) + 1152\right )}}{x - 36} \]

[In]

integrate((36*log(5)-3*x^2+216*x-432)/(x^2-72*x+1296),x, algorithm="fricas")

[Out]

-3*(x^2 - 36*x + 12*log(5) + 1152)/(x - 36)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {-432+216 x-3 x^2+36 \log (5)}{1296-72 x+x^2} \, dx=- 3 x - \frac {36 \log {\left (5 \right )} + 3456}{x - 36} \]

[In]

integrate((36*ln(5)-3*x**2+216*x-432)/(x**2-72*x+1296),x)

[Out]

-3*x - (36*log(5) + 3456)/(x - 36)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-432+216 x-3 x^2+36 \log (5)}{1296-72 x+x^2} \, dx=-3 \, x - \frac {36 \, {\left (\log \left (5\right ) + 96\right )}}{x - 36} \]

[In]

integrate((36*log(5)-3*x^2+216*x-432)/(x^2-72*x+1296),x, algorithm="maxima")

[Out]

-3*x - 36*(log(5) + 96)/(x - 36)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-432+216 x-3 x^2+36 \log (5)}{1296-72 x+x^2} \, dx=-3 \, x - \frac {36 \, {\left (\log \left (5\right ) + 96\right )}}{x - 36} \]

[In]

integrate((36*log(5)-3*x^2+216*x-432)/(x^2-72*x+1296),x, algorithm="giac")

[Out]

-3*x - 36*(log(5) + 96)/(x - 36)

Mupad [B] (verification not implemented)

Time = 17.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-432+216 x-3 x^2+36 \log (5)}{1296-72 x+x^2} \, dx=-3\,x-\frac {36\,\ln \left (5\right )+3456}{x-36} \]

[In]

int((216*x + 36*log(5) - 3*x^2 - 432)/(x^2 - 72*x + 1296),x)

[Out]

- 3*x - (36*log(5) + 3456)/(x - 36)