\(\int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 (4 x+x^2)} \, dx\) [6422]

Optimal result

Integrand size = 32, antiderivative size = 17 \[ \int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 \left (4 x+x^2\right )} \, dx=6 \log \left (\frac {x^2}{(4+x) (625+x)}\right ) \]

[Out]

6*ln(x^2/(x+625)/(4+x))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2023, 1608, 814} \[ \int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 \left (4 x+x^2\right )} \, dx=12 \log (x)-6 \log (x+4)-6 \log (x+625) \]

[In]

Int[(24*x + 625*(48 + 6*x))/(4*x^2 + x^3 + 625*(4*x + x^2)),x]

[Out]

12*Log[x] - 6*Log[4 + x] - 6*Log[625 + x]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 2023

Int[(u_)^(p_.)*(z_), x_Symbol] :> Int[ExpandToSum[z, x]*ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && BinomialQ[z,
x] && GeneralizedTrinomialQ[u, x] && EqQ[BinomialDegree[z, x] - GeneralizedTrinomialDegree[u, x], 0] &&  !(Bi
nomialMatchQ[z, x] && GeneralizedTrinomialMatchQ[u, x])

Rubi steps \begin{align*} \text {integral}& = \int \frac {30000+3774 x}{2500 x+629 x^2+x^3} \, dx \\ & = \int \frac {30000+3774 x}{x \left (2500+629 x+x^2\right )} \, dx \\ & = \int \left (\frac {12}{x}-\frac {6}{4+x}-\frac {6}{625+x}\right ) \, dx \\ & = 12 \log (x)-6 \log (4+x)-6 \log (625+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 \left (4 x+x^2\right )} \, dx=6 \left (2 \log (x)-\log \left (2500+629 x+x^2\right )\right ) \]

[In]

Integrate[(24*x + 625*(48 + 6*x))/(4*x^2 + x^3 + 625*(4*x + x^2)),x]

[Out]

6*(2*Log[x] - Log[2500 + 629*x + x^2])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

method	result	size
risch	\(12 \ln \left (x \right )-6 \ln \left (x^{2}+629 x +2500\right )\)	\(17\)
default	\(-6 \ln \left (x +625\right )+12 \ln \left (x \right )-6 \ln \left (4+x \right )\)	\(18\)
norman	\(-6 \ln \left (x +625\right )+12 \ln \left (x \right )-6 \ln \left (4+x \right )\)	\(18\)
parallelrisch	\(-6 \ln \left (x +625\right )+12 \ln \left (x \right )-6 \ln \left (4+x \right )\)	\(18\)

[In]

int((3774*x+30000)/(x^3+629*x^2+2500*x),x,method=_RETURNVERBOSE)

[Out]

12*ln(x)-6*ln(x^2+629*x+2500)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 \left (4 x+x^2\right )} \, dx=-6 \, \log \left (x^{2} + 629 \, x + 2500\right ) + 12 \, \log \left (x\right ) \]

[In]

integrate((3774*x+30000)/(x^3+629*x^2+2500*x),x, algorithm="fricas")

[Out]

-6*log(x^2 + 629*x + 2500) + 12*log(x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 \left (4 x+x^2\right )} \, dx=12 \log {\left (x \right )} - 6 \log {\left (x^{2} + 629 x + 2500 \right )} \]

[In]

integrate((3774*x+30000)/(x**3+629*x**2+2500*x),x)

[Out]

12*log(x) - 6*log(x**2 + 629*x + 2500)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 \left (4 x+x^2\right )} \, dx=-6 \, \log \left (x + 625\right ) - 6 \, \log \left (x + 4\right ) + 12 \, \log \left (x\right ) \]

[In]

integrate((3774*x+30000)/(x^3+629*x^2+2500*x),x, algorithm="maxima")

[Out]

-6*log(x + 625) - 6*log(x + 4) + 12*log(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 \left (4 x+x^2\right )} \, dx=-6 \, \log \left ({\left | x + 625 \right |}\right ) - 6 \, \log \left ({\left | x + 4 \right |}\right ) + 12 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((3774*x+30000)/(x^3+629*x^2+2500*x),x, algorithm="giac")

[Out]

-6*log(abs(x + 625)) - 6*log(abs(x + 4)) + 12*log(abs(x))

Mupad [B] (verification not implemented)

Time = 14.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {24 x+625 (48+6 x)}{4 x^2+x^3+625 \left (4 x+x^2\right )} \, dx=12\,\ln \left (x\right )-6\,\ln \left (x^2+629\,x+2500\right ) \]

[In]

int((3774*x + 30000)/(2500*x + 629*x^2 + x^3),x)

[Out]

12*log(x) - 6*log(629*x + x^2 + 2500)