\(\int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+(375 x+125 x^2) \log (x)+(75 x+25 x^2) \log (3+x)} \, dx\) [7735]

Optimal result

Integrand size = 60, antiderivative size = 21 \[ \int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+\left (375 x+125 x^2\right ) \log (x)+\left (75 x+25 x^2\right ) \log (3+x)} \, dx=-\log \left (-\frac {x^2}{25}+5 (x+\log (x))+\log (3+x)\right ) \]

[Out]

-ln(ln(3+x)+5*x+5*ln(x)-1/25*x^2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6820, 6816} \[ \int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+\left (375 x+125 x^2\right ) \log (x)+\left (75 x+25 x^2\right ) \log (3+x)} \, dx=-\log ((125-x) x+125 \log (x)+25 \log (x+3)) \]

[In]

Int[(-375 - 525*x - 119*x^2 + 2*x^3)/(375*x^2 + 122*x^3 - x^4 + (375*x + 125*x^2)*Log[x] + (75*x + 25*x^2)*Log
[3 + x]),x]

[Out]

-Log[(125 - x)*x + 125*Log[x] + 25*Log[3 + x]]

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {375+525 x+119 x^2-2 x^3}{x (3+x) ((-125+x) x-125 \log (x)-25 \log (3+x))} \, dx \\ & = -\log ((125-x) x+125 \log (x)+25 \log (3+x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+\left (375 x+125 x^2\right ) \log (x)+\left (75 x+25 x^2\right ) \log (3+x)} \, dx=-\log \left (-125 x+x^2-125 \log (x)-25 \log (3+x)\right ) \]

[In]

Integrate[(-375 - 525*x - 119*x^2 + 2*x^3)/(375*x^2 + 122*x^3 - x^4 + (375*x + 125*x^2)*Log[x] + (75*x + 25*x^
2)*Log[3 + x]),x]

[Out]

-Log[-125*x + x^2 - 125*Log[x] - 25*Log[3 + x]]

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\ln \left (\ln \left (3+x \right )+5 x +5 \ln \left (x \right )-\frac {x^{2}}{25}\right )\)	\(21\)
parallelrisch	\(-\ln \left (x^{2}-125 x -125 \ln \left (x \right )-25 \ln \left (3+x \right )\right )\)	\(21\)
default	\(-\ln \left (-x^{2}+125 \ln \left (x \right )+25 \ln \left (3+x \right )+125 x \right )\)	\(23\)

[In]

int((2*x^3-119*x^2-525*x-375)/((25*x^2+75*x)*ln(3+x)+(125*x^2+375*x)*ln(x)-x^4+122*x^3+375*x^2),x,method=_RETU
RNVERBOSE)

[Out]

-ln(ln(3+x)+5*x+5*ln(x)-1/25*x^2)

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+\left (375 x+125 x^2\right ) \log (x)+\left (75 x+25 x^2\right ) \log (3+x)} \, dx=-\log \left (-x^{2} + 125 \, x + 25 \, \log \left (x + 3\right ) + 125 \, \log \left (x\right )\right ) \]

[In]

integrate((2*x^3-119*x^2-525*x-375)/((25*x^2+75*x)*log(3+x)+(125*x^2+375*x)*log(x)-x^4+122*x^3+375*x^2),x, alg
orithm="fricas")

[Out]

-log(-x^2 + 125*x + 25*log(x + 3) + 125*log(x))

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+\left (375 x+125 x^2\right ) \log (x)+\left (75 x+25 x^2\right ) \log (3+x)} \, dx=- \log {\left (- \frac {x^{2}}{25} + 5 x + 5 \log {\left (x \right )} + \log {\left (x + 3 \right )} \right )} \]

[In]

integrate((2*x**3-119*x**2-525*x-375)/((25*x**2+75*x)*ln(3+x)+(125*x**2+375*x)*ln(x)-x**4+122*x**3+375*x**2),x
)

[Out]

-log(-x**2/25 + 5*x + 5*log(x) + log(x + 3))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+\left (375 x+125 x^2\right ) \log (x)+\left (75 x+25 x^2\right ) \log (3+x)} \, dx=-\log \left (-\frac {1}{25} \, x^{2} + 5 \, x + \log \left (x + 3\right ) + 5 \, \log \left (x\right )\right ) \]

[In]

integrate((2*x^3-119*x^2-525*x-375)/((25*x^2+75*x)*log(3+x)+(125*x^2+375*x)*log(x)-x^4+122*x^3+375*x^2),x, alg
orithm="maxima")

[Out]

-log(-1/25*x^2 + 5*x + log(x + 3) + 5*log(x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+\left (375 x+125 x^2\right ) \log (x)+\left (75 x+25 x^2\right ) \log (3+x)} \, dx=-\log \left (-x^{2} + 125 \, x + 25 \, \log \left (x + 3\right ) + 125 \, \log \left (x\right )\right ) \]

[In]

integrate((2*x^3-119*x^2-525*x-375)/((25*x^2+75*x)*log(3+x)+(125*x^2+375*x)*log(x)-x^4+122*x^3+375*x^2),x, alg
orithm="giac")

[Out]

-log(-x^2 + 125*x + 25*log(x + 3) + 125*log(x))

Mupad [B] (verification not implemented)

Time = 13.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-375-525 x-119 x^2+2 x^3}{375 x^2+122 x^3-x^4+\left (375 x+125 x^2\right ) \log (x)+\left (75 x+25 x^2\right ) \log (3+x)} \, dx=-\ln \left (5\,x+\ln \left (x+3\right )+5\,\ln \left (x\right )-\frac {x^2}{25}\right ) \]

[In]

int(-(525*x + 119*x^2 - 2*x^3 + 375)/(log(x + 3)*(75*x + 25*x^2) + log(x)*(375*x + 125*x^2) + 375*x^2 + 122*x^
3 - x^4),x)

[Out]

-log(5*x + log(x + 3) + 5*log(x) - x^2/25)