\(\int \frac {(-250 x^2+50 x^3) \log (x) \log (3+x)+(375 x+50 x^2-25 x^3+(-750 x-250 x^2) \log (x)) \log ^2(3+x)}{(-54000+14400 x+4320 x^2-1728 x^3+144 x^4) \log ^2(x)} \, dx\) [5656]

Optimal result

Integrand size = 79, antiderivative size = 24 \[ \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx=\frac {25 x^2 \log ^2(3+x)}{144 (5-x)^2 \log (x)} \]

[Out]

25/144*x^2/(5-x)^2*ln(3+x)^2/ln(x)

Rubi [F]

\[ \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx=\int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx \]

[In]

Int[((-250*x^2 + 50*x^3)*Log[x]*Log[3 + x] + (375*x + 50*x^2 - 25*x^3 + (-750*x - 250*x^2)*Log[x])*Log[3 + x]^
2)/((-54000 + 14400*x + 4320*x^2 - 1728*x^3 + 144*x^4)*Log[x]^2),x]

[Out]

(625*Defer[Int][Log[3 + x]/((-5 + x)^2*Log[x]), x])/576 + (1375*Defer[Int][Log[3 + x]/((-5 + x)*Log[x]), x])/4
608 + (25*Defer[Int][Log[3 + x]/((3 + x)*Log[x]), x])/512 - (125*Defer[Int][Log[3 + x]^2/((-5 + x)^2*Log[x]^2)
, x])/144 - (25*Defer[Int][Log[3 + x]^2/((-5 + x)*Log[x]^2), x])/144 - (625*Defer[Int][Log[3 + x]^2/((-5 + x)^
3*Log[x]), x])/72 - (125*Defer[Int][Log[3 + x]^2/((-5 + x)^2*Log[x]), x])/72

Rubi steps \begin{align*} \text {integral}& = \int \frac {25 x \log (3+x) \left (\left (-15-2 x+x^2\right ) \log (3+x)-2 \log (x) ((-5+x) x-5 (3+x) \log (3+x))\right )}{144 (5-x)^3 (3+x) \log ^2(x)} \, dx \\ & = \frac {25}{144} \int \frac {x \log (3+x) \left (\left (-15-2 x+x^2\right ) \log (3+x)-2 \log (x) ((-5+x) x-5 (3+x) \log (3+x))\right )}{(5-x)^3 (3+x) \log ^2(x)} \, dx \\ & = \frac {25}{144} \int \left (\frac {2 x^2 \log (3+x)}{(-5+x)^2 (3+x) \log (x)}-\frac {x (-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}\right ) \, dx \\ & = -\left (\frac {25}{144} \int \frac {x (-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx\right )+\frac {25}{72} \int \frac {x^2 \log (3+x)}{(-5+x)^2 (3+x) \log (x)} \, dx \\ & = -\left (\frac {25}{144} \int \left (\frac {5 (-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}+\frac {(-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^2 \log ^2(x)}\right ) \, dx\right )+\frac {25}{72} \int \left (\frac {25 \log (3+x)}{8 (-5+x)^2 \log (x)}+\frac {55 \log (3+x)}{64 (-5+x) \log (x)}+\frac {9 \log (3+x)}{64 (3+x) \log (x)}\right ) \, dx \\ & = \frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \frac {(-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}-\frac {125}{144} \int \frac {(-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx \\ & = \frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \left (-\frac {5 \log ^2(3+x)}{(-5+x)^2 \log ^2(x)}+\frac {x \log ^2(3+x)}{(-5+x)^2 \log ^2(x)}+\frac {10 \log ^2(3+x)}{(-5+x)^2 \log (x)}\right ) \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}-\frac {125}{144} \int \left (-\frac {5 \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}+\frac {x \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}+\frac {10 \log ^2(3+x)}{(-5+x)^3 \log (x)}\right ) \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx \\ & = \frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \frac {x \log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}+\frac {125}{144} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx-\frac {125}{144} \int \frac {x \log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx-\frac {125}{72} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log (x)} \, dx+\frac {625}{144} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx-\frac {625}{72} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log (x)} \, dx \\ & = \frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \left (\frac {5 \log ^2(3+x)}{(-5+x)^2 \log ^2(x)}+\frac {\log ^2(3+x)}{(-5+x) \log ^2(x)}\right ) \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}+\frac {125}{144} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx-\frac {125}{144} \int \left (\frac {5 \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}+\frac {\log ^2(3+x)}{(-5+x)^2 \log ^2(x)}\right ) \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx-\frac {125}{72} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log (x)} \, dx+\frac {625}{144} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx-\frac {625}{72} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log (x)} \, dx \\ & = \frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \frac {\log ^2(3+x)}{(-5+x) \log ^2(x)} \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}-\frac {125}{144} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx-\frac {125}{72} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log (x)} \, dx-\frac {625}{72} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx=\frac {25 x^2 \log ^2(3+x)}{144 (-5+x)^2 \log (x)} \]

[In]

Integrate[((-250*x^2 + 50*x^3)*Log[x]*Log[3 + x] + (375*x + 50*x^2 - 25*x^3 + (-750*x - 250*x^2)*Log[x])*Log[3
+ x]^2)/((-54000 + 14400*x + 4320*x^2 - 1728*x^3 + 144*x^4)*Log[x]^2),x]

[Out]

(25*x^2*Log[3 + x]^2)/(144*(-5 + x)^2*Log[x])

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08

method	result	size
risch	\(\frac {25 x^{2} \ln \left (3+x \right )^{2}}{144 \left (x^{2}-10 x +25\right ) \ln \left (x \right )}\)	\(26\)
parallelrisch	\(\frac {25 x^{2} \ln \left (3+x \right )^{2}}{144 \left (x^{2}-10 x +25\right ) \ln \left (x \right )}\)	\(26\)

[In]

int((((-250*x^2-750*x)*ln(x)-25*x^3+50*x^2+375*x)*ln(3+x)^2+(50*x^3-250*x^2)*ln(x)*ln(3+x))/(144*x^4-1728*x^3+
4320*x^2+14400*x-54000)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

25/144*x^2/(x^2-10*x+25)/ln(x)*ln(3+x)^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx=\frac {25 \, x^{2} \log \left (x + 3\right )^{2}}{144 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (x\right )} \]

[In]

integrate((((-250*x^2-750*x)*log(x)-25*x^3+50*x^2+375*x)*log(3+x)^2+(50*x^3-250*x^2)*log(x)*log(3+x))/(144*x^4
-1728*x^3+4320*x^2+14400*x-54000)/log(x)^2,x, algorithm="fricas")

[Out]

25/144*x^2*log(x + 3)^2/((x^2 - 10*x + 25)*log(x))

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((((-250*x**2-750*x)*ln(x)-25*x**3+50*x**2+375*x)*ln(3+x)**2+(50*x**3-250*x**2)*ln(x)*ln(3+x))/(144*x
**4-1728*x**3+4320*x**2+14400*x-54000)/ln(x)**2,x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx=\frac {25 \, x^{2} \log \left (x + 3\right )^{2}}{144 \, {\left (x^{2} - 10 \, x + 25\right )} \log \left (x\right )} \]

[In]

integrate((((-250*x^2-750*x)*log(x)-25*x^3+50*x^2+375*x)*log(3+x)^2+(50*x^3-250*x^2)*log(x)*log(3+x))/(144*x^4
-1728*x^3+4320*x^2+14400*x-54000)/log(x)^2,x, algorithm="maxima")

[Out]

25/144*x^2*log(x + 3)^2/((x^2 - 10*x + 25)*log(x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx=\frac {25 \, x^{2} \log \left (x + 3\right )^{2}}{144 \, {\left (x^{2} \log \left (x\right ) - 10 \, x \log \left (x\right ) + 25 \, \log \left (x\right )\right )}} \]

[In]

integrate((((-250*x^2-750*x)*log(x)-25*x^3+50*x^2+375*x)*log(3+x)^2+(50*x^3-250*x^2)*log(x)*log(3+x))/(144*x^4
-1728*x^3+4320*x^2+14400*x-54000)/log(x)^2,x, algorithm="giac")

[Out]

25/144*x^2*log(x + 3)^2/(x^2*log(x) - 10*x*log(x) + 25*log(x))

Mupad [B] (verification not implemented)

Time = 11.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx=\frac {25\,x^2\,{\ln \left (x+3\right )}^2}{144\,\ln \left (x\right )\,{\left (x-5\right )}^2} \]

[In]

int((log(x + 3)^2*(375*x - log(x)*(750*x + 250*x^2) + 50*x^2 - 25*x^3) - log(x + 3)*log(x)*(250*x^2 - 50*x^3))
/(log(x)^2*(14400*x + 4320*x^2 - 1728*x^3 + 144*x^4 - 54000)),x)

[Out]

(25*x^2*log(x + 3)^2)/(144*log(x)*(x - 5)^2)