\(\int \frac {-390625 x-19530625 x^2+15625 x^3+(781250 x+19531250 x^2) \log (x+25 x^2)}{x^2+25 x^3+(1250 x+31250 x^2) \log (x+25 x^2)+(390625+9765625 x) \log ^2(x+25 x^2)} \, dx\) [3332]

Optimal result

Integrand size = 78, antiderivative size = 24 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {46}{3}+\frac {x^2}{\frac {x}{625}+\log \left (x+25 x^2\right )} \]

[Out]

x^2/(1/625*x+ln(25*x^2+x))+46/3

Rubi [F]

\[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx \]

[In]

Int[(-390625*x - 19530625*x^2 + 15625*x^3 + (781250*x + 19531250*x^2)*Log[x + 25*x^2])/(x^2 + 25*x^3 + (1250*x
+ 31250*x^2)*Log[x + 25*x^2] + (390625 + 9765625*x)*Log[x + 25*x^2]^2),x]

[Out]

15625*Defer[Int][(x + 625*Log[x*(1 + 25*x)])^(-2), x] - 781250*Defer[Int][x/(x + 625*Log[x*(1 + 25*x)])^2, x]
- 625*Defer[Int][x^2/(x + 625*Log[x*(1 + 25*x)])^2, x] - 15625*Defer[Int][1/((1 + 25*x)*(x + 625*Log[x*(1 + 25
*x)])^2), x] + 1250*Defer[Int][x/(x + 625*Log[x*(1 + 25*x)]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {625 x \left (-625-31249 x+25 x^2+1250 (1+25 x) \log (x (1+25 x))\right )}{(1+25 x) (x+625 \log (x (1+25 x)))^2} \, dx \\ & = 625 \int \frac {x \left (-625-31249 x+25 x^2+1250 (1+25 x) \log (x (1+25 x))\right )}{(1+25 x) (x+625 \log (x (1+25 x)))^2} \, dx \\ & = 625 \int \left (-\frac {x \left (625+31251 x+25 x^2\right )}{(1+25 x) (x+625 \log (x (1+25 x)))^2}+\frac {2 x}{x+625 \log (x (1+25 x))}\right ) \, dx \\ & = -\left (625 \int \frac {x \left (625+31251 x+25 x^2\right )}{(1+25 x) (x+625 \log (x (1+25 x)))^2} \, dx\right )+1250 \int \frac {x}{x+625 \log (x (1+25 x))} \, dx \\ & = -\left (625 \int \left (-\frac {25}{(x+625 \log (x (1+25 x)))^2}+\frac {1250 x}{(x+625 \log (x (1+25 x)))^2}+\frac {x^2}{(x+625 \log (x (1+25 x)))^2}+\frac {25}{(1+25 x) (x+625 \log (x (1+25 x)))^2}\right ) \, dx\right )+1250 \int \frac {x}{x+625 \log (x (1+25 x))} \, dx \\ & = -\left (625 \int \frac {x^2}{(x+625 \log (x (1+25 x)))^2} \, dx\right )+1250 \int \frac {x}{x+625 \log (x (1+25 x))} \, dx+15625 \int \frac {1}{(x+625 \log (x (1+25 x)))^2} \, dx-15625 \int \frac {1}{(1+25 x) (x+625 \log (x (1+25 x)))^2} \, dx-781250 \int \frac {x}{(x+625 \log (x (1+25 x)))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625 x^2}{x+625 \log (x (1+25 x))} \]

[In]

Integrate[(-390625*x - 19530625*x^2 + 15625*x^3 + (781250*x + 19531250*x^2)*Log[x + 25*x^2])/(x^2 + 25*x^3 + (
1250*x + 31250*x^2)*Log[x + 25*x^2] + (390625 + 9765625*x)*Log[x + 25*x^2]^2),x]

[Out]

(625*x^2)/(x + 625*Log[x*(1 + 25*x)])

Maple [A] (verified)

Time = 2.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

method	result	size
norman	\(\frac {625 x^{2}}{625 \ln \left (25 x^{2}+x \right )+x}\)	\(20\)
risch	\(\frac {625 x^{2}}{625 \ln \left (25 x^{2}+x \right )+x}\)	\(20\)
parallelrisch	\(\frac {625 x^{2}}{625 \ln \left (25 x^{2}+x \right )+x}\)	\(20\)

[In]

int(((19531250*x^2+781250*x)*ln(25*x^2+x)+15625*x^3-19530625*x^2-390625*x)/((9765625*x+390625)*ln(25*x^2+x)^2+
(31250*x^2+1250*x)*ln(25*x^2+x)+25*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

625*x^2/(625*ln(25*x^2+x)+x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625 \, x^{2}}{x + 625 \, \log \left (25 \, x^{2} + x\right )} \]

[In]

integrate(((19531250*x^2+781250*x)*log(25*x^2+x)+15625*x^3-19530625*x^2-390625*x)/((9765625*x+390625)*log(25*x
^2+x)^2+(31250*x^2+1250*x)*log(25*x^2+x)+25*x^3+x^2),x, algorithm="fricas")

[Out]

625*x^2/(x + 625*log(25*x^2 + x))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.58 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {x^{2}}{\frac {x}{625} + \log {\left (25 x^{2} + x \right )}} \]

[In]

integrate(((19531250*x**2+781250*x)*ln(25*x**2+x)+15625*x**3-19530625*x**2-390625*x)/((9765625*x+390625)*ln(25
*x**2+x)**2+(31250*x**2+1250*x)*ln(25*x**2+x)+25*x**3+x**2),x)

[Out]

x**2/(x/625 + log(25*x**2 + x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625 \, x^{2}}{x + 625 \, \log \left (25 \, x + 1\right ) + 625 \, \log \left (x\right )} \]

[In]

integrate(((19531250*x^2+781250*x)*log(25*x^2+x)+15625*x^3-19530625*x^2-390625*x)/((9765625*x+390625)*log(25*x
^2+x)^2+(31250*x^2+1250*x)*log(25*x^2+x)+25*x^3+x^2),x, algorithm="maxima")

[Out]

625*x^2/(x + 625*log(25*x + 1) + 625*log(x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625 \, x^{2}}{x + 625 \, \log \left (25 \, x^{2} + x\right )} \]

[In]

integrate(((19531250*x^2+781250*x)*log(25*x^2+x)+15625*x^3-19530625*x^2-390625*x)/((9765625*x+390625)*log(25*x
^2+x)^2+(31250*x^2+1250*x)*log(25*x^2+x)+25*x^3+x^2),x, algorithm="giac")

[Out]

625*x^2/(x + 625*log(25*x^2 + x))

Mupad [B] (verification not implemented)

Time = 9.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-390625 x-19530625 x^2+15625 x^3+\left (781250 x+19531250 x^2\right ) \log \left (x+25 x^2\right )}{x^2+25 x^3+\left (1250 x+31250 x^2\right ) \log \left (x+25 x^2\right )+(390625+9765625 x) \log ^2\left (x+25 x^2\right )} \, dx=\frac {625\,\left (2500\,x+1562500\,\ln \left (25\,x^2+x\right )+x^2\right )}{x+625\,\ln \left (25\,x^2+x\right )} \]

[In]

int(-(390625*x - log(x + 25*x^2)*(781250*x + 19531250*x^2) + 19530625*x^2 - 15625*x^3)/(log(x + 25*x^2)*(1250*
x + 31250*x^2) + log(x + 25*x^2)^2*(9765625*x + 390625) + x^2 + 25*x^3),x)

[Out]

(625*(2500*x + 1562500*log(x + 25*x^2) + x^2))/(x + 625*log(x + 25*x^2))