\(\int \frac {100-160 x-20 x^2+6 x^3+e^x (150 x+30 x^2-10 x^3)}{(-25 x+30 x^2-x^4+e^x (-25 x^2+5 x^3)) \log ^2(\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x (50 x^5-50 x^6-10 x^7)}{625-250 x+25 x^2})} \, dx\) [6545]

Optimal result

Integrand size = 135, antiderivative size = 34 \[ \int \frac {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx=\frac {1}{\log \left (\frac {x^2 \left (-x+\left (1-e^x+\frac {x}{5}\right ) x^2\right )^2}{(-5+x)^2}\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx=\int \frac {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx \]

[In]

Int[(100 - 160*x - 20*x^2 + 6*x^3 + E^x*(150*x + 30*x^2 - 10*x^3))/((-25*x + 30*x^2 - x^4 + E^x*(-25*x^2 + 5*x
^3))*Log[(25*x^4 - 50*x^5 + 15*x^6 + 25*E^(2*x)*x^6 + 10*x^7 + x^8 + E^x*(50*x^5 - 50*x^6 - 10*x^7))/(625 - 25
0*x + 25*x^2)]^2),x]

[Out]

-2*Defer[Int][Log[(x^4*(-5 - 5*(-1 + E^x)*x + x^2)^2)/(25*(-5 + x)^2)]^(-2), x] + 2*Defer[Int][1/((-5 + x)*Log
[(x^4*(-5 - 5*(-1 + E^x)*x + x^2)^2)/(25*(-5 + x)^2)]^2), x] - 6*Defer[Int][1/(x*Log[(x^4*(-5 - 5*(-1 + E^x)*x
+ x^2)^2)/(25*(-5 + x)^2)]^2), x] - 10*Defer[Int][1/((-5 + 5*x - 5*E^x*x + x^2)*Log[(x^4*(-5 - 5*(-1 + E^x)*x
+ x^2)^2)/(25*(-5 + x)^2)]^2), x] - 10*Defer[Int][1/(x*(-5 + 5*x - 5*E^x*x + x^2)*Log[(x^4*(-5 - 5*(-1 + E^x)
*x + x^2)^2)/(25*(-5 + x)^2)]^2), x] + 8*Defer[Int][x/((-5 + 5*x - 5*E^x*x + x^2)*Log[(x^4*(-5 - 5*(-1 + E^x)*
x + x^2)^2)/(25*(-5 + x)^2)]^2), x] + 2*Defer[Int][x^2/((-5 + 5*x - 5*E^x*x + x^2)*Log[(x^4*(-5 - 5*(-1 + E^x)
*x + x^2)^2)/(25*(-5 + x)^2)]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (-50+\left (80-75 e^x\right ) x-5 \left (-2+3 e^x\right ) x^2+\left (-3+5 e^x\right ) x^3\right )}{(5-x) x \left (5+5 \left (-1+e^x\right ) x-x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx \\ & = 2 \int \frac {-50+\left (80-75 e^x\right ) x-5 \left (-2+3 e^x\right ) x^2+\left (-3+5 e^x\right ) x^3}{(5-x) x \left (5+5 \left (-1+e^x\right ) x-x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx \\ & = 2 \int \left (\frac {15+3 x-x^2}{(-5+x) x \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}+\frac {-5-5 x+4 x^2+x^3}{x \left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}\right ) \, dx \\ & = 2 \int \frac {15+3 x-x^2}{(-5+x) x \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx+2 \int \frac {-5-5 x+4 x^2+x^3}{x \left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx \\ & = 2 \int \left (-\frac {1}{\log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}+\frac {1}{(-5+x) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}-\frac {3}{x \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}\right ) \, dx+2 \int \left (-\frac {5}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}-\frac {5}{x \left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}+\frac {4 x}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}+\frac {x^2}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx\right )+2 \int \frac {1}{(-5+x) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx+2 \int \frac {x^2}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx-6 \int \frac {1}{x \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx+8 \int \frac {x}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx-10 \int \frac {1}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx-10 \int \frac {1}{x \left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx=\frac {1}{\log \left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \]

[In]

Integrate[(100 - 160*x - 20*x^2 + 6*x^3 + E^x*(150*x + 30*x^2 - 10*x^3))/((-25*x + 30*x^2 - x^4 + E^x*(-25*x^2
+ 5*x^3))*Log[(25*x^4 - 50*x^5 + 15*x^6 + 25*E^(2*x)*x^6 + 10*x^7 + x^8 + E^x*(50*x^5 - 50*x^6 - 10*x^7))/(62
5 - 250*x + 25*x^2)]^2),x]

[Out]

Log[(x^4*(-5 - 5*(-1 + E^x)*x + x^2)^2)/(25*(-5 + x)^2)]^(-1)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(31)=62\).

Time = 1.83 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.00

method	result	size
parallelrisch	\(\frac {1}{\ln \left (\frac {25 \,{\mathrm e}^{2 x} x^{6}+\left (-10 x^{7}-50 x^{6}+50 x^{5}\right ) {\mathrm e}^{x}+x^{8}+10 x^{7}+15 x^{6}-50 x^{5}+25 x^{4}}{25 x^{2}-250 x +625}\right )}\)	\(68\)
risch	\(\text {Expression too large to display}\)	\(818\)

[In]

int(((-10*x^3+30*x^2+150*x)*exp(x)+6*x^3-20*x^2-160*x+100)/((5*x^3-25*x^2)*exp(x)-x^4+30*x^2-25*x)/ln((25*x^6*
exp(x)^2+(-10*x^7-50*x^6+50*x^5)*exp(x)+x^8+10*x^7+15*x^6-50*x^5+25*x^4)/(25*x^2-250*x+625))^2,x,method=_RETUR
NVERBOSE)

[Out]

1/ln(1/25/(x^2-10*x+25)*(25*x^6*exp(x)^2+(-10*x^7-50*x^6+50*x^5)*exp(x)+x^8+10*x^7+15*x^6-50*x^5+25*x^4))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (30) = 60\).

Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.94 \[ \int \frac {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx=\frac {1}{\log \left (\frac {x^{8} + 10 \, x^{7} + 25 \, x^{6} e^{\left (2 \, x\right )} + 15 \, x^{6} - 50 \, x^{5} + 25 \, x^{4} - 10 \, {\left (x^{7} + 5 \, x^{6} - 5 \, x^{5}\right )} e^{x}}{25 \, {\left (x^{2} - 10 \, x + 25\right )}}\right )} \]

[In]

integrate(((-10*x^3+30*x^2+150*x)*exp(x)+6*x^3-20*x^2-160*x+100)/((5*x^3-25*x^2)*exp(x)-x^4+30*x^2-25*x)/log((
25*x^6*exp(x)^2+(-10*x^7-50*x^6+50*x^5)*exp(x)+x^8+10*x^7+15*x^6-50*x^5+25*x^4)/(25*x^2-250*x+625))^2,x, algor
ithm="fricas")

[Out]

1/log(1/25*(x^8 + 10*x^7 + 25*x^6*e^(2*x) + 15*x^6 - 50*x^5 + 25*x^4 - 10*(x^7 + 5*x^6 - 5*x^5)*e^x)/(x^2 - 10
*x + 25))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx=\frac {1}{\log {\left (\frac {x^{8} + 10 x^{7} + 25 x^{6} e^{2 x} + 15 x^{6} - 50 x^{5} + 25 x^{4} + \left (- 10 x^{7} - 50 x^{6} + 50 x^{5}\right ) e^{x}}{25 x^{2} - 250 x + 625} \right )}} \]

[In]

integrate(((-10*x**3+30*x**2+150*x)*exp(x)+6*x**3-20*x**2-160*x+100)/((5*x**3-25*x**2)*exp(x)-x**4+30*x**2-25*
x)/ln((25*x**6*exp(x)**2+(-10*x**7-50*x**6+50*x**5)*exp(x)+x**8+10*x**7+15*x**6-50*x**5+25*x**4)/(25*x**2-250*
x+625))**2,x)

[Out]

1/log((x**8 + 10*x**7 + 25*x**6*exp(2*x) + 15*x**6 - 50*x**5 + 25*x**4 + (-10*x**7 - 50*x**6 + 50*x**5)*exp(x)
)/(25*x**2 - 250*x + 625))

Maxima [A] (verification not implemented)

none

Time = 0.55 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx=-\frac {1}{2 \, {\left (\log \left (5\right ) - \log \left (-x^{2} + 5 \, x e^{x} - 5 \, x + 5\right ) + \log \left (x - 5\right ) - 2 \, \log \left (x\right )\right )}} \]

[In]

integrate(((-10*x^3+30*x^2+150*x)*exp(x)+6*x^3-20*x^2-160*x+100)/((5*x^3-25*x^2)*exp(x)-x^4+30*x^2-25*x)/log((
25*x^6*exp(x)^2+(-10*x^7-50*x^6+50*x^5)*exp(x)+x^8+10*x^7+15*x^6-50*x^5+25*x^4)/(25*x^2-250*x+625))^2,x, algor
ithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (30) = 60\).

Time = 1.51 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \frac {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx=\frac {1}{\log \left (\frac {x^{8} - 10 \, x^{7} e^{x} + 10 \, x^{7} + 25 \, x^{6} e^{\left (2 \, x\right )} - 50 \, x^{6} e^{x} + 15 \, x^{6} + 50 \, x^{5} e^{x} - 50 \, x^{5} + 25 \, x^{4}}{25 \, {\left (x^{2} - 10 \, x + 25\right )}}\right )} \]

[In]

integrate(((-10*x^3+30*x^2+150*x)*exp(x)+6*x^3-20*x^2-160*x+100)/((5*x^3-25*x^2)*exp(x)-x^4+30*x^2-25*x)/log((
25*x^6*exp(x)^2+(-10*x^7-50*x^6+50*x^5)*exp(x)+x^8+10*x^7+15*x^6-50*x^5+25*x^4)/(25*x^2-250*x+625))^2,x, algor
ithm="giac")

[Out]

1/log(1/25*(x^8 - 10*x^7*e^x + 10*x^7 + 25*x^6*e^(2*x) - 50*x^6*e^x + 15*x^6 + 50*x^5*e^x - 50*x^5 + 25*x^4)/(
x^2 - 10*x + 25))

Mupad [B] (verification not implemented)

Time = 12.81 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \frac {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx=\frac {1}{\ln \left (\frac {25\,x^6\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (10\,x^7+50\,x^6-50\,x^5\right )+25\,x^4-50\,x^5+15\,x^6+10\,x^7+x^8}{25\,x^2-250\,x+625}\right )} \]

[In]

int(-(6*x^3 - 20*x^2 - 160*x + exp(x)*(150*x + 30*x^2 - 10*x^3) + 100)/(log((25*x^6*exp(2*x) - exp(x)*(50*x^6
- 50*x^5 + 10*x^7) + 25*x^4 - 50*x^5 + 15*x^6 + 10*x^7 + x^8)/(25*x^2 - 250*x + 625))^2*(25*x + exp(x)*(25*x^2
- 5*x^3) - 30*x^2 + x^4)),x)

[Out]

1/log((25*x^6*exp(2*x) - exp(x)*(50*x^6 - 50*x^5 + 10*x^7) + 25*x^4 - 50*x^5 + 15*x^6 + 10*x^7 + x^8)/(25*x^2
- 250*x + 625))