\(\int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8)+(-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7) \log (\frac {1}{3} e^{\frac {3 x+e^x (5+x^3)}{5+x^3}})}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+(100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8) \log (\frac {1}{3} e^{\frac {3 x+e^x (5+x^3)}{5+x^3}})+(25+10 x^3+x^6) \log ^2(\frac {1}{3} e^{\frac {3 x+e^x (5+x^3)}{5+x^3}})} \, dx\) [4839]

Optimal result

Integrand size = 257, antiderivative size = 37 \[ \int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx=\frac {5-x}{-2+x-\frac {\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{x}} \]

[Out]

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

Rubi [F]

\[ \int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx=\int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx \]

[In]

Int[(75*x - 90*x^2 - 30*x^4 - 24*x^5 - 3*x^8 + E^x*(125*x - 25*x^2 + 50*x^4 - 10*x^5 + 5*x^7 - x^8) + (-125 +
50*x - 50*x^3 + 20*x^4 - 5*x^6 + 2*x^7)*Log[E^((3*x + E^x*(5 + x^3))/(5 + x^3))/3])/(100*x^2 - 100*x^3 + 25*x^
4 + 40*x^5 - 40*x^6 + 10*x^7 + 4*x^8 - 4*x^9 + x^10 + (100*x - 50*x^2 + 40*x^4 - 20*x^5 + 4*x^7 - 2*x^8)*Log[E
^((3*x + E^x*(5 + x^3))/(5 + x^3))/3] + (25 + 10*x^3 + x^6)*Log[E^((3*x + E^x*(5 + x^3))/(5 + x^3))/3]^2),x]

[Out]

10*Defer[Int][x/(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^2, x] + 5*Defer[Int][(E^x*x)/(-2*x + x^2 - Log
[E^(E^x + (3*x)/(5 + x^3))/3])^2, x] - 12*Defer[Int][x^2/(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^2, x]
- Defer[Int][(E^x*x^2)/(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^2, x] + 2*Defer[Int][x^3/(-2*x + x^2 -
Log[E^(E^x + (3*x)/(5 + x^3))/3])^2, x] + 2*Defer[Int][1/((-(-5)^(1/3) + x)*(-2*x + x^2 - Log[E^(E^x + (3*x)/
(5 + x^3))/3])^2), x] + 2*Defer[Int][1/((5^(1/3) + x)*(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^2), x] +
2*5^(2/3)*Defer[Int][1/((5^(1/3) + x)*(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^2), x] + 2*Defer[Int][1
/(((-1)^(2/3)*5^(1/3) + x)*(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^2), x] + 2*(-5)^(2/3)*Defer[Int][1/
((5^(1/3) - (-1)^(1/3)*x)*(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^2), x] - 2*(-1)^(1/3)*5^(2/3)*Defer[
Int][1/((5^(1/3) + (-1)^(2/3)*x)*(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^2), x] + 225*Defer[Int][x/((5
+ x^3)^2*(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^2), x] - 45*Defer[Int][x^2/((5 + x^3)^2*(-2*x + x^2
- Log[E^(E^x + (3*x)/(5 + x^3))/3])^2), x] + 5*Defer[Int][(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3])^(-1)
, x] - 2*Defer[Int][x/(-2*x + x^2 - Log[E^(E^x + (3*x)/(5 + x^3))/3]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-x \left (e^x (-5+x) \left (5+x^3\right )^2+3 \left (-25+30 x+10 x^3+8 x^4+x^7\right )\right )+(-5+2 x) \left (5+x^3\right )^2 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{\left (5+x^3\right )^2 \left ((-2+x) x-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ & = \int \left (-\frac {e^x (-5+x) x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8-125 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+50 x \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )-50 x^3 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+20 x^4 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )-5 x^6 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+2 x^7 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx \\ & = -\int \frac {e^x (-5+x) x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8-125 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+50 x \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )-50 x^3 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+20 x^4 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )-5 x^6 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )+2 x^7 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ & = -\int \left (-\frac {5 e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx+\int \frac {-3 x \left (-25+30 x+10 x^3+8 x^4+x^7\right )+(-5+2 x) \left (5+x^3\right )^2 \log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}{\left (5+x^3\right )^2 \left ((-2+x) x-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ & = 5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \left (\frac {x \left (325-315 x+50 x^2+70 x^3-114 x^4+20 x^5+10 x^6-12 x^7+2 x^8\right )}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {5-2 x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}\right ) \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ & = 5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \frac {x \left (325-315 x+50 x^2+70 x^3-114 x^4+20 x^5+10 x^6-12 x^7+2 x^8\right )}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \frac {5-2 x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx \\ & = 5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\int \left (\frac {10 x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}-\frac {12 x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {2 x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}-\frac {45 (-5+x) x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {6 (-5+x) x}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx+\int \left (\frac {5}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}-\frac {2 x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )}\right ) \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ & = 2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+6 \int \frac {(-5+x) x}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-45 \int \frac {(-5+x) x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ & = 2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+6 \int \left (-\frac {5 x}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {x^2}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-45 \int \left (-\frac {5 x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {x^2}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ & = 2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+6 \int \frac {x^2}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-30 \int \frac {x}{\left (5+x^3\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-45 \int \frac {x^2}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+225 \int \frac {x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ & = 2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+6 \int \left (\frac {1}{3 \left (-\sqrt [3]{-5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {1}{3 \left (\sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {1}{3 \left ((-1)^{2/3} \sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-30 \int \left (-\frac {1}{3 \sqrt [3]{5} \left (\sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}-\frac {(-1)^{2/3}}{3 \sqrt [3]{5} \left (\sqrt [3]{5}-\sqrt [3]{-1} x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}+\frac {\sqrt [3]{-\frac {1}{5}}}{3 \left (\sqrt [3]{5}+(-1)^{2/3} x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2}\right ) \, dx-45 \int \frac {x^2}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+225 \int \frac {x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ & = 2 \int \frac {x^3}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+2 \int \frac {1}{\left (-\sqrt [3]{-5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+2 \int \frac {1}{\left (\sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+2 \int \frac {1}{\left ((-1)^{2/3} \sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-2 \int \frac {x}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+5 \int \frac {e^x x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+5 \int \frac {1}{-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \, dx+10 \int \frac {x}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-45 \int \frac {x^2}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+225 \int \frac {x}{\left (5+x^3\right )^2 \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\left (2 (-5)^{2/3}\right ) \int \frac {1}{\left (\sqrt [3]{5}-\sqrt [3]{-1} x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx+\left (2\ 5^{2/3}\right ) \int \frac {1}{\left (\sqrt [3]{5}+x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\left (2 \sqrt [3]{-1} 5^{2/3}\right ) \int \frac {1}{\left (\sqrt [3]{5}+(-1)^{2/3} x\right ) \left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx-\int \frac {e^x x^2}{\left (-2 x+x^2-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx=-\frac {(-5+x) x}{(-2+x) x-\log \left (\frac {1}{3} e^{e^x+\frac {3 x}{5+x^3}}\right )} \]

[In]

Integrate[(75*x - 90*x^2 - 30*x^4 - 24*x^5 - 3*x^8 + E^x*(125*x - 25*x^2 + 50*x^4 - 10*x^5 + 5*x^7 - x^8) + (-
125 + 50*x - 50*x^3 + 20*x^4 - 5*x^6 + 2*x^7)*Log[E^((3*x + E^x*(5 + x^3))/(5 + x^3))/3])/(100*x^2 - 100*x^3 +
25*x^4 + 40*x^5 - 40*x^6 + 10*x^7 + 4*x^8 - 4*x^9 + x^10 + (100*x - 50*x^2 + 40*x^4 - 20*x^5 + 4*x^7 - 2*x^8)
*Log[E^((3*x + E^x*(5 + x^3))/(5 + x^3))/3] + (25 + 10*x^3 + x^6)*Log[E^((3*x + E^x*(5 + x^3))/(5 + x^3))/3]^2
),x]

[Out]

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

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.30

\[-\frac {2 \left (-5+x \right ) x}{2 x^{2}+2 \ln \left (3\right )-4 x -2 \ln \left ({\mathrm e}^{\frac {{\mathrm e}^{x} x^{3}+5 \,{\mathrm e}^{x}+3 x}{x^{3}+5}}\right )}\]

[In]

int(((2*x^7-5*x^6+20*x^4-50*x^3+50*x-125)*ln(1/3*exp(((x^3+5)*exp(x)+3*x)/(x^3+5)))+(-x^8+5*x^7-10*x^5+50*x^4-
25*x^2+125*x)*exp(x)-3*x^8-24*x^5-30*x^4-90*x^2+75*x)/((x^6+10*x^3+25)*ln(1/3*exp(((x^3+5)*exp(x)+3*x)/(x^3+5)
))^2+(-2*x^8+4*x^7-20*x^5+40*x^4-50*x^2+100*x)*ln(1/3*exp(((x^3+5)*exp(x)+3*x)/(x^3+5)))+x^10-4*x^9+4*x^8+10*x
^7-40*x^6+40*x^5+25*x^4-100*x^3+100*x^2),x)

[Out]

-2*(-5+x)*x/(2*x^2+2*ln(3)-4*x-2*ln(exp((exp(x)*x^3+5*exp(x)+3*x)/(x^3+5))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.49 \[ \int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx=-\frac {x^{5} - 5 \, x^{4} + 5 \, x^{2} - 25 \, x}{x^{5} - 2 \, x^{4} + 5 \, x^{2} - {\left (x^{3} + 5\right )} e^{x} + {\left (x^{3} + 5\right )} \log \left (3\right ) - 13 \, x} \]

[In]

integrate(((2*x^7-5*x^6+20*x^4-50*x^3+50*x-125)*log(1/3*exp(((x^3+5)*exp(x)+3*x)/(x^3+5)))+(-x^8+5*x^7-10*x^5+
50*x^4-25*x^2+125*x)*exp(x)-3*x^8-24*x^5-30*x^4-90*x^2+75*x)/((x^6+10*x^3+25)*log(1/3*exp(((x^3+5)*exp(x)+3*x)
/(x^3+5)))^2+(-2*x^8+4*x^7-20*x^5+40*x^4-50*x^2+100*x)*log(1/3*exp(((x^3+5)*exp(x)+3*x)/(x^3+5)))+x^10-4*x^9+4
*x^8+10*x^7-40*x^6+40*x^5+25*x^4-100*x^3+100*x^2),x, algorithm="fricas")

[Out]

-(x^5 - 5*x^4 + 5*x^2 - 25*x)/(x^5 - 2*x^4 + 5*x^2 - (x^3 + 5)*e^x + (x^3 + 5)*log(3) - 13*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).

Time = 0.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43 \[ \int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx=\frac {x^{5} - 5 x^{4} + 5 x^{2} - 25 x}{- x^{5} + 2 x^{4} - x^{3} \log {\left (3 \right )} - 5 x^{2} + 13 x + \left (x^{3} + 5\right ) e^{x} - 5 \log {\left (3 \right )}} \]

[In]

integrate(((2*x**7-5*x**6+20*x**4-50*x**3+50*x-125)*ln(1/3*exp(((x**3+5)*exp(x)+3*x)/(x**3+5)))+(-x**8+5*x**7-
10*x**5+50*x**4-25*x**2+125*x)*exp(x)-3*x**8-24*x**5-30*x**4-90*x**2+75*x)/((x**6+10*x**3+25)*ln(1/3*exp(((x**
3+5)*exp(x)+3*x)/(x**3+5)))**2+(-2*x**8+4*x**7-20*x**5+40*x**4-50*x**2+100*x)*ln(1/3*exp(((x**3+5)*exp(x)+3*x)
/(x**3+5)))+x**10-4*x**9+4*x**8+10*x**7-40*x**6+40*x**5+25*x**4-100*x**3+100*x**2),x)

[Out]

(x**5 - 5*x**4 + 5*x**2 - 25*x)/(-x**5 + 2*x**4 - x**3*log(3) - 5*x**2 + 13*x + (x**3 + 5)*exp(x) - 5*log(3))

Maxima [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx=-\frac {x^{5} - 5 \, x^{4} + 5 \, x^{2} - 25 \, x}{x^{5} - 2 \, x^{4} + x^{3} \log \left (3\right ) + 5 \, x^{2} - {\left (x^{3} + 5\right )} e^{x} - 13 \, x + 5 \, \log \left (3\right )} \]

[In]

integrate(((2*x^7-5*x^6+20*x^4-50*x^3+50*x-125)*log(1/3*exp(((x^3+5)*exp(x)+3*x)/(x^3+5)))+(-x^8+5*x^7-10*x^5+
50*x^4-25*x^2+125*x)*exp(x)-3*x^8-24*x^5-30*x^4-90*x^2+75*x)/((x^6+10*x^3+25)*log(1/3*exp(((x^3+5)*exp(x)+3*x)
/(x^3+5)))^2+(-2*x^8+4*x^7-20*x^5+40*x^4-50*x^2+100*x)*log(1/3*exp(((x^3+5)*exp(x)+3*x)/(x^3+5)))+x^10-4*x^9+4
*x^8+10*x^7-40*x^6+40*x^5+25*x^4-100*x^3+100*x^2),x, algorithm="maxima")

[Out]

-(x^5 - 5*x^4 + 5*x^2 - 25*x)/(x^5 - 2*x^4 + x^3*log(3) + 5*x^2 - (x^3 + 5)*e^x - 13*x + 5*log(3))

Giac [A] (verification not implemented)

none

Time = 0.57 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.59 \[ \int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx=-\frac {x^{5} - 5 \, x^{4} + 5 \, x^{2} - 25 \, x}{x^{5} - 2 \, x^{4} - x^{3} e^{x} + x^{3} \log \left (3\right ) + 5 \, x^{2} - 13 \, x - 5 \, e^{x} + 5 \, \log \left (3\right )} \]

[In]

integrate(((2*x^7-5*x^6+20*x^4-50*x^3+50*x-125)*log(1/3*exp(((x^3+5)*exp(x)+3*x)/(x^3+5)))+(-x^8+5*x^7-10*x^5+
50*x^4-25*x^2+125*x)*exp(x)-3*x^8-24*x^5-30*x^4-90*x^2+75*x)/((x^6+10*x^3+25)*log(1/3*exp(((x^3+5)*exp(x)+3*x)
/(x^3+5)))^2+(-2*x^8+4*x^7-20*x^5+40*x^4-50*x^2+100*x)*log(1/3*exp(((x^3+5)*exp(x)+3*x)/(x^3+5)))+x^10-4*x^9+4
*x^8+10*x^7-40*x^6+40*x^5+25*x^4-100*x^3+100*x^2),x, algorithm="giac")

[Out]

-(x^5 - 5*x^4 + 5*x^2 - 25*x)/(x^5 - 2*x^4 - x^3*e^x + x^3*log(3) + 5*x^2 - 13*x - 5*e^x + 5*log(3))

Mupad [B] (verification not implemented)

Time = 11.76 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.70 \[ \int \frac {75 x-90 x^2-30 x^4-24 x^5-3 x^8+e^x \left (125 x-25 x^2+50 x^4-10 x^5+5 x^7-x^8\right )+\left (-125+50 x-50 x^3+20 x^4-5 x^6+2 x^7\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )}{100 x^2-100 x^3+25 x^4+40 x^5-40 x^6+10 x^7+4 x^8-4 x^9+x^{10}+\left (100 x-50 x^2+40 x^4-20 x^5+4 x^7-2 x^8\right ) \log \left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )+\left (25+10 x^3+x^6\right ) \log ^2\left (\frac {1}{3} e^{\frac {3 x+e^x \left (5+x^3\right )}{5+x^3}}\right )} \, dx=-\frac {-x^5+5\,x^4-5\,x^2+25\,x}{13\,x-5\,\ln \left (3\right )+5\,{\mathrm {e}}^x+x^3\,{\mathrm {e}}^x-x^3\,\ln \left (3\right )-5\,x^2+2\,x^4-x^5} \]

[In]

int(-(90*x^2 - log(exp((3*x + exp(x)*(x^3 + 5))/(x^3 + 5))/3)*(50*x - 50*x^3 + 20*x^4 - 5*x^6 + 2*x^7 - 125) -
exp(x)*(125*x - 25*x^2 + 50*x^4 - 10*x^5 + 5*x^7 - x^8) - 75*x + 30*x^4 + 24*x^5 + 3*x^8)/(log(exp((3*x + exp
(x)*(x^3 + 5))/(x^3 + 5))/3)*(100*x - 50*x^2 + 40*x^4 - 20*x^5 + 4*x^7 - 2*x^8) + 100*x^2 - 100*x^3 + 25*x^4 +
40*x^5 - 40*x^6 + 10*x^7 + 4*x^8 - 4*x^9 + x^10 + log(exp((3*x + exp(x)*(x^3 + 5))/(x^3 + 5))/3)^2*(10*x^3 +
x^6 + 25)),x)

[Out]

-(25*x - 5*x^2 + 5*x^4 - x^5)/(13*x - 5*log(3) + 5*exp(x) + x^3*exp(x) - x^3*log(3) - 5*x^2 + 2*x^4 - x^5)