\(\int \frac {90+30 x+45 x^2+15 x^3+e^x (45-30 x-15 x^2)+e^{\frac {1}{5} (x+5 \log (3+x))} (-93-87 x-3 x^2+3 x^3+e^x (21+12 x))}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} (-30 x-10 x^2)} \, dx\) [4592]

Optimal result

Integrand size = 126, antiderivative size = 32 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3 \left (2+e^x-x-x^2\right )}{-x+e^{x/5} (3+x)} \]

[Out]

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

Rubi [F]

\[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx \]

[In]

Int[(90 + 30*x + 45*x^2 + 15*x^3 + E^x*(45 - 30*x - 15*x^2) + E^((x + 5*Log[3 + x])/5)*(-93 - 87*x - 3*x^2 + 3
*x^3 + E^x*(21 + 12*x)))/(15*x^2 + 5*x^3 + E^((2*(x + 5*Log[3 + x]))/5)*(15 + 5*x) + E^((x + 5*Log[3 + x])/5)*
(-30*x - 10*x^2)),x]

[Out]

(3*x^4)/(3 + x)^5 - (81*E^(x/5))/(3 + x)^4 + (81*E^(x/5))/(3 + x)^3 + (27*E^((2*x)/5))/(3 + x)^3 - (27*E^(x/5)
)/(3 + x)^2 - (18*E^((2*x)/5))/(3 + x)^2 - (9*E^((3*x)/5))/(3 + x)^2 + (3*E^(x/5))/(3 + x) + (3*E^((2*x)/5))/(
3 + x) + (3*E^((3*x)/5))/(3 + x) + (3*E^((4*x)/5))/(3 + x) - (29646*Defer[Int][1/((3 + x)^5*(3*E^(x/5) - x + E
^(x/5)*x)), x])/5 - 648*Defer[Int][1/((3 + x)^3*(3*E^(x/5) - x + E^(x/5)*x)), x] + 18*Defer[Int][1/((3 + x)*(3
*E^(x/5) - x + E^(x/5)*x)), x] + 27*Defer[Int][(x - E^(x/5)*(3 + x))^(-2), x] - (54*Defer[Int][x/(x - E^(x/5)*
(3 + x))^2, x])/5 + (3*Defer[Int][x^2/(x - E^(x/5)*(3 + x))^2, x])/5 + (3*Defer[Int][x^3/(x - E^(x/5)*(3 + x))
^2, x])/5 - 2187*Defer[Int][1/((3 + x)^6*(x - E^(x/5)*(3 + x))^2), x] + (16038*Defer[Int][1/((3 + x)^5*(x - E^
(x/5)*(3 + x))^2), x])/5 - (7776*Defer[Int][1/((3 + x)^4*(x - E^(x/5)*(3 + x))^2), x])/5 + 81*Defer[Int][1/((3
+ x)^3*(x - E^(x/5)*(3 + x))^2), x] + 189*Defer[Int][1/((3 + x)^2*(x - E^(x/5)*(3 + x))^2), x] - 108*Defer[In
t][1/((3 + x)*(x - E^(x/5)*(3 + x))^2), x] - (54*Defer[Int][(-x + E^(x/5)*(3 + x))^(-1), x])/5 - (12*Defer[Int
][x/(-x + E^(x/5)*(3 + x)), x])/5 + (3*Defer[Int][x^2/(-x + E^(x/5)*(3 + x)), x])/5 + 4374*Defer[Int][1/((3 +
x)^6*(-x + E^(x/5)*(3 + x))), x] + 2997*Defer[Int][1/((3 + x)^4*(-x + E^(x/5)*(3 + x))), x] + 36*Defer[Int][1/
((3 + x)^2*(-x + E^(x/5)*(3 + x))), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 \left (-5 e^x (-1+x)+e^{6 x/5} (7+4 x)+5 \left (2+x^2\right )+e^{x/5} \left (-31-29 x-x^2+x^3\right )\right )}{5 \left (x-e^{x/5} (3+x)\right )^2} \, dx \\ & = \frac {3}{5} \int \frac {-5 e^x (-1+x)+e^{6 x/5} (7+4 x)+5 \left (2+x^2\right )+e^{x/5} \left (-31-29 x-x^2+x^3\right )}{\left (x-e^{x/5} (3+x)\right )^2} \, dx \\ & = \frac {3}{5} \int \left (-\frac {5 (-12+x) x^3}{(3+x)^6}+\frac {e^{4 x/5} (7+4 x)}{(3+x)^2}+\frac {e^{x/5} x^2 \left (45-2 x+x^2\right )}{(3+x)^5}+\frac {e^{2 x/5} x \left (30+x+2 x^2\right )}{(3+x)^4}+\frac {e^{3 x/5} \left (15+4 x+3 x^2\right )}{(3+x)^3}+\frac {-7533-19602 x-20358 x^2-10782 x^3-2865 x^4-294 x^5+45 x^6+14 x^7+x^8}{(3+x)^6 \left (3 e^{x/5}-x+e^{x/5} x\right )}+\frac {7290+7047 x-3807 x^2-7668 x^3-3357 x^4-45 x^5+396 x^6+135 x^7+19 x^8+x^9}{(3+x)^6 \left (3 e^{x/5}-x+e^{x/5} x\right )^2}\right ) \, dx \\ & = \frac {3}{5} \int \frac {e^{4 x/5} (7+4 x)}{(3+x)^2} \, dx+\frac {3}{5} \int \frac {e^{x/5} x^2 \left (45-2 x+x^2\right )}{(3+x)^5} \, dx+\frac {3}{5} \int \frac {e^{2 x/5} x \left (30+x+2 x^2\right )}{(3+x)^4} \, dx+\frac {3}{5} \int \frac {e^{3 x/5} \left (15+4 x+3 x^2\right )}{(3+x)^3} \, dx+\frac {3}{5} \int \frac {-7533-19602 x-20358 x^2-10782 x^3-2865 x^4-294 x^5+45 x^6+14 x^7+x^8}{(3+x)^6 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+\frac {3}{5} \int \frac {7290+7047 x-3807 x^2-7668 x^3-3357 x^4-45 x^5+396 x^6+135 x^7+19 x^8+x^9}{(3+x)^6 \left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx-3 \int \frac {(-12+x) x^3}{(3+x)^6} \, dx \\ & = \frac {3 x^4}{(3+x)^5}+\frac {3 e^{4 x/5}}{3+x}+\frac {3}{5} \int \left (\frac {540 e^{x/5}}{(3+x)^5}-\frac {432 e^{x/5}}{(3+x)^4}+\frac {117 e^{x/5}}{(3+x)^3}-\frac {14 e^{x/5}}{(3+x)^2}+\frac {e^{x/5}}{3+x}\right ) \, dx+\frac {3}{5} \int \left (-\frac {135 e^{2 x/5}}{(3+x)^4}+\frac {78 e^{2 x/5}}{(3+x)^3}-\frac {17 e^{2 x/5}}{(3+x)^2}+\frac {2 e^{2 x/5}}{3+x}\right ) \, dx+\frac {3}{5} \int \left (\frac {30 e^{3 x/5}}{(3+x)^3}-\frac {14 e^{3 x/5}}{(3+x)^2}+\frac {3 e^{3 x/5}}{3+x}\right ) \, dx+\frac {3}{5} \int \frac {7290+7047 x-3807 x^2-7668 x^3-3357 x^4-45 x^5+396 x^6+135 x^7+19 x^8+x^9}{(3+x)^6 \left (x-e^{x/5} (3+x)\right )^2} \, dx+\frac {3}{5} \int \left (-\frac {18}{3 e^{x/5}-x+e^{x/5} x}-\frac {4 x}{3 e^{x/5}-x+e^{x/5} x}+\frac {x^2}{3 e^{x/5}-x+e^{x/5} x}+\frac {7290}{(3+x)^6 \left (3 e^{x/5}-x+e^{x/5} x\right )}-\frac {9882}{(3+x)^5 \left (3 e^{x/5}-x+e^{x/5} x\right )}+\frac {4995}{(3+x)^4 \left (3 e^{x/5}-x+e^{x/5} x\right )}-\frac {1080}{(3+x)^3 \left (3 e^{x/5}-x+e^{x/5} x\right )}+\frac {60}{(3+x)^2 \left (3 e^{x/5}-x+e^{x/5} x\right )}+\frac {30}{(3+x) \left (3 e^{x/5}-x+e^{x/5} x\right )}\right ) \, dx \\ & = \frac {3 x^4}{(3+x)^5}+\frac {3 e^{4 x/5}}{3+x}+\frac {3}{5} \int \frac {e^{x/5}}{3+x} \, dx+\frac {3}{5} \int \frac {x^2}{3 e^{x/5}-x+e^{x/5} x} \, dx+\frac {3}{5} \int \left (\frac {45}{\left (3 e^{x/5}-x+e^{x/5} x\right )^2}-\frac {18 x}{\left (3 e^{x/5}-x+e^{x/5} x\right )^2}+\frac {x^2}{\left (3 e^{x/5}-x+e^{x/5} x\right )^2}+\frac {x^3}{\left (3 e^{x/5}-x+e^{x/5} x\right )^2}-\frac {3645}{(3+x)^6 \left (3 e^{x/5}-x+e^{x/5} x\right )^2}+\frac {5346}{(3+x)^5 \left (3 e^{x/5}-x+e^{x/5} x\right )^2}-\frac {2592}{(3+x)^4 \left (3 e^{x/5}-x+e^{x/5} x\right )^2}+\frac {135}{(3+x)^3 \left (3 e^{x/5}-x+e^{x/5} x\right )^2}+\frac {315}{(3+x)^2 \left (3 e^{x/5}-x+e^{x/5} x\right )^2}-\frac {180}{(3+x) \left (3 e^{x/5}-x+e^{x/5} x\right )^2}\right ) \, dx+\frac {6}{5} \int \frac {e^{2 x/5}}{3+x} \, dx+\frac {9}{5} \int \frac {e^{3 x/5}}{3+x} \, dx-\frac {12}{5} \int \frac {x}{3 e^{x/5}-x+e^{x/5} x} \, dx-\frac {42}{5} \int \frac {e^{x/5}}{(3+x)^2} \, dx-\frac {42}{5} \int \frac {e^{3 x/5}}{(3+x)^2} \, dx-\frac {51}{5} \int \frac {e^{2 x/5}}{(3+x)^2} \, dx-\frac {54}{5} \int \frac {1}{3 e^{x/5}-x+e^{x/5} x} \, dx+18 \int \frac {e^{3 x/5}}{(3+x)^3} \, dx+18 \int \frac {1}{(3+x) \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+36 \int \frac {1}{(3+x)^2 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+\frac {234}{5} \int \frac {e^{2 x/5}}{(3+x)^3} \, dx+\frac {351}{5} \int \frac {e^{x/5}}{(3+x)^3} \, dx-81 \int \frac {e^{2 x/5}}{(3+x)^4} \, dx-\frac {1296}{5} \int \frac {e^{x/5}}{(3+x)^4} \, dx+324 \int \frac {e^{x/5}}{(3+x)^5} \, dx-648 \int \frac {1}{(3+x)^3 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+2997 \int \frac {1}{(3+x)^4 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+4374 \int \frac {1}{(3+x)^6 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx-\frac {29646}{5} \int \frac {1}{(3+x)^5 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx \\ & = \frac {3 x^4}{(3+x)^5}-\frac {81 e^{x/5}}{(3+x)^4}+\frac {432 e^{x/5}}{5 (3+x)^3}+\frac {27 e^{2 x/5}}{(3+x)^3}-\frac {351 e^{x/5}}{10 (3+x)^2}-\frac {117 e^{2 x/5}}{5 (3+x)^2}-\frac {9 e^{3 x/5}}{(3+x)^2}+\frac {42 e^{x/5}}{5 (3+x)}+\frac {51 e^{2 x/5}}{5 (3+x)}+\frac {42 e^{3 x/5}}{5 (3+x)}+\frac {3 e^{4 x/5}}{3+x}+\frac {3 \text {Ei}\left (\frac {3+x}{5}\right )}{5 e^{3/5}}+\frac {6 \text {Ei}\left (\frac {2 (3+x)}{5}\right )}{5 e^{6/5}}+\frac {9 \text {Ei}\left (\frac {3 (3+x)}{5}\right )}{5 e^{9/5}}+\frac {3}{5} \int \frac {x^2}{\left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx+\frac {3}{5} \int \frac {x^3}{\left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx+\frac {3}{5} \int \frac {x^2}{-x+e^{x/5} (3+x)} \, dx-\frac {42}{25} \int \frac {e^{x/5}}{3+x} \, dx-\frac {12}{5} \int \frac {x}{-x+e^{x/5} (3+x)} \, dx-\frac {102}{25} \int \frac {e^{2 x/5}}{3+x} \, dx-\frac {126}{25} \int \frac {e^{3 x/5}}{3+x} \, dx+\frac {27}{5} \int \frac {e^{3 x/5}}{(3+x)^2} \, dx+\frac {351}{50} \int \frac {e^{x/5}}{(3+x)^2} \, dx+\frac {234}{25} \int \frac {e^{2 x/5}}{(3+x)^2} \, dx-\frac {54}{5} \int \frac {e^{2 x/5}}{(3+x)^3} \, dx-\frac {54}{5} \int \frac {x}{\left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx-\frac {54}{5} \int \frac {1}{-x+e^{x/5} (3+x)} \, dx+\frac {81}{5} \int \frac {e^{x/5}}{(3+x)^4} \, dx-\frac {432}{25} \int \frac {e^{x/5}}{(3+x)^3} \, dx+18 \int \frac {1}{(3+x) \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+27 \int \frac {1}{\left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx+36 \int \frac {1}{(3+x)^2 \left (-x+e^{x/5} (3+x)\right )} \, dx+81 \int \frac {1}{(3+x)^3 \left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx-108 \int \frac {1}{(3+x) \left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx+189 \int \frac {1}{(3+x)^2 \left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx-648 \int \frac {1}{(3+x)^3 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx-\frac {7776}{5} \int \frac {1}{(3+x)^4 \left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx-2187 \int \frac {1}{(3+x)^6 \left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx+2997 \int \frac {1}{(3+x)^4 \left (-x+e^{x/5} (3+x)\right )} \, dx+\frac {16038}{5} \int \frac {1}{(3+x)^5 \left (3 e^{x/5}-x+e^{x/5} x\right )^2} \, dx+4374 \int \frac {1}{(3+x)^6 \left (-x+e^{x/5} (3+x)\right )} \, dx-\frac {29646}{5} \int \frac {1}{(3+x)^5 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx \\ & = \frac {3 x^4}{(3+x)^5}-\frac {81 e^{x/5}}{(3+x)^4}+\frac {81 e^{x/5}}{(3+x)^3}+\frac {27 e^{2 x/5}}{(3+x)^3}-\frac {1323 e^{x/5}}{50 (3+x)^2}-\frac {18 e^{2 x/5}}{(3+x)^2}-\frac {9 e^{3 x/5}}{(3+x)^2}+\frac {69 e^{x/5}}{50 (3+x)}+\frac {21 e^{2 x/5}}{25 (3+x)}+\frac {3 e^{3 x/5}}{3+x}+\frac {3 e^{4 x/5}}{3+x}-\frac {27 \text {Ei}\left (\frac {3+x}{5}\right )}{25 e^{3/5}}-\frac {72 \text {Ei}\left (\frac {2 (3+x)}{5}\right )}{25 e^{6/5}}-\frac {81 \text {Ei}\left (\frac {3 (3+x)}{5}\right )}{25 e^{9/5}}+\frac {3}{5} \int \frac {x^2}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+\frac {3}{5} \int \frac {x^3}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+\frac {3}{5} \int \frac {x^2}{-x+e^{x/5} (3+x)} \, dx+\frac {27}{25} \int \frac {e^{x/5}}{(3+x)^3} \, dx+\frac {351}{250} \int \frac {e^{x/5}}{3+x} \, dx-\frac {216}{125} \int \frac {e^{x/5}}{(3+x)^2} \, dx-\frac {54}{25} \int \frac {e^{2 x/5}}{(3+x)^2} \, dx-\frac {12}{5} \int \frac {x}{-x+e^{x/5} (3+x)} \, dx+\frac {81}{25} \int \frac {e^{3 x/5}}{3+x} \, dx+\frac {468}{125} \int \frac {e^{2 x/5}}{3+x} \, dx-\frac {54}{5} \int \frac {x}{\left (x-e^{x/5} (3+x)\right )^2} \, dx-\frac {54}{5} \int \frac {1}{-x+e^{x/5} (3+x)} \, dx+18 \int \frac {1}{(3+x) \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+27 \int \frac {1}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+36 \int \frac {1}{(3+x)^2 \left (-x+e^{x/5} (3+x)\right )} \, dx+81 \int \frac {1}{(3+x)^3 \left (x-e^{x/5} (3+x)\right )^2} \, dx-108 \int \frac {1}{(3+x) \left (x-e^{x/5} (3+x)\right )^2} \, dx+189 \int \frac {1}{(3+x)^2 \left (x-e^{x/5} (3+x)\right )^2} \, dx-648 \int \frac {1}{(3+x)^3 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx-\frac {7776}{5} \int \frac {1}{(3+x)^4 \left (x-e^{x/5} (3+x)\right )^2} \, dx-2187 \int \frac {1}{(3+x)^6 \left (x-e^{x/5} (3+x)\right )^2} \, dx+2997 \int \frac {1}{(3+x)^4 \left (-x+e^{x/5} (3+x)\right )} \, dx+\frac {16038}{5} \int \frac {1}{(3+x)^5 \left (x-e^{x/5} (3+x)\right )^2} \, dx+4374 \int \frac {1}{(3+x)^6 \left (-x+e^{x/5} (3+x)\right )} \, dx-\frac {29646}{5} \int \frac {1}{(3+x)^5 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx \\ & = \frac {3 x^4}{(3+x)^5}-\frac {81 e^{x/5}}{(3+x)^4}+\frac {81 e^{x/5}}{(3+x)^3}+\frac {27 e^{2 x/5}}{(3+x)^3}-\frac {27 e^{x/5}}{(3+x)^2}-\frac {18 e^{2 x/5}}{(3+x)^2}-\frac {9 e^{3 x/5}}{(3+x)^2}+\frac {777 e^{x/5}}{250 (3+x)}+\frac {3 e^{2 x/5}}{3+x}+\frac {3 e^{3 x/5}}{3+x}+\frac {3 e^{4 x/5}}{3+x}+\frac {81 \text {Ei}\left (\frac {3+x}{5}\right )}{250 e^{3/5}}+\frac {108 \text {Ei}\left (\frac {2 (3+x)}{5}\right )}{125 e^{6/5}}+\frac {27}{250} \int \frac {e^{x/5}}{(3+x)^2} \, dx-\frac {216}{625} \int \frac {e^{x/5}}{3+x} \, dx+\frac {3}{5} \int \frac {x^2}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+\frac {3}{5} \int \frac {x^3}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+\frac {3}{5} \int \frac {x^2}{-x+e^{x/5} (3+x)} \, dx-\frac {108}{125} \int \frac {e^{2 x/5}}{3+x} \, dx-\frac {12}{5} \int \frac {x}{-x+e^{x/5} (3+x)} \, dx-\frac {54}{5} \int \frac {x}{\left (x-e^{x/5} (3+x)\right )^2} \, dx-\frac {54}{5} \int \frac {1}{-x+e^{x/5} (3+x)} \, dx+18 \int \frac {1}{(3+x) \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+27 \int \frac {1}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+36 \int \frac {1}{(3+x)^2 \left (-x+e^{x/5} (3+x)\right )} \, dx+81 \int \frac {1}{(3+x)^3 \left (x-e^{x/5} (3+x)\right )^2} \, dx-108 \int \frac {1}{(3+x) \left (x-e^{x/5} (3+x)\right )^2} \, dx+189 \int \frac {1}{(3+x)^2 \left (x-e^{x/5} (3+x)\right )^2} \, dx-648 \int \frac {1}{(3+x)^3 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx-\frac {7776}{5} \int \frac {1}{(3+x)^4 \left (x-e^{x/5} (3+x)\right )^2} \, dx-2187 \int \frac {1}{(3+x)^6 \left (x-e^{x/5} (3+x)\right )^2} \, dx+2997 \int \frac {1}{(3+x)^4 \left (-x+e^{x/5} (3+x)\right )} \, dx+\frac {16038}{5} \int \frac {1}{(3+x)^5 \left (x-e^{x/5} (3+x)\right )^2} \, dx+4374 \int \frac {1}{(3+x)^6 \left (-x+e^{x/5} (3+x)\right )} \, dx-\frac {29646}{5} \int \frac {1}{(3+x)^5 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx \\ & = \frac {3 x^4}{(3+x)^5}-\frac {81 e^{x/5}}{(3+x)^4}+\frac {81 e^{x/5}}{(3+x)^3}+\frac {27 e^{2 x/5}}{(3+x)^3}-\frac {27 e^{x/5}}{(3+x)^2}-\frac {18 e^{2 x/5}}{(3+x)^2}-\frac {9 e^{3 x/5}}{(3+x)^2}+\frac {3 e^{x/5}}{3+x}+\frac {3 e^{2 x/5}}{3+x}+\frac {3 e^{3 x/5}}{3+x}+\frac {3 e^{4 x/5}}{3+x}-\frac {27 \text {Ei}\left (\frac {3+x}{5}\right )}{1250 e^{3/5}}+\frac {27 \int \frac {e^{x/5}}{3+x} \, dx}{1250}+\frac {3}{5} \int \frac {x^2}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+\frac {3}{5} \int \frac {x^3}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+\frac {3}{5} \int \frac {x^2}{-x+e^{x/5} (3+x)} \, dx-\frac {12}{5} \int \frac {x}{-x+e^{x/5} (3+x)} \, dx-\frac {54}{5} \int \frac {x}{\left (x-e^{x/5} (3+x)\right )^2} \, dx-\frac {54}{5} \int \frac {1}{-x+e^{x/5} (3+x)} \, dx+18 \int \frac {1}{(3+x) \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+27 \int \frac {1}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+36 \int \frac {1}{(3+x)^2 \left (-x+e^{x/5} (3+x)\right )} \, dx+81 \int \frac {1}{(3+x)^3 \left (x-e^{x/5} (3+x)\right )^2} \, dx-108 \int \frac {1}{(3+x) \left (x-e^{x/5} (3+x)\right )^2} \, dx+189 \int \frac {1}{(3+x)^2 \left (x-e^{x/5} (3+x)\right )^2} \, dx-648 \int \frac {1}{(3+x)^3 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx-\frac {7776}{5} \int \frac {1}{(3+x)^4 \left (x-e^{x/5} (3+x)\right )^2} \, dx-2187 \int \frac {1}{(3+x)^6 \left (x-e^{x/5} (3+x)\right )^2} \, dx+2997 \int \frac {1}{(3+x)^4 \left (-x+e^{x/5} (3+x)\right )} \, dx+\frac {16038}{5} \int \frac {1}{(3+x)^5 \left (x-e^{x/5} (3+x)\right )^2} \, dx+4374 \int \frac {1}{(3+x)^6 \left (-x+e^{x/5} (3+x)\right )} \, dx-\frac {29646}{5} \int \frac {1}{(3+x)^5 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx \\ & = \frac {3 x^4}{(3+x)^5}-\frac {81 e^{x/5}}{(3+x)^4}+\frac {81 e^{x/5}}{(3+x)^3}+\frac {27 e^{2 x/5}}{(3+x)^3}-\frac {27 e^{x/5}}{(3+x)^2}-\frac {18 e^{2 x/5}}{(3+x)^2}-\frac {9 e^{3 x/5}}{(3+x)^2}+\frac {3 e^{x/5}}{3+x}+\frac {3 e^{2 x/5}}{3+x}+\frac {3 e^{3 x/5}}{3+x}+\frac {3 e^{4 x/5}}{3+x}+\frac {3}{5} \int \frac {x^2}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+\frac {3}{5} \int \frac {x^3}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+\frac {3}{5} \int \frac {x^2}{-x+e^{x/5} (3+x)} \, dx-\frac {12}{5} \int \frac {x}{-x+e^{x/5} (3+x)} \, dx-\frac {54}{5} \int \frac {x}{\left (x-e^{x/5} (3+x)\right )^2} \, dx-\frac {54}{5} \int \frac {1}{-x+e^{x/5} (3+x)} \, dx+18 \int \frac {1}{(3+x) \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx+27 \int \frac {1}{\left (x-e^{x/5} (3+x)\right )^2} \, dx+36 \int \frac {1}{(3+x)^2 \left (-x+e^{x/5} (3+x)\right )} \, dx+81 \int \frac {1}{(3+x)^3 \left (x-e^{x/5} (3+x)\right )^2} \, dx-108 \int \frac {1}{(3+x) \left (x-e^{x/5} (3+x)\right )^2} \, dx+189 \int \frac {1}{(3+x)^2 \left (x-e^{x/5} (3+x)\right )^2} \, dx-648 \int \frac {1}{(3+x)^3 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx-\frac {7776}{5} \int \frac {1}{(3+x)^4 \left (x-e^{x/5} (3+x)\right )^2} \, dx-2187 \int \frac {1}{(3+x)^6 \left (x-e^{x/5} (3+x)\right )^2} \, dx+2997 \int \frac {1}{(3+x)^4 \left (-x+e^{x/5} (3+x)\right )} \, dx+\frac {16038}{5} \int \frac {1}{(3+x)^5 \left (x-e^{x/5} (3+x)\right )^2} \, dx+4374 \int \frac {1}{(3+x)^6 \left (-x+e^{x/5} (3+x)\right )} \, dx-\frac {29646}{5} \int \frac {1}{(3+x)^5 \left (3 e^{x/5}-x+e^{x/5} x\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 7.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3 \left (2+e^x-x-x^2\right )}{-x+e^{x/5} (3+x)} \]

[In]

Integrate[(90 + 30*x + 45*x^2 + 15*x^3 + E^x*(45 - 30*x - 15*x^2) + E^((x + 5*Log[3 + x])/5)*(-93 - 87*x - 3*x
^2 + 3*x^3 + E^x*(21 + 12*x)))/(15*x^2 + 5*x^3 + E^((2*(x + 5*Log[3 + x]))/5)*(15 + 5*x) + E^((x + 5*Log[3 + x
])/5)*(-30*x - 10*x^2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {15 x^{2}-30+15 x -15 \,{\mathrm e}^{x}}{5 x -5 \,{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}}\)	\(32\)
default	\(\frac {-6+3 \,{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}+3 x^{2}}{x -{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}}+\frac {3 \,{\mathrm e}^{x}}{x \,{\mathrm e}^{\frac {x}{5}}-x +3 \,{\mathrm e}^{\frac {x}{5}}}\)	\(62\)
parts	\(\frac {-6+3 \,{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}+3 x^{2}}{x -{\mathrm e}^{\ln \left (3+x \right )+\frac {x}{5}}}+\frac {3 \,{\mathrm e}^{x}}{x \,{\mathrm e}^{\frac {x}{5}}-x +3 \,{\mathrm e}^{\frac {x}{5}}}\)	\(62\)
risch	\(\frac {3 \,{\mathrm e}^{x} x^{3}-3 x^{5}+27 \,{\mathrm e}^{x} x^{2}-30 x^{4}+81 \,{\mathrm e}^{x} x -102 x^{3}+81 \,{\mathrm e}^{x}-108 x^{2}+81 x +162}{\left (x \,{\mathrm e}^{\frac {x}{5}}-x +3 \,{\mathrm e}^{\frac {x}{5}}\right ) \left (3+x \right )^{3}}\)	\(73\)

[In]

int((((12*x+21)*exp(x)+3*x^3-3*x^2-87*x-93)*exp(ln(3+x)+1/5*x)+(-15*x^2-30*x+45)*exp(x)+15*x^3+45*x^2+30*x+90)
/((5*x+15)*exp(ln(3+x)+1/5*x)^2+(-10*x^2-30*x)*exp(ln(3+x)+1/5*x)+5*x^3+15*x^2),x,method=_RETURNVERBOSE)

[Out]

1/5*(15*x^2-30+15*x-15*exp(x))/(x-exp(ln(3+x)+1/5*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (27) = 54\).

Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.41 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3 \, {\left (x^{7} + 16 \, x^{6} + 103 \, x^{5} + 330 \, x^{4} + 495 \, x^{3} + 108 \, x^{2} - 567 \, x - e^{\left (x + 5 \, \log \left (x + 3\right )\right )} - 486\right )}}{x^{6} + 15 \, x^{5} + 90 \, x^{4} + 270 \, x^{3} + 405 \, x^{2} - {\left (x^{5} + 15 \, x^{4} + 90 \, x^{3} + 270 \, x^{2} + 405 \, x + 243\right )} e^{\left (\frac {1}{5} \, x + \log \left (x + 3\right )\right )} + 243 \, x} \]

[In]

integrate((((12*x+21)*exp(x)+3*x^3-3*x^2-87*x-93)*exp(log(3+x)+1/5*x)+(-15*x^2-30*x+45)*exp(x)+15*x^3+45*x^2+3
0*x+90)/((5*x+15)*exp(log(3+x)+1/5*x)^2+(-10*x^2-30*x)*exp(log(3+x)+1/5*x)+5*x^3+15*x^2),x, algorithm="fricas"
)

[Out]

3*(x^7 + 16*x^6 + 103*x^5 + 330*x^4 + 495*x^3 + 108*x^2 - 567*x - e^(x + 5*log(x + 3)) - 486)/(x^6 + 15*x^5 +
90*x^4 + 270*x^3 + 405*x^2 - (x^5 + 15*x^4 + 90*x^3 + 270*x^2 + 405*x + 243)*e^(1/5*x + log(x + 3)) + 243*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (22) = 44\).

Time = 0.32 (sec) , antiderivative size = 366, normalized size of antiderivative = 11.44 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3 x^{4}}{x^{5} + 15 x^{4} + 90 x^{3} + 270 x^{2} + 405 x + 243} + \frac {\left (3 x^{9} + 54 x^{8} + 405 x^{7} + 1620 x^{6} + 3645 x^{5} + 4374 x^{4} + 2187 x^{3}\right ) e^{\frac {x}{5}} + \left (3 x^{9} + 63 x^{8} + 567 x^{7} + 2835 x^{6} + 8505 x^{5} + 15309 x^{4} + 15309 x^{3} + 6561 x^{2}\right ) e^{\frac {2 x}{5}} + \left (3 x^{9} + 72 x^{8} + 756 x^{7} + 4536 x^{6} + 17010 x^{5} + 40824 x^{4} + 61236 x^{3} + 52488 x^{2} + 19683 x\right ) e^{\frac {3 x}{5}} + \left (3 x^{9} + 81 x^{8} + 972 x^{7} + 6804 x^{6} + 30618 x^{5} + 91854 x^{4} + 183708 x^{3} + 236196 x^{2} + 177147 x + 59049\right ) e^{\frac {4 x}{5}}}{x^{10} + 30 x^{9} + 405 x^{8} + 3240 x^{7} + 17010 x^{6} + 61236 x^{5} + 153090 x^{4} + 262440 x^{3} + 295245 x^{2} + 196830 x + 59049} + \frac {- 3 x^{7} - 48 x^{6} - 306 x^{5} - 990 x^{4} - 1485 x^{3} - 324 x^{2} + 1701 x + 1458}{- x^{6} - 15 x^{5} - 90 x^{4} - 270 x^{3} - 405 x^{2} - 243 x + \left (x^{6} + 18 x^{5} + 135 x^{4} + 540 x^{3} + 1215 x^{2} + 1458 x + 729\right ) e^{\frac {x}{5}}} \]

[In]

integrate((((12*x+21)*exp(x)+3*x**3-3*x**2-87*x-93)*exp(ln(3+x)+1/5*x)+(-15*x**2-30*x+45)*exp(x)+15*x**3+45*x*
*2+30*x+90)/((5*x+15)*exp(ln(3+x)+1/5*x)**2+(-10*x**2-30*x)*exp(ln(3+x)+1/5*x)+5*x**3+15*x**2),x)

[Out]

3*x**4/(x**5 + 15*x**4 + 90*x**3 + 270*x**2 + 405*x + 243) + ((3*x**9 + 54*x**8 + 405*x**7 + 1620*x**6 + 3645*
x**5 + 4374*x**4 + 2187*x**3)*exp(x/5) + (3*x**9 + 63*x**8 + 567*x**7 + 2835*x**6 + 8505*x**5 + 15309*x**4 + 1
5309*x**3 + 6561*x**2)*exp(2*x/5) + (3*x**9 + 72*x**8 + 756*x**7 + 4536*x**6 + 17010*x**5 + 40824*x**4 + 61236
*x**3 + 52488*x**2 + 19683*x)*exp(3*x/5) + (3*x**9 + 81*x**8 + 972*x**7 + 6804*x**6 + 30618*x**5 + 91854*x**4
+ 183708*x**3 + 236196*x**2 + 177147*x + 59049)*exp(4*x/5))/(x**10 + 30*x**9 + 405*x**8 + 3240*x**7 + 17010*x*
*6 + 61236*x**5 + 153090*x**4 + 262440*x**3 + 295245*x**2 + 196830*x + 59049) + (-3*x**7 - 48*x**6 - 306*x**5
- 990*x**4 - 1485*x**3 - 324*x**2 + 1701*x + 1458)/(-x**6 - 15*x**5 - 90*x**4 - 270*x**3 - 405*x**2 - 243*x +
(x**6 + 18*x**5 + 135*x**4 + 540*x**3 + 1215*x**2 + 1458*x + 729)*exp(x/5))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=-\frac {3 \, {\left (x^{2} + x - e^{x} - 2\right )}}{{\left (x + 3\right )} e^{\left (\frac {1}{5} \, x\right )} - x} \]

[In]

integrate((((12*x+21)*exp(x)+3*x^3-3*x^2-87*x-93)*exp(log(3+x)+1/5*x)+(-15*x^2-30*x+45)*exp(x)+15*x^3+45*x^2+3
0*x+90)/((5*x+15)*exp(log(3+x)+1/5*x)^2+(-10*x^2-30*x)*exp(log(3+x)+1/5*x)+5*x^3+15*x^2),x, algorithm="maxima"
)

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (27) = 54\).

Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.38 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=-\frac {3 \, {\left (x^{5} + 10 \, x^{4} - x^{3} e^{x} + 34 \, x^{3} - 9 \, x^{2} e^{x} + 36 \, x^{2} - 27 \, x e^{x} - 27 \, x - 27 \, e^{x} - 54\right )}}{x^{4} e^{\left (\frac {1}{5} \, x\right )} - x^{4} + 12 \, x^{3} e^{\left (\frac {1}{5} \, x\right )} - 9 \, x^{3} + 54 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} - 27 \, x^{2} + 108 \, x e^{\left (\frac {1}{5} \, x\right )} - 27 \, x + 81 \, e^{\left (\frac {1}{5} \, x\right )}} \]

[In]

integrate((((12*x+21)*exp(x)+3*x^3-3*x^2-87*x-93)*exp(log(3+x)+1/5*x)+(-15*x^2-30*x+45)*exp(x)+15*x^3+45*x^2+3
0*x+90)/((5*x+15)*exp(log(3+x)+1/5*x)^2+(-10*x^2-30*x)*exp(log(3+x)+1/5*x)+5*x^3+15*x^2),x, algorithm="giac")

[Out]

-3*(x^5 + 10*x^4 - x^3*e^x + 34*x^3 - 9*x^2*e^x + 36*x^2 - 27*x*e^x - 27*x - 27*e^x - 54)/(x^4*e^(1/5*x) - x^4
+ 12*x^3*e^(1/5*x) - 9*x^3 + 54*x^2*e^(1/5*x) - 27*x^2 + 108*x*e^(1/5*x) - 27*x + 81*e^(1/5*x))

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {90+30 x+45 x^2+15 x^3+e^x \left (45-30 x-15 x^2\right )+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-93-87 x-3 x^2+3 x^3+e^x (21+12 x)\right )}{15 x^2+5 x^3+e^{\frac {2}{5} (x+5 \log (3+x))} (15+5 x)+e^{\frac {1}{5} (x+5 \log (3+x))} \left (-30 x-10 x^2\right )} \, dx=\frac {3\,x-3\,{\mathrm {e}}^x+3\,x^2-6}{x-{\mathrm {e}}^{x/5}\,\left (x+3\right )} \]

[In]

int((30*x - exp(x/5 + log(x + 3))*(87*x - exp(x)*(12*x + 21) + 3*x^2 - 3*x^3 + 93) - exp(x)*(30*x + 15*x^2 - 4
5) + 45*x^2 + 15*x^3 + 90)/(exp((2*x)/5 + 2*log(x + 3))*(5*x + 15) - exp(x/5 + log(x + 3))*(30*x + 10*x^2) + 1
5*x^2 + 5*x^3),x)

[Out]

(3*x - 3*exp(x) + 3*x^2 - 6)/(x - exp(x/5)*(x + 3))