Integrand size = 80, antiderivative size = 30 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=e^x+\frac {1}{x}-x^2 \left (x+\frac {2 x^2}{2 x+x^2}\right )^2 \] Output:
exp(x)+1/x-(2*x^2/(x^2+2*x)+x)^2*x^2
Time = 1.92 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=e^x+\frac {1}{x}+4 x^2-4 x^3-x^4-\frac {63}{(2+x)^2}-\frac {x^2}{4 (2+x)^2}+\frac {63}{2+x} \] Input:
Integrate[(-8 - 12*x - 6*x^2 - x^3 - 128*x^5 - 112*x^6 - 36*x^7 - 4*x^8 + E^x*(8*x^2 + 12*x^3 + 6*x^4 + x^5))/(8*x^2 + 12*x^3 + 6*x^4 + x^5),x]
Output:
E^x + x^(-1) + 4*x^2 - 4*x^3 - x^4 - 63/(2 + x)^2 - x^2/(4*(2 + x)^2) + 63 /(2 + x)
Time = 0.78 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2026, 2007, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-4 x^8-36 x^7-112 x^6-128 x^5-x^3-6 x^2+e^x \left (x^5+6 x^4+12 x^3+8 x^2\right )-12 x-8}{x^5+6 x^4+12 x^3+8 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-4 x^8-36 x^7-112 x^6-128 x^5-x^3-6 x^2+e^x \left (x^5+6 x^4+12 x^3+8 x^2\right )-12 x-8}{x^2 \left (x^3+6 x^2+12 x+8\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-4 x^8-36 x^7-112 x^6-128 x^5-x^3-6 x^2+e^x \left (x^5+6 x^4+12 x^3+8 x^2\right )-12 x-8}{x^2 (x+2)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4 x^6}{(x+2)^3}-\frac {36 x^5}{(x+2)^3}-\frac {112 x^4}{(x+2)^3}-\frac {128 x^3}{(x+2)^3}-\frac {8}{(x+2)^3 x^2}-\frac {x}{(x+2)^3}+e^x-\frac {6}{(x+2)^3}-\frac {12}{(x+2)^3 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -x^4-4 x^3-\frac {x^2}{4 (x+2)^2}+4 x^2+e^x+\frac {63}{x+2}-\frac {63}{(x+2)^2}+\frac {1}{x}\) |
Input:
Int[(-8 - 12*x - 6*x^2 - x^3 - 128*x^5 - 112*x^6 - 36*x^7 - 4*x^8 + E^x*(8 *x^2 + 12*x^3 + 6*x^4 + x^5))/(8*x^2 + 12*x^3 + 6*x^4 + x^5),x]
Output:
E^x + x^(-1) + 4*x^2 - 4*x^3 - x^4 - 63/(2 + x)^2 - x^2/(4*(2 + x)^2) + 63 /(2 + x)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 1.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20
method | result | size |
default | \({\mathrm e}^{x}+\frac {1}{x}-\frac {64}{\left (2+x \right )^{2}}+\frac {64}{2+x}+4 x^{2}-4 x^{3}-x^{4}\) | \(36\) |
parts | \({\mathrm e}^{x}+\frac {1}{x}-\frac {64}{\left (2+x \right )^{2}}+\frac {64}{2+x}+4 x^{2}-4 x^{3}-x^{4}\) | \(36\) |
risch | \(-x^{4}-4 x^{3}+4 x^{2}+\frac {65 x^{2}+68 x +4}{x \left (x^{2}+4 x +4\right )}+{\mathrm e}^{x}\) | \(43\) |
norman | \(\frac {4+x^{2}+4 x +{\mathrm e}^{x} x^{3}-16 x^{5}-8 x^{6}-x^{7}+4 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{2}}{x \left (2+x \right )^{2}}\) | \(51\) |
parallelrisch | \(-\frac {x^{7}+8 x^{6}+16 x^{5}-{\mathrm e}^{x} x^{3}-4-4 \,{\mathrm e}^{x} x^{2}-x^{2}-4 \,{\mathrm e}^{x} x -4 x}{x \left (x^{2}+4 x +4\right )}\) | \(58\) |
orering | \(\frac {\left (x^{11}+12 x^{10}+40 x^{9}-24 x^{8}-280 x^{7}+127 x^{6}+2296 x^{5}+4074 x^{4}+2032 x^{3}+32 x^{2}+64 x +32\right ) \left (\left (x^{5}+6 x^{4}+12 x^{3}+8 x^{2}\right ) {\mathrm e}^{x}-4 x^{8}-36 x^{7}-112 x^{6}-128 x^{5}-x^{3}-6 x^{2}-12 x -8\right )}{\left (4 x^{10}+32 x^{9}+64 x^{8}-120 x^{7}-640 x^{6}-767 x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+80 x +32\right ) \left (x^{5}+6 x^{4}+12 x^{3}+8 x^{2}\right )}-\frac {\left (x^{9}+6 x^{8}-4 x^{7}-80 x^{6}+767 x^{4}+1529 x^{3}+1006 x^{2}-20 x -8\right ) x \left (2+x \right ) \left (\frac {\left (5 x^{4}+24 x^{3}+36 x^{2}+16 x \right ) {\mathrm e}^{x}+\left (x^{5}+6 x^{4}+12 x^{3}+8 x^{2}\right ) {\mathrm e}^{x}-32 x^{7}-252 x^{6}-672 x^{5}-640 x^{4}-3 x^{2}-12 x -12}{x^{5}+6 x^{4}+12 x^{3}+8 x^{2}}-\frac {\left (\left (x^{5}+6 x^{4}+12 x^{3}+8 x^{2}\right ) {\mathrm e}^{x}-4 x^{8}-36 x^{7}-112 x^{6}-128 x^{5}-x^{3}-6 x^{2}-12 x -8\right ) \left (5 x^{4}+24 x^{3}+36 x^{2}+16 x \right )}{\left (x^{5}+6 x^{4}+12 x^{3}+8 x^{2}\right )^{2}}\right )}{4 x^{10}+32 x^{9}+64 x^{8}-120 x^{7}-640 x^{6}-767 x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+80 x +32}\) | \(478\) |
Input:
int(((x^5+6*x^4+12*x^3+8*x^2)*exp(x)-4*x^8-36*x^7-112*x^6-128*x^5-x^3-6*x^ 2-12*x-8)/(x^5+6*x^4+12*x^3+8*x^2),x,method=_RETURNVERBOSE)
Output:
exp(x)+1/x-64/(2+x)^2+64/(2+x)+4*x^2-4*x^3-x^4
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.00 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=-\frac {x^{7} + 8 \, x^{6} + 16 \, x^{5} - 16 \, x^{3} - 65 \, x^{2} - {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} e^{x} - 68 \, x - 4}{x^{3} + 4 \, x^{2} + 4 \, x} \] Input:
integrate(((x^5+6*x^4+12*x^3+8*x^2)*exp(x)-4*x^8-36*x^7-112*x^6-128*x^5-x^ 3-6*x^2-12*x-8)/(x^5+6*x^4+12*x^3+8*x^2),x, algorithm="fricas")
Output:
-(x^7 + 8*x^6 + 16*x^5 - 16*x^3 - 65*x^2 - (x^3 + 4*x^2 + 4*x)*e^x - 68*x - 4)/(x^3 + 4*x^2 + 4*x)
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=- x^{4} - 4 x^{3} + 4 x^{2} - \frac {- 65 x^{2} - 68 x - 4}{x^{3} + 4 x^{2} + 4 x} + e^{x} \] Input:
integrate(((x**5+6*x**4+12*x**3+8*x**2)*exp(x)-4*x**8-36*x**7-112*x**6-128 *x**5-x**3-6*x**2-12*x-8)/(x**5+6*x**4+12*x**3+8*x**2),x)
Output:
-x**4 - 4*x**3 + 4*x**2 - (-65*x**2 - 68*x - 4)/(x**3 + 4*x**2 + 4*x) + ex p(x)
\[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=\int { -\frac {4 \, x^{8} + 36 \, x^{7} + 112 \, x^{6} + 128 \, x^{5} + x^{3} + 6 \, x^{2} - {\left (x^{5} + 6 \, x^{4} + 12 \, x^{3} + 8 \, x^{2}\right )} e^{x} + 12 \, x + 8}{x^{5} + 6 \, x^{4} + 12 \, x^{3} + 8 \, x^{2}} \,d x } \] Input:
integrate(((x^5+6*x^4+12*x^3+8*x^2)*exp(x)-4*x^8-36*x^7-112*x^6-128*x^5-x^ 3-6*x^2-12*x-8)/(x^5+6*x^4+12*x^3+8*x^2),x, algorithm="maxima")
Output:
-x^4 - 4*x^3 + 4*x^2 + (x^3 + 6*x^2)*e^x/(x^3 + 6*x^2 + 12*x + 8) + (3*x^2 + 9*x + 4)/(x^3 + 4*x^2 + 4*x) - 128*(6*x + 11)/(x^2 + 4*x + 4) + 576*(5* x + 9)/(x^2 + 4*x + 4) - 896*(4*x + 7)/(x^2 + 4*x + 4) + 512*(3*x + 5)/(x^ 2 + 4*x + 4) - 3*(x + 3)/(x^2 + 4*x + 4) + (x + 1)/(x^2 + 4*x + 4) + 12*e^ x/(x^2 + 4*x + 4) - 8*e^(-2)*exp_integral_e(3, -x - 2)/(x + 2)^2 + 3/(x^2 + 4*x + 4) - 24*integrate(x*e^x/(x^4 + 8*x^3 + 24*x^2 + 32*x + 16), x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=-\frac {x^{7} + 8 \, x^{6} + 16 \, x^{5} - x^{3} e^{x} - 16 \, x^{3} - 4 \, x^{2} e^{x} - 65 \, x^{2} - 4 \, x e^{x} - 68 \, x - 4}{x^{3} + 4 \, x^{2} + 4 \, x} \] Input:
integrate(((x^5+6*x^4+12*x^3+8*x^2)*exp(x)-4*x^8-36*x^7-112*x^6-128*x^5-x^ 3-6*x^2-12*x-8)/(x^5+6*x^4+12*x^3+8*x^2),x, algorithm="giac")
Output:
-(x^7 + 8*x^6 + 16*x^5 - x^3*e^x - 16*x^3 - 4*x^2*e^x - 65*x^2 - 4*x*e^x - 68*x - 4)/(x^3 + 4*x^2 + 4*x)
Time = 2.93 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx={\mathrm {e}}^x+4\,x^2-4\,x^3-x^4+\frac {65\,x^2+68\,x+4}{x\,{\left (x+2\right )}^2} \] Input:
int(-(12*x - exp(x)*(8*x^2 + 12*x^3 + 6*x^4 + x^5) + 6*x^2 + x^3 + 128*x^5 + 112*x^6 + 36*x^7 + 4*x^8 + 8)/(8*x^2 + 12*x^3 + 6*x^4 + x^5),x)
Output:
exp(x) + 4*x^2 - 4*x^3 - x^4 + (68*x + 65*x^2 + 4)/(x*(x + 2)^2)
Time = 0.16 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {-8-12 x-6 x^2-x^3-128 x^5-112 x^6-36 x^7-4 x^8+e^x \left (8 x^2+12 x^3+6 x^4+x^5\right )}{8 x^2+12 x^3+6 x^4+x^5} \, dx=\frac {4 e^{x} x^{3}+16 e^{x} x^{2}+16 e^{x} x -4 x^{7}-32 x^{6}-64 x^{5}-x^{3}+12 x +16}{4 x \left (x^{2}+4 x +4\right )} \] Input:
int(((x^5+6*x^4+12*x^3+8*x^2)*exp(x)-4*x^8-36*x^7-112*x^6-128*x^5-x^3-6*x^ 2-12*x-8)/(x^5+6*x^4+12*x^3+8*x^2),x)
Output:
(4*e**x*x**3 + 16*e**x*x**2 + 16*e**x*x - 4*x**7 - 32*x**6 - 64*x**5 - x** 3 + 12*x + 16)/(4*x*(x**2 + 4*x + 4))