Integrand size = 62, antiderivative size = 34 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=\frac {3 \left (-\frac {e^{12+x}}{x^2}+x \left (-x^2+\frac {3}{x (1+x)}\right )\right )}{x} \] Output:
3*((3/x/(1+x)-x^2)*x-exp(12)*exp(x)/x^2)/x
Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-\frac {3 e^{12+x}}{x^3}+\frac {9}{x}-3 x^2-\frac {9}{1+x} \] Input:
Integrate[(-9*x^2 - 18*x^3 - 6*x^5 - 12*x^6 - 6*x^7 + E^(12 + x)*(9 + 15*x + 3*x^2 - 3*x^3))/(x^4 + 2*x^5 + x^6),x]
Output:
(-3*E^(12 + x))/x^3 + 9/x - 3*x^2 - 9/(1 + x)
Time = 0.66 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, 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.
\(\displaystyle \int \frac {-6 x^7-12 x^6-6 x^5-18 x^3-9 x^2+e^{x+12} \left (-3 x^3+3 x^2+15 x+9\right )}{x^6+2 x^5+x^4} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-6 x^7-12 x^6-6 x^5-18 x^3-9 x^2+e^{x+12} \left (-3 x^3+3 x^2+15 x+9\right )}{x^4 \left (x^2+2 x+1\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-6 x^7-12 x^6-6 x^5-18 x^3-9 x^2+e^{x+12} \left (-3 x^3+3 x^2+15 x+9\right )}{x^4 (x+1)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {3 e^{x+12} (x-3)}{x^4}-\frac {6 x^3}{(x+1)^2}-\frac {12 x^2}{(x+1)^2}-\frac {9}{(x+1)^2 x^2}-\frac {6 x}{(x+1)^2}-\frac {18}{(x+1)^2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 e^{x+12}}{x^3}-3 x^2-\frac {9}{x+1}+\frac {9}{x}\) |
Input:
Int[(-9*x^2 - 18*x^3 - 6*x^5 - 12*x^6 - 6*x^7 + E^(12 + x)*(9 + 15*x + 3*x ^2 - 3*x^3))/(x^4 + 2*x^5 + x^6),x]
Output:
(-3*E^(12 + x))/x^3 + 9/x - 3*x^2 - 9/(1 + 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 = 0.55 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-3 x^{2}+\frac {9}{x \left (1+x \right )}-\frac {3 \,{\mathrm e}^{x +12}}{x^{3}}\) | \(26\) |
parts | \(\frac {9}{x}-\frac {9}{1+x}-3 x^{2}-\frac {3 \,{\mathrm e}^{12} {\mathrm e}^{x}}{x^{3}}\) | \(28\) |
norman | \(\frac {9 x^{2}-3 x^{5}-3 x^{6}-3 \,{\mathrm e}^{12} {\mathrm e}^{x}-3 \,{\mathrm e}^{x} {\mathrm e}^{12} x}{x^{3} \left (1+x \right )}\) | \(39\) |
parallelrisch | \(-\frac {3 x^{6}+3 x^{5}+3 \,{\mathrm e}^{x} {\mathrm e}^{12} x +3 \,{\mathrm e}^{12} {\mathrm e}^{x}-9 x^{2}}{x^{3} \left (1+x \right )}\) | \(40\) |
default | \(\frac {9}{x}-\frac {9}{1+x}-3 x^{2}+9 \,{\mathrm e}^{12} \left (-\frac {{\mathrm e}^{x}}{3 x^{3}}+\frac {5 \,{\mathrm e}^{x}}{6 x^{2}}-\frac {13 \,{\mathrm e}^{x}}{6 x}+\frac {11 \,\operatorname {expIntegral}_{1}\left (-x \right )}{6}-5 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-1-x \right )-\frac {{\mathrm e}^{x}}{1+x}\right )+15 \,{\mathrm e}^{12} \left (-\frac {3 \,\operatorname {expIntegral}_{1}\left (-x \right )}{2}+4 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-1-x \right )+\frac {3 \,{\mathrm e}^{x}}{2 x}-\frac {{\mathrm e}^{x}}{2 x^{2}}+\frac {{\mathrm e}^{x}}{1+x}\right )+3 \,{\mathrm e}^{12} \left (\operatorname {expIntegral}_{1}\left (-x \right )-3 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-1-x \right )-\frac {{\mathrm e}^{x}}{x}-\frac {{\mathrm e}^{x}}{1+x}\right )-3 \,{\mathrm e}^{12} \left (-\operatorname {expIntegral}_{1}\left (-x \right )+2 \,{\mathrm e}^{-1} \operatorname {expIntegral}_{1}\left (-1-x \right )+\frac {{\mathrm e}^{x}}{1+x}\right )\) | \(185\) |
orering | \(\frac {x \left (x^{8}-3 x^{7}+4 x^{6}-14 x^{5}-6 x^{4}+193 x^{3}+285 x^{2}+72 x -30\right ) \left (\left (-3 x^{3}+3 x^{2}+15 x +9\right ) {\mathrm e}^{12} {\mathrm e}^{x}-6 x^{7}-12 x^{6}-6 x^{5}-18 x^{3}-9 x^{2}\right )}{\left (2 x^{8}-8 x^{7}-6 x^{6}+50 x^{5}+82 x^{4}+21 x^{3}-15 x^{2}+42 x +18\right ) \left (x^{6}+2 x^{5}+x^{4}\right )}-\frac {\left (x^{6}-3 x^{5}-14 x^{3}-48 x^{2}-27 x +6\right ) x^{2} \left (1+x \right ) \left (\frac {\left (-9 x^{2}+6 x +15\right ) {\mathrm e}^{12} {\mathrm e}^{x}+\left (-3 x^{3}+3 x^{2}+15 x +9\right ) {\mathrm e}^{12} {\mathrm e}^{x}-42 x^{6}-72 x^{5}-30 x^{4}-54 x^{2}-18 x}{x^{6}+2 x^{5}+x^{4}}-\frac {\left (\left (-3 x^{3}+3 x^{2}+15 x +9\right ) {\mathrm e}^{12} {\mathrm e}^{x}-6 x^{7}-12 x^{6}-6 x^{5}-18 x^{3}-9 x^{2}\right ) \left (6 x^{5}+10 x^{4}+4 x^{3}\right )}{\left (x^{6}+2 x^{5}+x^{4}\right )^{2}}\right )}{2 x^{8}-8 x^{7}-6 x^{6}+50 x^{5}+82 x^{4}+21 x^{3}-15 x^{2}+42 x +18}\) | \(370\) |
Input:
int(((-3*x^3+3*x^2+15*x+9)*exp(12)*exp(x)-6*x^7-12*x^6-6*x^5-18*x^3-9*x^2) /(x^6+2*x^5+x^4),x,method=_RETURNVERBOSE)
Output:
-3*x^2+9/x/(1+x)-3/x^3*exp(x+12)
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-\frac {3 \, {\left (x^{6} + x^{5} - 3 \, x^{2} + {\left (x + 1\right )} e^{\left (x + 12\right )}\right )}}{x^{4} + x^{3}} \] Input:
integrate(((-3*x^3+3*x^2+15*x+9)*exp(12)*exp(x)-6*x^7-12*x^6-6*x^5-18*x^3- 9*x^2)/(x^6+2*x^5+x^4),x, algorithm="fricas")
Output:
-3*(x^6 + x^5 - 3*x^2 + (x + 1)*e^(x + 12))/(x^4 + x^3)
Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=- 3 x^{2} + \frac {9}{x^{2} + x} - \frac {3 e^{12} e^{x}}{x^{3}} \] Input:
integrate(((-3*x**3+3*x**2+15*x+9)*exp(12)*exp(x)-6*x**7-12*x**6-6*x**5-18 *x**3-9*x**2)/(x**6+2*x**5+x**4),x)
Output:
-3*x**2 + 9/(x**2 + x) - 3*exp(12)*exp(x)/x**3
Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-3 \, x^{2} + \frac {9 \, {\left (2 \, x + 1\right )}}{x^{2} + x} - \frac {18}{x + 1} - \frac {3 \, e^{\left (x + 12\right )}}{x^{3}} \] Input:
integrate(((-3*x^3+3*x^2+15*x+9)*exp(12)*exp(x)-6*x^7-12*x^6-6*x^5-18*x^3- 9*x^2)/(x^6+2*x^5+x^4),x, algorithm="maxima")
Output:
-3*x^2 + 9*(2*x + 1)/(x^2 + x) - 18/(x + 1) - 3*e^(x + 12)/x^3
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-\frac {3 \, {\left (x^{6} + x^{5} - 3 \, x^{2} + x e^{\left (x + 12\right )} + e^{\left (x + 12\right )}\right )}}{x^{4} + x^{3}} \] Input:
integrate(((-3*x^3+3*x^2+15*x+9)*exp(12)*exp(x)-6*x^7-12*x^6-6*x^5-18*x^3- 9*x^2)/(x^6+2*x^5+x^4),x, algorithm="giac")
Output:
-3*(x^6 + x^5 - 3*x^2 + x*e^(x + 12) + e^(x + 12))/(x^4 + x^3)
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=-3\,x^2-\frac {3\,{\mathrm {e}}^{x+12}+3\,x\,{\mathrm {e}}^{x+12}-9\,x^2}{x^3\,\left (x+1\right )} \] Input:
int(-(9*x^2 + 18*x^3 + 6*x^5 + 12*x^6 + 6*x^7 - exp(12)*exp(x)*(15*x + 3*x ^2 - 3*x^3 + 9))/(x^4 + 2*x^5 + x^6),x)
Output:
- 3*x^2 - (3*exp(x + 12) + 3*x*exp(x + 12) - 9*x^2)/(x^3*(x + 1))
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {-9 x^2-18 x^3-6 x^5-12 x^6-6 x^7+e^{12+x} \left (9+15 x+3 x^2-3 x^3\right )}{x^4+2 x^5+x^6} \, dx=\frac {-3 e^{x} e^{12} x -3 e^{x} e^{12}-3 x^{6}-3 x^{5}+9 x^{2}}{x^{3} \left (x +1\right )} \] Input:
int(((-3*x^3+3*x^2+15*x+9)*exp(12)*exp(x)-6*x^7-12*x^6-6*x^5-18*x^3-9*x^2) /(x^6+2*x^5+x^4),x)
Output:
(3*( - e**x*e**12*x - e**x*e**12 - x**6 - x**5 + 3*x**2))/(x**3*(x + 1))