Integrand size = 78, antiderivative size = 28 \[ \int \frac {e^{2 x} \left (8-20 x+24 x^2-14 x^3+2 x^4\right )+e^{3+x} \left (-5184 x^3+3888 x^4-972 x^5+81 x^6\right )}{-5184 x^3+3888 x^4-972 x^5+81 x^6} \, dx=e^{3+x}+\frac {e^{2 x} (-1+x)^2}{81 (-4+x)^2 x^2} \] Output:
1/81*exp(x)^2*(-1+x)^2/(-4+x)^2/x^2+exp(3+x)
Time = 0.59 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {e^{2 x} \left (8-20 x+24 x^2-14 x^3+2 x^4\right )+e^{3+x} \left (-5184 x^3+3888 x^4-972 x^5+81 x^6\right )}{-5184 x^3+3888 x^4-972 x^5+81 x^6} \, dx=e^{3+x}+\frac {2}{81} e^{2 x} \left (\frac {9}{32 (-4+x)^2}+\frac {3}{64 (-4+x)}+\frac {1}{32 x^2}-\frac {3}{64 x}\right ) \] Input:
Integrate[(E^(2*x)*(8 - 20*x + 24*x^2 - 14*x^3 + 2*x^4) + E^(3 + x)*(-5184 *x^3 + 3888*x^4 - 972*x^5 + 81*x^6))/(-5184*x^3 + 3888*x^4 - 972*x^5 + 81* x^6),x]
Output:
E^(3 + x) + (2*E^(2*x)*(9/(32*(-4 + x)^2) + 3/(64*(-4 + x)) + 1/(32*x^2) - 3/(64*x)))/81
Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(28)=56\).
Time = 0.98 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2026, 2007, 7239, 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.
\(\displaystyle \int \frac {e^{2 x} \left (2 x^4-14 x^3+24 x^2-20 x+8\right )+e^{x+3} \left (81 x^6-972 x^5+3888 x^4-5184 x^3\right )}{81 x^6-972 x^5+3888 x^4-5184 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{2 x} \left (2 x^4-14 x^3+24 x^2-20 x+8\right )+e^{x+3} \left (81 x^6-972 x^5+3888 x^4-5184 x^3\right )}{x^3 \left (81 x^3-972 x^2+3888 x-5184\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{2 x} \left (2 x^4-14 x^3+24 x^2-20 x+8\right )+e^{x+3} \left (81 x^6-972 x^5+3888 x^4-5184 x^3\right )}{x^3 \left (3 \sqrt [3]{3} x-12 \sqrt [3]{3}\right )^3}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\frac {2 e^{2 x} \left (x^4-7 x^3+12 x^2-10 x+4\right )}{81 (x-4)^3 x^3}+e^{x+3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{2 x}}{1296 x^2}+e^{x+3}-\frac {e^{2 x}}{864 x}-\frac {e^{2 x}}{864 (4-x)}+\frac {e^{2 x}}{144 (4-x)^2}\) |
Input:
Int[(E^(2*x)*(8 - 20*x + 24*x^2 - 14*x^3 + 2*x^4) + E^(3 + x)*(-5184*x^3 + 3888*x^4 - 972*x^5 + 81*x^6))/(-5184*x^3 + 3888*x^4 - 972*x^5 + 81*x^6),x ]
Output:
E^(3 + x) + E^(2*x)/(144*(4 - x)^2) - E^(2*x)/(864*(4 - x)) + E^(2*x)/(129 6*x^2) - E^(2*x)/(864*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])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {\left (x^{2}-2 x +1\right ) {\mathrm e}^{2 x}}{81 \left (x -4\right )^{2} x^{2}}+{\mathrm e}^{3+x}\) | \(28\) |
parts | \(\frac {{\mathrm e}^{2 x}}{864 x -3456}+\frac {{\mathrm e}^{2 x}}{144 \left (x -4\right )^{2}}-\frac {{\mathrm e}^{2 x}}{864 x}+\frac {{\mathrm e}^{2 x}}{1296 x^{2}}+{\mathrm e}^{3+x}\) | \(46\) |
norman | \(\frac {{\mathrm e}^{x} {\mathrm e}^{3} x^{4}+\frac {{\mathrm e}^{2 x}}{81}-\frac {2 x \,{\mathrm e}^{2 x}}{81}+\frac {{\mathrm e}^{2 x} x^{2}}{81}+16 x^{2} {\mathrm e}^{3} {\mathrm e}^{x}-8 \,{\mathrm e}^{x} {\mathrm e}^{3} x^{3}}{\left (x -4\right )^{2} x^{2}}\) | \(59\) |
parallelrisch | \(\frac {648 \,{\mathrm e}^{3+x} x^{4}-5184 x^{3} {\mathrm e}^{3+x}+8 \,{\mathrm e}^{2 x} x^{2}+10368 x^{2} {\mathrm e}^{3+x}-16 x \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{2 x}}{648 x^{2} \left (x^{2}-8 x +16\right )}\) | \(66\) |
default | \(\frac {{\mathrm e}^{2 x}}{864 x -3456}-\frac {{\mathrm e}^{2 x}}{864 x}+\frac {{\mathrm e}^{2 x}}{144 \left (x -4\right )^{2}}+\frac {{\mathrm e}^{2 x}}{1296 x^{2}}-64 \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{x}}{2 \left (x -4\right )^{2}}-\frac {{\mathrm e}^{x}}{2 \left (x -4\right )}-\frac {{\mathrm e}^{4} \operatorname {expIntegral}_{1}\left (-x +4\right )}{2}\right )+48 \,{\mathrm e}^{3} \left (-\frac {3 \,{\mathrm e}^{x}}{x -4}-3 \,{\mathrm e}^{4} \operatorname {expIntegral}_{1}\left (-x +4\right )-\frac {2 \,{\mathrm e}^{x}}{\left (x -4\right )^{2}}\right )-12 \,{\mathrm e}^{3} \left (-\frac {16 \,{\mathrm e}^{x}}{x -4}-17 \,{\mathrm e}^{4} \operatorname {expIntegral}_{1}\left (-x +4\right )-\frac {8 \,{\mathrm e}^{x}}{\left (x -4\right )^{2}}\right )+{\mathrm e}^{3} \left ({\mathrm e}^{x}-\frac {80 \,{\mathrm e}^{x}}{x -4}-92 \,{\mathrm e}^{4} \operatorname {expIntegral}_{1}\left (-x +4\right )-\frac {32 \,{\mathrm e}^{x}}{\left (x -4\right )^{2}}\right )\) | \(179\) |
orering | \(\frac {\left (3 x^{6}-38 x^{5}+161 x^{4}-288 x^{3}+356 x^{2}-272 x +96\right ) \left (\left (81 x^{6}-972 x^{5}+3888 x^{4}-5184 x^{3}\right ) {\mathrm e}^{3+x}+\left (2 x^{4}-14 x^{3}+24 x^{2}-20 x +8\right ) {\mathrm e}^{2 x}\right )}{2 \left (x^{6}-13 x^{5}+57 x^{4}-106 x^{3}+142 x^{2}-120 x +48\right ) \left (81 x^{6}-972 x^{5}+3888 x^{4}-5184 x^{3}\right )}-\frac {\left (x^{4}-8 x^{3}+15 x^{2}-16 x +8\right ) \left (x -4\right ) x \left (\frac {\left (486 x^{5}-4860 x^{4}+15552 x^{3}-15552 x^{2}\right ) {\mathrm e}^{3+x}+\left (81 x^{6}-972 x^{5}+3888 x^{4}-5184 x^{3}\right ) {\mathrm e}^{3+x}+\left (8 x^{3}-42 x^{2}+48 x -20\right ) {\mathrm e}^{2 x}+2 \left (2 x^{4}-14 x^{3}+24 x^{2}-20 x +8\right ) {\mathrm e}^{2 x}}{81 x^{6}-972 x^{5}+3888 x^{4}-5184 x^{3}}-\frac {\left (\left (81 x^{6}-972 x^{5}+3888 x^{4}-5184 x^{3}\right ) {\mathrm e}^{3+x}+\left (2 x^{4}-14 x^{3}+24 x^{2}-20 x +8\right ) {\mathrm e}^{2 x}\right ) \left (486 x^{5}-4860 x^{4}+15552 x^{3}-15552 x^{2}\right )}{\left (81 x^{6}-972 x^{5}+3888 x^{4}-5184 x^{3}\right )^{2}}\right )}{2 \left (x^{6}-13 x^{5}+57 x^{4}-106 x^{3}+142 x^{2}-120 x +48\right )}\) | \(415\) |
Input:
int(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x +8)*exp(x)^2)/(81*x^6-972*x^5+3888*x^4-5184*x^3),x,method=_RETURNVERBOSE)
Output:
1/81*(x^2-2*x+1)/(x-4)^2/x^2*exp(2*x)+exp(3+x)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {e^{2 x} \left (8-20 x+24 x^2-14 x^3+2 x^4\right )+e^{3+x} \left (-5184 x^3+3888 x^4-972 x^5+81 x^6\right )}{-5184 x^3+3888 x^4-972 x^5+81 x^6} \, dx=\frac {{\left ({\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x + 6\right )} + 81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (x + 9\right )}\right )} e^{\left (-6\right )}}{81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )}} \] Input:
integrate(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^ 2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+3888*x^4-5184*x^3),x, algorithm="frica s")
Output:
1/81*((x^2 - 2*x + 1)*e^(2*x + 6) + 81*(x^4 - 8*x^3 + 16*x^2)*e^(x + 9))*e ^(-6)/(x^4 - 8*x^3 + 16*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {e^{2 x} \left (8-20 x+24 x^2-14 x^3+2 x^4\right )+e^{3+x} \left (-5184 x^3+3888 x^4-972 x^5+81 x^6\right )}{-5184 x^3+3888 x^4-972 x^5+81 x^6} \, dx=\frac {\left (x^{2} - 2 x + 1\right ) e^{2 x} + \left (81 x^{4} e^{3} - 648 x^{3} e^{3} + 1296 x^{2} e^{3}\right ) \sqrt {e^{2 x}}}{81 x^{4} - 648 x^{3} + 1296 x^{2}} \] Input:
integrate(((81*x**6-972*x**5+3888*x**4-5184*x**3)*exp(3+x)+(2*x**4-14*x**3 +24*x**2-20*x+8)*exp(x)**2)/(81*x**6-972*x**5+3888*x**4-5184*x**3),x)
Output:
((x**2 - 2*x + 1)*exp(2*x) + (81*x**4*exp(3) - 648*x**3*exp(3) + 1296*x**2 *exp(3))*sqrt(exp(2*x)))/(81*x**4 - 648*x**3 + 1296*x**2)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04 \[ \int \frac {e^{2 x} \left (8-20 x+24 x^2-14 x^3+2 x^4\right )+e^{3+x} \left (-5184 x^3+3888 x^4-972 x^5+81 x^6\right )}{-5184 x^3+3888 x^4-972 x^5+81 x^6} \, dx=\frac {{\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + 81 \, {\left (x^{4} e^{3} - 8 \, x^{3} e^{3} + 16 \, x^{2} e^{3}\right )} e^{x}}{81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )}} \] Input:
integrate(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^ 2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+3888*x^4-5184*x^3),x, algorithm="maxim a")
Output:
1/81*((x^2 - 2*x + 1)*e^(2*x) + 81*(x^4*e^3 - 8*x^3*e^3 + 16*x^2*e^3)*e^x) /(x^4 - 8*x^3 + 16*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.36 \[ \int \frac {e^{2 x} \left (8-20 x+24 x^2-14 x^3+2 x^4\right )+e^{3+x} \left (-5184 x^3+3888 x^4-972 x^5+81 x^6\right )}{-5184 x^3+3888 x^4-972 x^5+81 x^6} \, dx=\frac {81 \, {\left (x + 3\right )}^{4} e^{\left (x + 9\right )} - 1620 \, {\left (x + 3\right )}^{3} e^{\left (x + 9\right )} + {\left (x + 3\right )}^{2} e^{\left (2 \, x + 6\right )} + 11502 \, {\left (x + 3\right )}^{2} e^{\left (x + 9\right )} - 8 \, {\left (x + 3\right )} e^{\left (2 \, x + 6\right )} - 34020 \, {\left (x + 3\right )} e^{\left (x + 9\right )} + 16 \, e^{\left (2 \, x + 6\right )} + 35721 \, e^{\left (x + 9\right )}}{81 \, {\left ({\left (x + 3\right )}^{4} e^{6} - 20 \, {\left (x + 3\right )}^{3} e^{6} + 142 \, {\left (x + 3\right )}^{2} e^{6} - 420 \, {\left (x + 3\right )} e^{6} + 441 \, e^{6}\right )}} \] Input:
integrate(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^ 2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+3888*x^4-5184*x^3),x, algorithm="giac" )
Output:
1/81*(81*(x + 3)^4*e^(x + 9) - 1620*(x + 3)^3*e^(x + 9) + (x + 3)^2*e^(2*x + 6) + 11502*(x + 3)^2*e^(x + 9) - 8*(x + 3)*e^(2*x + 6) - 34020*(x + 3)* e^(x + 9) + 16*e^(2*x + 6) + 35721*e^(x + 9))/((x + 3)^4*e^6 - 20*(x + 3)^ 3*e^6 + 142*(x + 3)^2*e^6 - 420*(x + 3)*e^6 + 441*e^6)
Time = 3.44 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {e^{2 x} \left (8-20 x+24 x^2-14 x^3+2 x^4\right )+e^{3+x} \left (-5184 x^3+3888 x^4-972 x^5+81 x^6\right )}{-5184 x^3+3888 x^4-972 x^5+81 x^6} \, dx=\frac {{\mathrm {e}}^{x-3}\,\left ({\mathrm {e}}^{x+3}-2\,x\,{\mathrm {e}}^{x+3}+x^2\,{\mathrm {e}}^{x+3}+1296\,x^2\,{\mathrm {e}}^6-648\,x^3\,{\mathrm {e}}^6+81\,x^4\,{\mathrm {e}}^6\right )}{81\,x^2\,{\left (x-4\right )}^2} \] Input:
int(-(exp(2*x)*(24*x^2 - 20*x - 14*x^3 + 2*x^4 + 8) - exp(x + 3)*(5184*x^3 - 3888*x^4 + 972*x^5 - 81*x^6))/(5184*x^3 - 3888*x^4 + 972*x^5 - 81*x^6), x)
Output:
(exp(x - 3)*(exp(x + 3) - 2*x*exp(x + 3) + x^2*exp(x + 3) + 1296*x^2*exp(6 ) - 648*x^3*exp(6) + 81*x^4*exp(6)))/(81*x^2*(x - 4)^2)
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {e^{2 x} \left (8-20 x+24 x^2-14 x^3+2 x^4\right )+e^{3+x} \left (-5184 x^3+3888 x^4-972 x^5+81 x^6\right )}{-5184 x^3+3888 x^4-972 x^5+81 x^6} \, dx=\frac {e^{x} \left (e^{x} x^{2}-2 e^{x} x +e^{x}+81 e^{3} x^{4}-648 e^{3} x^{3}+1296 e^{3} x^{2}\right )}{81 x^{2} \left (x^{2}-8 x +16\right )} \] Input:
int(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x +8)*exp(x)^2)/(81*x^6-972*x^5+3888*x^4-5184*x^3),x)
Output:
(e**x*(e**x*x**2 - 2*e**x*x + e**x + 81*e**3*x**4 - 648*e**3*x**3 + 1296*e **3*x**2))/(81*x**2*(x**2 - 8*x + 16))