Integrand size = 93, antiderivative size = 34 \[ \int \frac {-144-240 x-144 x^2-40 x^3-12 x^4-6 x^5-x^6+e^{10+4 x} \left (-16+16 x+16 x^2\right )+e^{5+2 x} \left (-96-32 x+64 x^2+32 x^3\right )}{8 x^3+12 x^4+6 x^5+x^6} \, dx=-x+\left (2+\frac {2 \left (-4-\frac {3+e^{5+2 x}}{x}-x\right )}{2+x}\right )^2 \] Output:
(2+2/(2+x)*(-4-x-(3+exp(5+2*x))/x))^2-x
Time = 2.59 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {-144-240 x-144 x^2-40 x^3-12 x^4-6 x^5-x^6+e^{10+4 x} \left (-16+16 x+16 x^2\right )+e^{5+2 x} \left (-96-32 x+64 x^2+32 x^3\right )}{8 x^3+12 x^4+6 x^5+x^6} \, dx=-\frac {-36-4 e^{10+4 x}-48 x-16 x^2+4 x^3+4 x^4+x^5-8 e^{5+2 x} (3+2 x)}{x^2 (2+x)^2} \] Input:
Integrate[(-144 - 240*x - 144*x^2 - 40*x^3 - 12*x^4 - 6*x^5 - x^6 + E^(10 + 4*x)*(-16 + 16*x + 16*x^2) + E^(5 + 2*x)*(-96 - 32*x + 64*x^2 + 32*x^3)) /(8*x^3 + 12*x^4 + 6*x^5 + x^6),x]
Output:
-((-36 - 4*E^(10 + 4*x) - 48*x - 16*x^2 + 4*x^3 + 4*x^4 + x^5 - 8*E^(5 + 2 *x)*(3 + 2*x))/(x^2*(2 + x)^2))
Leaf count is larger than twice the leaf count of optimal. \(139\) vs. \(2(34)=68\).
Time = 2.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, 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.
\(\displaystyle \int \frac {-x^6-6 x^5-12 x^4-40 x^3-144 x^2+e^{4 x+10} \left (16 x^2+16 x-16\right )+e^{2 x+5} \left (32 x^3+64 x^2-32 x-96\right )-240 x-144}{x^6+6 x^5+12 x^4+8 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-x^6-6 x^5-12 x^4-40 x^3-144 x^2+e^{4 x+10} \left (16 x^2+16 x-16\right )+e^{2 x+5} \left (32 x^3+64 x^2-32 x-96\right )-240 x-144}{x^3 \left (x^3+6 x^2+12 x+8\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-x^6-6 x^5-12 x^4-40 x^3-144 x^2+e^{4 x+10} \left (16 x^2+16 x-16\right )+e^{2 x+5} \left (32 x^3+64 x^2-32 x-96\right )-240 x-144}{x^3 (x+2)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {x^3}{(x+2)^3}-\frac {144}{(x+2)^3 x^3}-\frac {6 x^2}{(x+2)^3}-\frac {240}{(x+2)^3 x^2}+\frac {16 e^{4 x+10} \left (x^2+x-1\right )}{(x+2)^3 x^3}+\frac {32 e^{2 x+5} \left (x^3+2 x^2-x-3\right )}{(x+2)^3 x^3}-\frac {12 x}{(x+2)^3}-\frac {40}{(x+2)^3}-\frac {144}{(x+2)^3 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 x^2}{(x+2)^2}+\frac {6 e^{2 x+5}}{x^2}+\frac {e^{4 x+10}}{x^2}+\frac {9}{x^2}-x+\frac {2 e^{2 x+5}}{x+2}+\frac {e^{4 x+10}}{x+2}-\frac {15}{x+2}-\frac {2 e^{2 x+5}}{(x+2)^2}+\frac {e^{4 x+10}}{(x+2)^2}+\frac {13}{(x+2)^2}-\frac {2 e^{2 x+5}}{x}-\frac {e^{4 x+10}}{x}+\frac {3}{x}\) |
Input:
Int[(-144 - 240*x - 144*x^2 - 40*x^3 - 12*x^4 - 6*x^5 - x^6 + E^(10 + 4*x) *(-16 + 16*x + 16*x^2) + E^(5 + 2*x)*(-96 - 32*x + 64*x^2 + 32*x^3))/(8*x^ 3 + 12*x^4 + 6*x^5 + x^6),x]
Output:
9/x^2 + (6*E^(5 + 2*x))/x^2 + E^(10 + 4*x)/x^2 + 3/x - (2*E^(5 + 2*x))/x - E^(10 + 4*x)/x - x + 13/(2 + x)^2 - (2*E^(5 + 2*x))/(2 + x)^2 + E^(10 + 4 *x)/(2 + x)^2 - (3*x^2)/(2 + x)^2 - 15/(2 + x) + (2*E^(5 + 2*x))/(2 + x) + E^(10 + 4*x)/(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 = 1.43 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.79
method | result | size |
parallelrisch | \(-\frac {x^{5}-36-12 x^{3}-32 x^{2}-16 \,{\mathrm e}^{5+2 x} x -4 \,{\mathrm e}^{4 x +10}-48 x -24 \,{\mathrm e}^{5+2 x}}{x^{2} \left (x^{2}+4 x +4\right )}\) | \(61\) |
risch | \(-x +\frac {16 x^{2}+48 x +36}{x^{2} \left (x^{2}+4 x +4\right )}+\frac {4 \,{\mathrm e}^{4 x +10}}{x^{2} \left (2+x \right )^{2}}+\frac {8 \left (3+2 x \right ) {\mathrm e}^{5+2 x}}{x^{2} \left (2+x \right )^{2}}\) | \(68\) |
parts | \(-x +\frac {1}{\left (2+x \right )^{2}}-\frac {3}{2+x}+\frac {9}{x^{2}}+\frac {3}{x}-\frac {{\mathrm e}^{4 x +10}}{x}+\frac {{\mathrm e}^{4 x +10}}{x^{2}}+\frac {4 \,{\mathrm e}^{4 x +10}}{\left (4+2 x \right )^{2}}+\frac {2 \,{\mathrm e}^{4 x +10}}{4+2 x}-\frac {2 \,{\mathrm e}^{5+2 x}}{x}+\frac {6 \,{\mathrm e}^{5+2 x}}{x^{2}}-\frac {8 \,{\mathrm e}^{5+2 x}}{\left (4+2 x \right )^{2}}+\frac {4 \,{\mathrm e}^{5+2 x}}{4+2 x}\) | \(138\) |
derivativedivides | \(\frac {9}{x^{2}}+\frac {3}{x}+\frac {4}{\left (4+2 x \right )^{2}}-\frac {6}{4+2 x}-\frac {5}{2}-x +\frac {6 \,{\mathrm e}^{5+2 x}}{x^{2}}-\frac {2 \,{\mathrm e}^{5+2 x}}{x}-\frac {8 \,{\mathrm e}^{5+2 x}}{\left (4+2 x \right )^{2}}+\frac {4 \,{\mathrm e}^{5+2 x}}{4+2 x}+\frac {{\mathrm e}^{4 x +10}}{x^{2}}-\frac {{\mathrm e}^{4 x +10}}{x}+\frac {4 \,{\mathrm e}^{4 x +10}}{\left (4+2 x \right )^{2}}+\frac {2 \,{\mathrm e}^{4 x +10}}{4+2 x}\) | \(145\) |
default | \(\frac {9}{x^{2}}+\frac {3}{x}+\frac {4}{\left (4+2 x \right )^{2}}-\frac {6}{4+2 x}-\frac {5}{2}-x +\frac {6 \,{\mathrm e}^{5+2 x}}{x^{2}}-\frac {2 \,{\mathrm e}^{5+2 x}}{x}-\frac {8 \,{\mathrm e}^{5+2 x}}{\left (4+2 x \right )^{2}}+\frac {4 \,{\mathrm e}^{5+2 x}}{4+2 x}+\frac {{\mathrm e}^{4 x +10}}{x^{2}}-\frac {{\mathrm e}^{4 x +10}}{x}+\frac {4 \,{\mathrm e}^{4 x +10}}{\left (4+2 x \right )^{2}}+\frac {2 \,{\mathrm e}^{4 x +10}}{4+2 x}\) | \(145\) |
orering | \(\text {Expression too large to display}\) | \(1119\) |
Input:
int(((16*x^2+16*x-16)*exp(5+2*x)^2+(32*x^3+64*x^2-32*x-96)*exp(5+2*x)-x^6- 6*x^5-12*x^4-40*x^3-144*x^2-240*x-144)/(x^6+6*x^5+12*x^4+8*x^3),x,method=_ RETURNVERBOSE)
Output:
-(x^5-36-12*x^3-32*x^2-16*exp(5+2*x)*x-4*exp(5+2*x)^2-48*x-24*exp(5+2*x))/ x^2/(x^2+4*x+4)
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.82 \[ \int \frac {-144-240 x-144 x^2-40 x^3-12 x^4-6 x^5-x^6+e^{10+4 x} \left (-16+16 x+16 x^2\right )+e^{5+2 x} \left (-96-32 x+64 x^2+32 x^3\right )}{8 x^3+12 x^4+6 x^5+x^6} \, dx=-\frac {x^{5} + 4 \, x^{4} + 4 \, x^{3} - 16 \, x^{2} - 8 \, {\left (2 \, x + 3\right )} e^{\left (2 \, x + 5\right )} - 48 \, x - 4 \, e^{\left (4 \, x + 10\right )} - 36}{x^{4} + 4 \, x^{3} + 4 \, x^{2}} \] Input:
integrate(((16*x^2+16*x-16)*exp(5+2*x)^2+(32*x^3+64*x^2-32*x-96)*exp(5+2*x )-x^6-6*x^5-12*x^4-40*x^3-144*x^2-240*x-144)/(x^6+6*x^5+12*x^4+8*x^3),x, a lgorithm="fricas")
Output:
-(x^5 + 4*x^4 + 4*x^3 - 16*x^2 - 8*(2*x + 3)*e^(2*x + 5) - 48*x - 4*e^(4*x + 10) - 36)/(x^4 + 4*x^3 + 4*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (26) = 52\).
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.91 \[ \int \frac {-144-240 x-144 x^2-40 x^3-12 x^4-6 x^5-x^6+e^{10+4 x} \left (-16+16 x+16 x^2\right )+e^{5+2 x} \left (-96-32 x+64 x^2+32 x^3\right )}{8 x^3+12 x^4+6 x^5+x^6} \, dx=- x + \frac {\left (4 x^{4} + 16 x^{3} + 16 x^{2}\right ) e^{4 x + 10} + \left (16 x^{5} + 88 x^{4} + 160 x^{3} + 96 x^{2}\right ) e^{2 x + 5}}{x^{8} + 8 x^{7} + 24 x^{6} + 32 x^{5} + 16 x^{4}} - \frac {- 16 x^{2} - 48 x - 36}{x^{4} + 4 x^{3} + 4 x^{2}} \] Input:
integrate(((16*x**2+16*x-16)*exp(5+2*x)**2+(32*x**3+64*x**2-32*x-96)*exp(5 +2*x)-x**6-6*x**5-12*x**4-40*x**3-144*x**2-240*x-144)/(x**6+6*x**5+12*x**4 +8*x**3),x)
Output:
-x + ((4*x**4 + 16*x**3 + 16*x**2)*exp(4*x + 10) + (16*x**5 + 88*x**4 + 16 0*x**3 + 96*x**2)*exp(2*x + 5))/(x**8 + 8*x**7 + 24*x**6 + 32*x**5 + 16*x* *4) - (-16*x**2 - 48*x - 36)/(x**4 + 4*x**3 + 4*x**2)
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (31) = 62\).
Time = 0.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 5.29 \[ \int \frac {-144-240 x-144 x^2-40 x^3-12 x^4-6 x^5-x^6+e^{10+4 x} \left (-16+16 x+16 x^2\right )+e^{5+2 x} \left (-96-32 x+64 x^2+32 x^3\right )}{8 x^3+12 x^4+6 x^5+x^6} \, dx=-x - \frac {18 \, {\left (3 \, x^{3} + 9 \, x^{2} + 4 \, x - 2\right )}}{x^{4} + 4 \, x^{3} + 4 \, x^{2}} + \frac {30 \, {\left (3 \, x^{2} + 9 \, x + 4\right )}}{x^{3} + 4 \, x^{2} + 4 \, x} + \frac {4 \, {\left (2 \, {\left (2 \, x e^{5} + 3 \, e^{5}\right )} e^{\left (2 \, x\right )} + e^{\left (4 \, x + 10\right )}\right )}}{x^{4} + 4 \, x^{3} + 4 \, x^{2}} + \frac {4 \, {\left (3 \, x + 5\right )}}{x^{2} + 4 \, x + 4} - \frac {12 \, {\left (2 \, x + 3\right )}}{x^{2} + 4 \, x + 4} - \frac {36 \, {\left (x + 3\right )}}{x^{2} + 4 \, x + 4} + \frac {12 \, {\left (x + 1\right )}}{x^{2} + 4 \, x + 4} + \frac {20}{x^{2} + 4 \, x + 4} \] Input:
integrate(((16*x^2+16*x-16)*exp(5+2*x)^2+(32*x^3+64*x^2-32*x-96)*exp(5+2*x )-x^6-6*x^5-12*x^4-40*x^3-144*x^2-240*x-144)/(x^6+6*x^5+12*x^4+8*x^3),x, a lgorithm="maxima")
Output:
-x - 18*(3*x^3 + 9*x^2 + 4*x - 2)/(x^4 + 4*x^3 + 4*x^2) + 30*(3*x^2 + 9*x + 4)/(x^3 + 4*x^2 + 4*x) + 4*(2*(2*x*e^5 + 3*e^5)*e^(2*x) + e^(4*x + 10))/ (x^4 + 4*x^3 + 4*x^2) + 4*(3*x + 5)/(x^2 + 4*x + 4) - 12*(2*x + 3)/(x^2 + 4*x + 4) - 36*(x + 3)/(x^2 + 4*x + 4) + 12*(x + 1)/(x^2 + 4*x + 4) + 20/(x ^2 + 4*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (31) = 62\).
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.94 \[ \int \frac {-144-240 x-144 x^2-40 x^3-12 x^4-6 x^5-x^6+e^{10+4 x} \left (-16+16 x+16 x^2\right )+e^{5+2 x} \left (-96-32 x+64 x^2+32 x^3\right )}{8 x^3+12 x^4+6 x^5+x^6} \, dx=-\frac {x^{5} + 4 \, x^{4} + 4 \, x^{3} - 16 \, x^{2} - 16 \, x e^{\left (2 \, x + 5\right )} - 48 \, x - 4 \, e^{\left (4 \, x + 10\right )} - 24 \, e^{\left (2 \, x + 5\right )} - 36}{x^{4} + 4 \, x^{3} + 4 \, x^{2}} \] Input:
integrate(((16*x^2+16*x-16)*exp(5+2*x)^2+(32*x^3+64*x^2-32*x-96)*exp(5+2*x )-x^6-6*x^5-12*x^4-40*x^3-144*x^2-240*x-144)/(x^6+6*x^5+12*x^4+8*x^3),x, a lgorithm="giac")
Output:
-(x^5 + 4*x^4 + 4*x^3 - 16*x^2 - 16*x*e^(2*x + 5) - 48*x - 4*e^(4*x + 10) - 24*e^(2*x + 5) - 36)/(x^4 + 4*x^3 + 4*x^2)
Time = 2.85 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int \frac {-144-240 x-144 x^2-40 x^3-12 x^4-6 x^5-x^6+e^{10+4 x} \left (-16+16 x+16 x^2\right )+e^{5+2 x} \left (-96-32 x+64 x^2+32 x^3\right )}{8 x^3+12 x^4+6 x^5+x^6} \, dx=\frac {24\,{\mathrm {e}}^{2\,x+5}+4\,{\mathrm {e}}^{4\,x+10}+x\,\left (16\,{\mathrm {e}}^{2\,x+5}+48\right )+16\,x^2+36}{x^2\,{\left (x+2\right )}^2}-x \] Input:
int(-(240*x - exp(4*x + 10)*(16*x + 16*x^2 - 16) + exp(2*x + 5)*(32*x - 64 *x^2 - 32*x^3 + 96) + 144*x^2 + 40*x^3 + 12*x^4 + 6*x^5 + x^6 + 144)/(8*x^ 3 + 12*x^4 + 6*x^5 + x^6),x)
Output:
(24*exp(2*x + 5) + 4*exp(4*x + 10) + x*(16*exp(2*x + 5) + 48) + 16*x^2 + 3 6)/(x^2*(x + 2)^2) - x
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {-144-240 x-144 x^2-40 x^3-12 x^4-6 x^5-x^6+e^{10+4 x} \left (-16+16 x+16 x^2\right )+e^{5+2 x} \left (-96-32 x+64 x^2+32 x^3\right )}{8 x^3+12 x^4+6 x^5+x^6} \, dx=\frac {4 e^{4 x} e^{10}+16 e^{2 x} e^{5} x +24 e^{2 x} e^{5}-x^{5}-3 x^{4}+20 x^{2}+48 x +36}{x^{2} \left (x^{2}+4 x +4\right )} \] Input:
int(((16*x^2+16*x-16)*exp(5+2*x)^2+(32*x^3+64*x^2-32*x-96)*exp(5+2*x)-x^6- 6*x^5-12*x^4-40*x^3-144*x^2-240*x-144)/(x^6+6*x^5+12*x^4+8*x^3),x)
Output:
(4*e**(4*x)*e**10 + 16*e**(2*x)*e**5*x + 24*e**(2*x)*e**5 - x**5 - 3*x**4 + 20*x**2 + 48*x + 36)/(x**2*(x**2 + 4*x + 4))