Integrand size = 40, antiderivative size = 22 \[ \int \frac {-256+e^x (-8-64 x)+e^{2 x} (18+16 x)}{4913+6936 x+3264 x^2+512 x^3} \, dx=\frac {\left (-4+e^x\right )^2}{\left (1-x^2+(4+x)^2\right )^2} \] Output:
(exp(x)-4)^2/((4+x)^2+1-x^2)^2
Time = 0.73 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-256+e^x (-8-64 x)+e^{2 x} (18+16 x)}{4913+6936 x+3264 x^2+512 x^3} \, dx=\frac {\left (-4+e^x\right )^2}{(17+8 x)^2} \] Input:
Integrate[(-256 + E^x*(-8 - 64*x) + E^(2*x)*(18 + 16*x))/(4913 + 6936*x + 3264*x^2 + 512*x^3),x]
Output:
(-4 + E^x)^2/(17 + 8*x)^2
Time = 0.52 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2007, 7292, 27, 25, 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.
\(\displaystyle \int \frac {e^x (-64 x-8)+e^{2 x} (16 x+18)-256}{512 x^3+3264 x^2+6936 x+4913} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^x (-64 x-8)+e^{2 x} (16 x+18)-256}{(8 x+17)^3}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 \left (4-e^x\right ) \left (-8 e^x x-9 e^x-32\right )}{(8 x+17)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {\left (4-e^x\right ) \left (8 e^x x+9 e^x+32\right )}{(8 x+17)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {\left (4-e^x\right ) \left (8 e^x x+9 e^x+32\right )}{(8 x+17)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {4 e^x (8 x+1)}{(8 x+17)^3}-\frac {e^{2 x} (8 x+9)}{(8 x+17)^3}+\frac {128}{(8 x+17)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (\frac {4 e^x}{(8 x+17)^2}-\frac {e^{2 x}}{2 (8 x+17)^2}-\frac {8}{(8 x+17)^2}\right )\) |
Input:
Int[(-256 + E^x*(-8 - 64*x) + E^(2*x)*(18 + 16*x))/(4913 + 6936*x + 3264*x ^2 + 512*x^3),x]
Output:
-2*(-8/(17 + 8*x)^2 + (4*E^x)/(17 + 8*x)^2 - E^(2*x)/(2*(17 + 8*x)^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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]
Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
norman | \(\frac {{\mathrm e}^{2 x}+16-8 \,{\mathrm e}^{x}}{\left (8 x +17\right )^{2}}\) | \(19\) |
parallelrisch | \(\frac {1024+64 \,{\mathrm e}^{2 x}-512 \,{\mathrm e}^{x}}{4096 x^{2}+17408 x +18496}\) | \(27\) |
default | \(\frac {16}{\left (8 x +17\right )^{2}}-\frac {{\mathrm e}^{x}}{8 \left (x +\frac {17}{8}\right )^{2}}+\frac {{\mathrm e}^{2 x}}{64 \left (x +\frac {17}{8}\right )^{2}}\) | \(31\) |
parts | \(\frac {16}{\left (8 x +17\right )^{2}}-\frac {{\mathrm e}^{x}}{8 \left (x +\frac {17}{8}\right )^{2}}+\frac {{\mathrm e}^{2 x}}{64 \left (x +\frac {17}{8}\right )^{2}}\) | \(31\) |
risch | \(\frac {1}{4 x^{2}+17 x +\frac {289}{16}}+\frac {{\mathrm e}^{2 x}}{\left (8 x +17\right )^{2}}-\frac {8 \,{\mathrm e}^{x}}{\left (8 x +17\right )^{2}}\) | \(37\) |
orering | \(-\frac {\left (512 x^{4}+1280 x^{3}+1640 x^{2}+2034 x +1615\right ) \left (\left (16 x +18\right ) {\mathrm e}^{2 x}+\left (-64 x -8\right ) {\mathrm e}^{x}-256\right )}{110 \left (8 x +5\right ) \left (512 x^{3}+3264 x^{2}+6936 x +4913\right )}+\frac {\left (8 x +17\right ) \left (96 x^{3}+196 x^{2}+191 x +85\right ) \left (\frac {16 \,{\mathrm e}^{2 x}+2 \left (16 x +18\right ) {\mathrm e}^{2 x}-64 \,{\mathrm e}^{x}+\left (-64 x -8\right ) {\mathrm e}^{x}}{512 x^{3}+3264 x^{2}+6936 x +4913}-\frac {\left (\left (16 x +18\right ) {\mathrm e}^{2 x}+\left (-64 x -8\right ) {\mathrm e}^{x}-256\right ) \left (1536 x^{2}+6528 x +6936\right )}{\left (512 x^{3}+3264 x^{2}+6936 x +4913\right )^{2}}\right )}{880 x +550}-\frac {\left (8 x +17\right )^{2} x \left (4 x +5\right ) \left (\frac {64 \,{\mathrm e}^{2 x}+4 \left (16 x +18\right ) {\mathrm e}^{2 x}-128 \,{\mathrm e}^{x}+\left (-64 x -8\right ) {\mathrm e}^{x}}{512 x^{3}+3264 x^{2}+6936 x +4913}-\frac {2 \left (16 \,{\mathrm e}^{2 x}+2 \left (16 x +18\right ) {\mathrm e}^{2 x}-64 \,{\mathrm e}^{x}+\left (-64 x -8\right ) {\mathrm e}^{x}\right ) \left (1536 x^{2}+6528 x +6936\right )}{\left (512 x^{3}+3264 x^{2}+6936 x +4913\right )^{2}}+\frac {2 \left (\left (16 x +18\right ) {\mathrm e}^{2 x}+\left (-64 x -8\right ) {\mathrm e}^{x}-256\right ) \left (1536 x^{2}+6528 x +6936\right )^{2}}{\left (512 x^{3}+3264 x^{2}+6936 x +4913\right )^{3}}-\frac {\left (\left (16 x +18\right ) {\mathrm e}^{2 x}+\left (-64 x -8\right ) {\mathrm e}^{x}-256\right ) \left (3072 x +6528\right )}{\left (512 x^{3}+3264 x^{2}+6936 x +4913\right )^{2}}\right )}{110 \left (8 x +5\right )}\) | \(420\) |
Input:
int(((16*x+18)*exp(x)^2+(-64*x-8)*exp(x)-256)/(512*x^3+3264*x^2+6936*x+491 3),x,method=_RETURNVERBOSE)
Output:
(exp(x)^2+16-8*exp(x))/(8*x+17)^2
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-256+e^x (-8-64 x)+e^{2 x} (18+16 x)}{4913+6936 x+3264 x^2+512 x^3} \, dx=\frac {e^{\left (2 \, x\right )} - 8 \, e^{x} + 16}{64 \, x^{2} + 272 \, x + 289} \] Input:
integrate(((16*x+18)*exp(x)^2+(-64*x-8)*exp(x)-256)/(512*x^3+3264*x^2+6936 *x+4913),x, algorithm="fricas")
Output:
(e^(2*x) - 8*e^x + 16)/(64*x^2 + 272*x + 289)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {-256+e^x (-8-64 x)+e^{2 x} (18+16 x)}{4913+6936 x+3264 x^2+512 x^3} \, dx=\frac {\left (- 512 x^{2} - 2176 x - 2312\right ) e^{x} + \left (64 x^{2} + 272 x + 289\right ) e^{2 x}}{4096 x^{4} + 34816 x^{3} + 110976 x^{2} + 157216 x + 83521} + \frac {256}{1024 x^{2} + 4352 x + 4624} \] Input:
integrate(((16*x+18)*exp(x)**2+(-64*x-8)*exp(x)-256)/(512*x**3+3264*x**2+6 936*x+4913),x)
Output:
((-512*x**2 - 2176*x - 2312)*exp(x) + (64*x**2 + 272*x + 289)*exp(2*x))/(4 096*x**4 + 34816*x**3 + 110976*x**2 + 157216*x + 83521) + 256/(1024*x**2 + 4352*x + 4624)
\[ \int \frac {-256+e^x (-8-64 x)+e^{2 x} (18+16 x)}{4913+6936 x+3264 x^2+512 x^3} \, dx=\int { \frac {2 \, {\left ({\left (8 \, x + 9\right )} e^{\left (2 \, x\right )} - 4 \, {\left (8 \, x + 1\right )} e^{x} - 128\right )}}{512 \, x^{3} + 3264 \, x^{2} + 6936 \, x + 4913} \,d x } \] Input:
integrate(((16*x+18)*exp(x)^2+(-64*x-8)*exp(x)-256)/(512*x^3+3264*x^2+6936 *x+4913),x, algorithm="maxima")
Output:
((8*x + 17)*e^(2*x) - 64*x*e^x)/(512*x^3 + 3264*x^2 + 6936*x + 4913) + e^( -17/8)*exp_integral_e(3, -x - 17/8)/(8*x + 17)^2 + 16/(64*x^2 + 272*x + 28 9) - 2*integrate(32*(16*x - 17)*e^x/(4096*x^4 + 34816*x^3 + 110976*x^2 + 1 57216*x + 83521), x)
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-256+e^x (-8-64 x)+e^{2 x} (18+16 x)}{4913+6936 x+3264 x^2+512 x^3} \, dx=\frac {e^{\left (2 \, x\right )} - 8 \, e^{x} + 16}{64 \, x^{2} + 272 \, x + 289} \] Input:
integrate(((16*x+18)*exp(x)^2+(-64*x-8)*exp(x)-256)/(512*x^3+3264*x^2+6936 *x+4913),x, algorithm="giac")
Output:
(e^(2*x) - 8*e^x + 16)/(64*x^2 + 272*x + 289)
Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {-256+e^x (-8-64 x)+e^{2 x} (18+16 x)}{4913+6936 x+3264 x^2+512 x^3} \, dx=\frac {{\left ({\mathrm {e}}^x-4\right )}^2}{{\left (8\,x+17\right )}^2} \] Input:
int(-(exp(x)*(64*x + 8) - exp(2*x)*(16*x + 18) + 256)/(6936*x + 3264*x^2 + 512*x^3 + 4913),x)
Output:
(exp(x) - 4)^2/(8*x + 17)^2
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {-256+e^x (-8-64 x)+e^{2 x} (18+16 x)}{4913+6936 x+3264 x^2+512 x^3} \, dx=\frac {e^{2 x}-8 e^{x}+16}{64 x^{2}+272 x +289} \] Input:
int(((16*x+18)*exp(x)^2+(-64*x-8)*exp(x)-256)/(512*x^3+3264*x^2+6936*x+491 3),x)
Output:
(e**(2*x) - 8*e**x + 16)/(64*x**2 + 272*x + 289)