Integrand size = 42, antiderivative size = 24 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {e^x x}{4 \left (5+\frac {4 (4-x)^2}{x^2}\right )} \] Output:
1/4*exp(x)*x/(4/x^2*(4-x)^2+5)
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {e^x x^3}{4 \left (64-32 x+9 x^2\right )} \] Input:
Integrate[(E^x*(192*x^2 - 23*x^4 + 9*x^5))/(16384 - 16384*x + 8704*x^2 - 2 304*x^3 + 324*x^4),x]
Output:
(E^x*x^3)/(4*(64 - 32*x + 9*x^2))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.47 (sec) , antiderivative size = 548, normalized size of antiderivative = 22.83, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2028, 2463, 27, 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.
\(\displaystyle \int \frac {e^x \left (9 x^5-23 x^4+192 x^2\right )}{324 x^4-2304 x^3+8704 x^2-16384 x+16384} \, dx\) |
\(\Big \downarrow \) 2028 |
\(\displaystyle \int \frac {e^x x^2 \left (9 x^3-23 x^2+192\right )}{324 x^4-2304 x^3+8704 x^2-16384 x+16384}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {e^x x^2 \left (9 x^3-23 x^2+192\right )}{4 \left (9 x^2-32 x+64\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {e^x x^2 \left (9 x^3-23 x^2+192\right )}{\left (9 x^2-32 x+64\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{4} \int \left (\frac {e^x x}{9}+\frac {41 e^x}{81}+\frac {64 e^x (7 x-39)}{81 \left (9 x^2-32 x+64\right )}+\frac {2048 e^x (11 x-4)}{81 \left (9 x^2-32 x+64\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (\frac {4 \left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )}{3645}-\frac {704 \left (2-i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )}{3645}+\frac {28 i e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )}{81 \sqrt {5}}+\frac {32}{405} e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )+\frac {4 \left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )}{3645}-\frac {704 \left (2+i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )}{3645}-\frac {28 i e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )}{81 \sqrt {5}}+\frac {32}{405} e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )+\frac {e^x x}{9}+\frac {32 e^x}{81}-\frac {704 \left (2-i \sqrt {5}\right ) e^x}{405 \left (-9 x+8 \left (2-i \sqrt {5}\right )\right )}+\frac {32 e^x}{45 \left (-9 x+8 \left (2-i \sqrt {5}\right )\right )}-\frac {704 \left (2+i \sqrt {5}\right ) e^x}{405 \left (-9 x+8 \left (2+i \sqrt {5}\right )\right )}+\frac {32 e^x}{45 \left (-9 x+8 \left (2+i \sqrt {5}\right )\right )}\right )\) |
Input:
Int[(E^x*(192*x^2 - 23*x^4 + 9*x^5))/(16384 - 16384*x + 8704*x^2 - 2304*x^ 3 + 324*x^4),x]
Output:
((32*E^x)/81 + (32*E^x)/(45*(8*(2 - I*Sqrt[5]) - 9*x)) - (704*(2 - I*Sqrt[ 5])*E^x)/(405*(8*(2 - I*Sqrt[5]) - 9*x)) + (32*E^x)/(45*(8*(2 + I*Sqrt[5]) - 9*x)) - (704*(2 + I*Sqrt[5])*E^x)/(405*(8*(2 + I*Sqrt[5]) - 9*x)) + (E^ x*x)/9 + (32*E^((8*(2 - I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 - I*Sqrt[5]) + 9*x)/9])/405 + (((28*I)/81)*E^((8*(2 - I*Sqrt[5]))/9)*ExpIntegralEi[(-8*( 2 - I*Sqrt[5]) + 9*x)/9])/Sqrt[5] - (704*(2 - I*Sqrt[5])*E^((8*(2 - I*Sqrt [5]))/9)*ExpIntegralEi[(-8*(2 - I*Sqrt[5]) + 9*x)/9])/3645 + (4*(280 - (23 9*I)*Sqrt[5])*E^((8*(2 - I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 - I*Sqrt[5]) + 9*x)/9])/3645 + (32*E^((8*(2 + I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 + I*S qrt[5]) + 9*x)/9])/405 - (((28*I)/81)*E^((8*(2 + I*Sqrt[5]))/9)*ExpIntegra lEi[(-8*(2 + I*Sqrt[5]) + 9*x)/9])/Sqrt[5] - (704*(2 + I*Sqrt[5])*E^((8*(2 + I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 + I*Sqrt[5]) + 9*x)/9])/3645 + (4*( 280 + (239*I)*Sqrt[5])*E^((8*(2 + I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 + I* Sqrt[5]) + 9*x)/9])/3645)/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\) | \(20\) |
norman | \(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\) | \(20\) |
risch | \(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\) | \(20\) |
parallelrisch | \(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\) | \(20\) |
orering | \(\frac {\left (9 x^{2}-32 x +64\right ) x \left (9 x^{5}-23 x^{4}+192 x^{2}\right ) {\mathrm e}^{x}}{\left (9 x^{3}-23 x^{2}+192\right ) \left (324 x^{4}-2304 x^{3}+8704 x^{2}-16384 x +16384\right )}\) | \(67\) |
default | \(-\frac {8 \,{\mathrm e}^{x} \left (x +16\right )}{15 \left (9 x^{2}-32 x +64\right )}-\frac {23 \,{\mathrm e}^{x}}{324}+\frac {184 \,{\mathrm e}^{x} \left (79 x -176\right )}{3645 \left (9 x^{2}-32 x +64\right )}+\frac {\left (9 x +55\right ) {\mathrm e}^{x}}{324}-\frac {128 \,{\mathrm e}^{x} \left (59 x -316\right )}{3645 \left (9 x^{2}-32 x +64\right )}\) | \(76\) |
Input:
int((9*x^5-23*x^4+192*x^2)*exp(x)/(324*x^4-2304*x^3+8704*x^2-16384*x+16384 ),x,method=_RETURNVERBOSE)
Output:
1/4*x^3*exp(x)/(9*x^2-32*x+64)
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^{3} e^{x}}{4 \, {\left (9 \, x^{2} - 32 \, x + 64\right )}} \] Input:
integrate((9*x^5-23*x^4+192*x^2)*exp(x)/(324*x^4-2304*x^3+8704*x^2-16384*x +16384),x, algorithm="fricas")
Output:
1/4*x^3*e^x/(9*x^2 - 32*x + 64)
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^{3} e^{x}}{36 x^{2} - 128 x + 256} \] Input:
integrate((9*x**5-23*x**4+192*x**2)*exp(x)/(324*x**4-2304*x**3+8704*x**2-1 6384*x+16384),x)
Output:
x**3*exp(x)/(36*x**2 - 128*x + 256)
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^{3} e^{x}}{4 \, {\left (9 \, x^{2} - 32 \, x + 64\right )}} \] Input:
integrate((9*x^5-23*x^4+192*x^2)*exp(x)/(324*x^4-2304*x^3+8704*x^2-16384*x +16384),x, algorithm="maxima")
Output:
1/4*x^3*e^x/(9*x^2 - 32*x + 64)
Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^{3} e^{x}}{4 \, {\left (9 \, x^{2} - 32 \, x + 64\right )}} \] Input:
integrate((9*x^5-23*x^4+192*x^2)*exp(x)/(324*x^4-2304*x^3+8704*x^2-16384*x +16384),x, algorithm="giac")
Output:
1/4*x^3*e^x/(9*x^2 - 32*x + 64)
Time = 3.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {x^3\,{\mathrm {e}}^x}{4\,\left (9\,x^2-32\,x+64\right )} \] Input:
int((exp(x)*(192*x^2 - 23*x^4 + 9*x^5))/(8704*x^2 - 16384*x - 2304*x^3 + 3 24*x^4 + 16384),x)
Output:
(x^3*exp(x))/(4*(9*x^2 - 32*x + 64))
Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^x \left (192 x^2-23 x^4+9 x^5\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx=\frac {e^{x} x^{3}}{36 x^{2}-128 x +256} \] Input:
int((9*x^5-23*x^4+192*x^2)*exp(x)/(324*x^4-2304*x^3+8704*x^2-16384*x+16384 ),x)
Output:
(e**x*x**3)/(4*(9*x**2 - 32*x + 64))