Integrand size = 30, antiderivative size = 18 \[ \int \frac {e^{9+\frac {4}{8 x+e x}} (-4+16 x+2 e x)}{8+e} \, dx=e^{9+\frac {4}{(8+e) x}} x^2 \]
Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{9+\frac {4}{8 x+e x}} (-4+16 x+2 e x)}{8+e} \, dx=e^{9+\frac {4}{8 x+e x}} x^2 \]
Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6, 27, 27, 2656, 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.
\(\displaystyle \int \frac {e^{\frac {4}{e x+8 x}+9} (2 e x+16 x-4)}{8+e} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^{\frac {4}{e x+8 x}+9} ((16+2 e) x-4)}{8+e}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -2 e^{9+\frac {4}{(8+e) x}} (2-(8+e) x)dx}{8+e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 \int e^{9+\frac {4}{(8+e) x}} (2-(8+e) x)dx}{8+e}\) |
\(\Big \downarrow \) 2656 |
\(\displaystyle -\frac {2 \int \left ((-8-e) e^{9+\frac {4}{(8+e) x}} x+2 e^{9+\frac {4}{(8+e) x}}\right )dx}{8+e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^{\frac {4}{(8+e) x}+9} x^2\) |
3.9.75.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[Px, x]
Time = 2.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
gosper | \(x^{2} {\mathrm e}^{9+\frac {4}{\left ({\mathrm e}+8\right ) x}}\) | \(19\) |
norman | \(x^{2} {\mathrm e}^{4} {\mathrm e}^{5} {\mathrm e}^{\frac {4}{x \,{\mathrm e}+8 x}}\) | \(22\) |
risch | \(x^{2} {\mathrm e}^{\frac {9 x \,{\mathrm e}+72 x +4}{x \left ({\mathrm e}+8\right )}}\) | \(26\) |
parallelrisch | \(\frac {{\mathrm e}^{4} {\mathrm e}^{5} \left ({\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} {\mathrm e}+8 \,{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2}\right )}{{\mathrm e}+8}\) | \(48\) |
derivativedivides | \(-\frac {16 \,{\mathrm e}^{4} {\mathrm e}^{5} \left (\frac {x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}}{4}+\operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right )+\frac {-\frac {{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} \left ({\mathrm e}+8\right )^{2}}{2}-2 x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}-8 \,\operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{{\mathrm e}+8}+\frac {2 \,{\mathrm e} \left (-\frac {{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} \left ({\mathrm e}+8\right )^{2}}{32}-\frac {x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}}{8}-\frac {\operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{2}\right )}{{\mathrm e}+8}\right )}{\left ({\mathrm e}+8\right )^{2}}\) | \(180\) |
default | \(-\frac {16 \,{\mathrm e}^{4} {\mathrm e}^{5} \left (\frac {x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}}{4}+\operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right )+\frac {-\frac {{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} \left ({\mathrm e}+8\right )^{2}}{2}-2 x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}-8 \,\operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{{\mathrm e}+8}+\frac {2 \,{\mathrm e} \left (-\frac {{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} \left ({\mathrm e}+8\right )^{2}}{32}-\frac {x \left ({\mathrm e}+8\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}}{8}-\frac {\operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{2}\right )}{{\mathrm e}+8}\right )}{\left ({\mathrm e}+8\right )^{2}}\) | \(180\) |
parts | \(\frac {2 \,{\mathrm e}^{4} {\mathrm e}^{5} {\mathrm e}^{2} {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2}}{\left ({\mathrm e}+8\right )^{2}}+\frac {32 \,{\mathrm e}^{4} {\mathrm e}^{5} {\mathrm e} \,{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2}}{\left ({\mathrm e}+8\right )^{2}}-\frac {4 \,{\mathrm e}^{4} {\mathrm e}^{5} x \,{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} {\mathrm e}}{\left ({\mathrm e}+8\right )^{2}}+\frac {128 \,{\mathrm e}^{4} {\mathrm e}^{5} {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2}}{\left ({\mathrm e}+8\right )^{2}}-\frac {32 \,{\mathrm e}^{4} {\mathrm e}^{5} x \,{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}}{\left ({\mathrm e}+8\right )^{2}}+\frac {8 \,{\mathrm e}^{4} {\mathrm e}^{5} {\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right ) x}{\left ({\mathrm e}+8\right )^{2}}+\frac {64 \,{\mathrm e}^{4} {\mathrm e}^{5} \operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right ) x}{\left ({\mathrm e}+8\right )^{2}}-\frac {16 \,{\mathrm e}^{4} {\mathrm e}^{5} \operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{\left ({\mathrm e}+8\right )^{2}}-\frac {{\mathrm e}^{5} {\mathrm e}^{2} {\mathrm e}^{4} x^{2} {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}+16 \,{\mathrm e}^{5} {\mathrm e} \,{\mathrm e}^{4} x^{2} {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}}-4 x \,{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} {\mathrm e}^{4} {\mathrm e}^{5} {\mathrm e}+8 \,{\mathrm e}^{4} {\mathrm e}^{5} \operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right ) x \,{\mathrm e}+64 \,{\mathrm e}^{4} {\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} x^{2} {\mathrm e}^{5}-32 x \,{\mathrm e}^{\frac {4}{\left ({\mathrm e}+8\right ) x}} {\mathrm e}^{4} {\mathrm e}^{5}+64 \,{\mathrm e}^{4} {\mathrm e}^{5} \operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right ) x -16 \,{\mathrm e}^{4} {\mathrm e}^{5} \operatorname {Ei}_{1}\left (-\frac {4}{\left ({\mathrm e}+8\right ) x}\right )}{\left ({\mathrm e}+8\right )^{2}}\) | \(398\) |
int((2*x*exp(1)+16*x-4)*exp(4)*exp(5)*exp(4/(x*exp(1)+8*x))/(exp(1)+8),x,m ethod=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {e^{9+\frac {4}{8 x+e x}} (-4+16 x+2 e x)}{8+e} \, dx=x^{2} e^{\left (\frac {9 \, x e + 72 \, x + 4}{x e + 8 \, x}\right )} \]
Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{9+\frac {4}{8 x+e x}} (-4+16 x+2 e x)}{8+e} \, dx=x^{2} e^{9} e^{\frac {4}{e x + 8 x}} \]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 4.28 \[ \int \frac {e^{9+\frac {4}{8 x+e x}} (-4+16 x+2 e x)}{8+e} \, dx=\frac {16 \, {\left (\frac {e^{9} \Gamma \left (-1, -\frac {4}{x {\left (e + 8\right )}}\right )}{e + 8} + \frac {2 \, e^{10} \Gamma \left (-2, -\frac {4}{x {\left (e + 8\right )}}\right )}{{\left (e + 8\right )}^{2}} + \frac {16 \, e^{9} \Gamma \left (-2, -\frac {4}{x {\left (e + 8\right )}}\right )}{{\left (e + 8\right )}^{2}}\right )}}{e + 8} \]
16*(e^9*gamma(-1, -4/(x*(e + 8)))/(e + 8) + 2*e^10*gamma(-2, -4/(x*(e + 8) ))/(e + 8)^2 + 16*e^9*gamma(-2, -4/(x*(e + 8)))/(e + 8)^2)/(e + 8)
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (18) = 36\).
Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 10.06 \[ \int \frac {e^{9+\frac {4}{8 x+e x}} (-4+16 x+2 e x)}{8+e} \, dx=\frac {16 \, e^{\left (\frac {9 \, x e + 72 \, x + 4}{x e + 8 \, x}\right )}}{\frac {{\left (9 \, x e + 72 \, x + 4\right )}^{2} e^{2}}{{\left (x e + 8 \, x\right )}^{2}} - \frac {18 \, {\left (9 \, x e + 72 \, x + 4\right )} e^{2}}{x e + 8 \, x} + \frac {16 \, {\left (9 \, x e + 72 \, x + 4\right )}^{2} e}{{\left (x e + 8 \, x\right )}^{2}} - \frac {288 \, {\left (9 \, x e + 72 \, x + 4\right )} e}{x e + 8 \, x} + \frac {64 \, {\left (9 \, x e + 72 \, x + 4\right )}^{2}}{{\left (x e + 8 \, x\right )}^{2}} - \frac {1152 \, {\left (9 \, x e + 72 \, x + 4\right )}}{x e + 8 \, x} + 81 \, e^{2} + 1296 \, e + 5184} \]
16*e^((9*x*e + 72*x + 4)/(x*e + 8*x))/((9*x*e + 72*x + 4)^2*e^2/(x*e + 8*x )^2 - 18*(9*x*e + 72*x + 4)*e^2/(x*e + 8*x) + 16*(9*x*e + 72*x + 4)^2*e/(x *e + 8*x)^2 - 288*(9*x*e + 72*x + 4)*e/(x*e + 8*x) + 64*(9*x*e + 72*x + 4) ^2/(x*e + 8*x)^2 - 1152*(9*x*e + 72*x + 4)/(x*e + 8*x) + 81*e^2 + 1296*e + 5184)
Time = 9.38 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{9+\frac {4}{8 x+e x}} (-4+16 x+2 e x)}{8+e} \, dx=x^2\,{\mathrm {e}}^9\,{\mathrm {e}}^{\frac {4}{8\,x+x\,\mathrm {e}}} \]