\(\int \frac {e^{1+6 x} (30 e^{8-2 x}-20 e^{4-x})}{16+25 e^{8-2 x}-40 e^{4-x}} \, dx\) [4483]

Optimal result

Integrand size = 49, antiderivative size = 19 \[ \int \frac {e^{1+6 x} \left (30 e^{8-2 x}-20 e^{4-x}\right )}{16+25 e^{8-2 x}-40 e^{4-x}} \, dx=\frac {e^{1+6 x}}{5-4 e^{-4+x}} \]

[Out]

exp(6*x+1)/(5-4/exp(-x+4))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2320, 12, 75} \[ \int \frac {e^{1+6 x} \left (30 e^{8-2 x}-20 e^{4-x}\right )}{16+25 e^{8-2 x}-40 e^{4-x}} \, dx=\frac {e^{6 x+5}}{5 e^4-4 e^x} \]

[In]

Int[(E^(1 + 6*x)*(30*E^(8 - 2*x) - 20*E^(4 - x)))/(16 + 25*E^(8 - 2*x) - 40*E^(4 - x)),x]

[Out]

E^(5 + 6*x)/(5*E^4 - 4*E^x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {10 e^5 \left (3 e^4-2 x\right ) x^5}{\left (5 e^4-4 x\right )^2} \, dx,x,e^x\right ) \\ & = \left (10 e^5\right ) \text {Subst}\left (\int \frac {\left (3 e^4-2 x\right ) x^5}{\left (5 e^4-4 x\right )^2} \, dx,x,e^x\right ) \\ & = \frac {e^{5+6 x}}{5 e^4-4 e^x} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(19)=38\).

Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \frac {e^{1+6 x} \left (30 e^{8-2 x}-20 e^{4-x}\right )}{16+25 e^{8-2 x}-40 e^{4-x}} \, dx=-\frac {15625 e^{29}-12500 e^{25+x}+4096 e^{5+6 x}}{4096 \left (-5 e^4+4 e^x\right )} \]

[In]

Integrate[(E^(1 + 6*x)*(30*E^(8 - 2*x) - 20*E^(4 - x)))/(16 + 25*E^(8 - 2*x) - 40*E^(4 - x)),x]

[Out]

-1/4096*(15625*E^29 - 12500*E^(25 + x) + 4096*E^(5 + 6*x))/(-5*E^4 + 4*E^x)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05

method	result	size
risch	\(\frac {{\mathrm e}^{5 x +5}}{5 \,{\mathrm e}^{-x +4}-4}\)	\(20\)
norman	\(\frac {{\mathrm e}^{25} {\mathrm e}^{5 x -20}}{5 \,{\mathrm e}^{-x +4}-4}\)	\(24\)
parallelrisch	\(\frac {{\mathrm e}^{-x +4} {\mathrm e}^{6 x +1}}{5 \,{\mathrm e}^{-x +4}-4}\)	\(26\)
default	\(\frac {-\frac {15 \,{\mathrm e} \,{\mathrm e}^{8} {\mathrm e}^{5 x}}{8}-\frac {125 \,{\mathrm e} \,{\mathrm e}^{12} {\mathrm e}^{4 x}}{32}-\frac {625 \,{\mathrm e} \,{\mathrm e}^{16} {\mathrm e}^{3 x}}{64}-\frac {9375 \,{\mathrm e}^{20} {\mathrm e} \,{\mathrm e}^{2 x}}{256}+\frac {234375 \,{\mathrm e}^{28} {\mathrm e}}{2048}}{5 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{x}}+\frac {{\mathrm e} \,{\mathrm e}^{4} {\mathrm e}^{6 x}+\frac {15 \,{\mathrm e} \,{\mathrm e}^{8} {\mathrm e}^{5 x}}{8}+\frac {125 \,{\mathrm e} \,{\mathrm e}^{12} {\mathrm e}^{4 x}}{32}+\frac {625 \,{\mathrm e} \,{\mathrm e}^{16} {\mathrm e}^{3 x}}{64}+\frac {9375 \,{\mathrm e}^{20} {\mathrm e} \,{\mathrm e}^{2 x}}{256}-\frac {234375 \,{\mathrm e}^{28} {\mathrm e}}{2048}}{5 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{x}}\)	\(152\)

[In]

int((30*exp(-x+4)^2-20*exp(-x+4))*exp(6*x+1)/(25*exp(-x+4)^2-40*exp(-x+4)+16),x,method=_RETURNVERBOSE)

[Out]

exp(5*x+5)/(5*exp(-x+4)-4)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {e^{1+6 x} \left (30 e^{8-2 x}-20 e^{4-x}\right )}{16+25 e^{8-2 x}-40 e^{4-x}} \, dx=-\frac {e^{25}}{4 \, e^{\left (-5 \, x + 20\right )} - 5 \, e^{\left (-6 \, x + 24\right )}} \]

[In]

integrate((30*exp(-x+4)^2-20*exp(-x+4))*exp(6*x+1)/(25*exp(-x+4)^2-40*exp(-x+4)+16),x, algorithm="fricas")

[Out]

-e^25/(4*e^(-5*x + 20) - 5*e^(-6*x + 24))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (14) = 28\).

Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 4.00 \[ \int \frac {e^{1+6 x} \left (30 e^{8-2 x}-20 e^{4-x}\right )}{16+25 e^{8-2 x}-40 e^{4-x}} \, dx=- \frac {625 e^{25} e^{x - 4}}{1024} - \frac {125 e^{25} e^{2 x - 8}}{256} - \frac {25 e^{25} e^{3 x - 12}}{64} - \frac {5 e^{25} e^{4 x - 16}}{16} - \frac {e^{25} e^{5 x - 20}}{4} + \frac {3125 e^{25}}{5120 e^{4 - x} - 4096} \]

[In]

integrate((30*exp(-x+4)**2-20*exp(-x+4))*exp(6*x+1)/(25*exp(-x+4)**2-40*exp(-x+4)+16),x)

[Out]

-625*exp(25)*exp(x - 4)/1024 - 125*exp(25)*exp(2*x - 8)/256 - 25*exp(25)*exp(3*x - 12)/64 - 5*exp(25)*exp(4*x
- 16)/16 - exp(25)*exp(5*x - 20)/4 + 3125*exp(25)/(5120*exp(4 - x) - 4096)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {e^{1+6 x} \left (30 e^{8-2 x}-20 e^{4-x}\right )}{16+25 e^{8-2 x}-40 e^{4-x}} \, dx=-\frac {e^{25}}{4 \, e^{\left (-5 \, x + 20\right )} - 5 \, e^{\left (-6 \, x + 24\right )}} \]

[In]

integrate((30*exp(-x+4)^2-20*exp(-x+4))*exp(6*x+1)/(25*exp(-x+4)^2-40*exp(-x+4)+16),x, algorithm="maxima")

[Out]

-e^25/(4*e^(-5*x + 20) - 5*e^(-6*x + 24))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.26 \[ \int \frac {e^{1+6 x} \left (30 e^{8-2 x}-20 e^{4-x}\right )}{16+25 e^{8-2 x}-40 e^{4-x}} \, dx=-\frac {1}{1024} \, {\left (256 \, e^{25} + 320 \, e^{\left (-x + 29\right )} + 400 \, e^{\left (-2 \, x + 33\right )} + 500 \, e^{\left (-3 \, x + 37\right )} + 625 \, e^{\left (-4 \, x + 41\right )}\right )} e^{\left (5 \, x - 20\right )} + \frac {3125 \, e^{25}}{1024 \, {\left (5 \, e^{\left (-x + 4\right )} - 4\right )}} \]

[In]

integrate((30*exp(-x+4)^2-20*exp(-x+4))*exp(6*x+1)/(25*exp(-x+4)^2-40*exp(-x+4)+16),x, algorithm="giac")

[Out]

-1/1024*(256*e^25 + 320*e^(-x + 29) + 400*e^(-2*x + 33) + 500*e^(-3*x + 37) + 625*e^(-4*x + 41))*e^(5*x - 20)
+ 3125/1024*e^25/(5*e^(-x + 4) - 4)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.89 \[ \int \frac {e^{1+6 x} \left (30 e^{8-2 x}-20 e^{4-x}\right )}{16+25 e^{8-2 x}-40 e^{4-x}} \, dx=\frac {3125\,{\mathrm {e}}^{25}}{1024\,\left (5\,{\mathrm {e}}^{4-x}-4\right )}-\frac {{\mathrm {e}}^{5\,x+5}}{4}-\frac {5\,{\mathrm {e}}^{4\,x+9}}{16}-\frac {25\,{\mathrm {e}}^{3\,x+13}}{64}-\frac {125\,{\mathrm {e}}^{2\,x+17}}{256}-\frac {625\,{\mathrm {e}}^{x+21}}{1024} \]

[In]

int(-(exp(6*x + 1)*(20*exp(4 - x) - 30*exp(8 - 2*x)))/(25*exp(8 - 2*x) - 40*exp(4 - x) + 16),x)

[Out]

(3125*exp(25))/(1024*(5*exp(4 - x) - 4)) - exp(5*x + 5)/4 - (5*exp(4*x + 9))/16 - (25*exp(3*x + 13))/64 - (125
*exp(2*x + 17))/256 - (625*exp(x + 21))/1024