∫ −256+ex(−8−64x)+e2x(18+16x)_______________________

Optimal. Leaf size=22 \[ \frac {\left (-4+e^x\right )^2}{\left (1-x^2+(4+x)^2\right )^2} \]

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Rubi [A] time = 0.33, antiderivative size = 35, normalized size of antiderivative = 1.59, number of steps used = 6, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6688, 12, 6742, 2197} \begin {gather*} -\frac {8 e^x}{(8 x+17)^2}+\frac {e^{2 x}}{(8 x+17)^2}+\frac {16}{(8 x+17)^2} \end {gather*}

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

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],

e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c

*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (4-e^x\right ) \left (-32-e^x (9+8 x)\right )}{(17+8 x)^3} \, dx\\ &=2 \int \frac {\left (4-e^x\right ) \left (-32-e^x (9+8 x)\right )}{(17+8 x)^3} \, dx\\ &=2 \int \left (-\frac {128}{(17+8 x)^3}-\frac {4 e^x (1+8 x)}{(17+8 x)^3}+\frac {e^{2 x} (9+8 x)}{(17+8 x)^3}\right ) \, dx\\ &=\frac {16}{(17+8 x)^2}+2 \int \frac {e^{2 x} (9+8 x)}{(17+8 x)^3} \, dx-8 \int \frac {e^x (1+8 x)}{(17+8 x)^3} \, dx\\ &=\frac {16}{(17+8 x)^2}-\frac {8 e^x}{(17+8 x)^2}+\frac {e^{2 x}}{(17+8 x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.14, size = 15, normalized size = 0.68 \begin {gather*} \frac {\left (-4+e^x\right )^2}{(17+8 x)^2} \end {gather*}

Integrate[(-256 + E^x*(-8 - 64*x) + E^(2*x)*(18 + 16*x))/(4913 + 6936*x + 3264*x^2 + 512*x^3),x]

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fricas [A] time = 0.57, size = 23, normalized size = 1.05 \begin {gather*} \frac {e^{\left (2 \, x\right )} - 8 \, e^{x} + 16}{64 \, x^{2} + 272 \, x + 289} \end {gather*}

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")

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giac [A] time = 0.22, size = 23, normalized size = 1.05 \begin {gather*} \frac {e^{\left (2 \, x\right )} - 8 \, e^{x} + 16}{64 \, x^{2} + 272 \, x + 289} \end {gather*}

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")

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method	result	size

norman	\(\frac {{\mathrm e}^{2 x}+16-8 \,{\mathrm e}^{x}}{\left (8 x +17\right )^{2}}\)	\(19\)
default	\(\frac {16}{\left (8 x +17\right )^{2}}-\frac {{\mathrm e}^{x}}{8 \left (x +\frac {17}{8}\right )^{2}}+\frac {{\mathrm e}^{2 x}}{64 \left (x +\frac {17}{8}\right )^{2}}\)	\(31\)
risch	\(\frac {1}{4 x^{2}+17 x +\frac {289}{16}}+\frac {{\mathrm e}^{2 x}}{\left (8 x +17\right )^{2}}-\frac {8 \,{\mathrm e}^{x}}{\left (8 x +17\right )^{2}}\)	\(37\)

int(((16*x+18)*exp(x)^2+(-64*x-8)*exp(x)-256)/(512*x^3+3264*x^2+6936*x+4913),x,method=_RETURNVERBOSE)

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {{\left (8 \, x + 17\right )} e^{\left (2 \, x\right )} - 64 \, x e^{x}}{512 \, x^{3} + 3264 \, x^{2} + 6936 \, x + 4913} + \frac {e^{\left (-\frac {17}{8}\right )} E_{3}\left (-x - \frac {17}{8}\right )}{{\left (8 \, x + 17\right )}^{2}} + \frac {16}{64 \, x^{2} + 272 \, x + 289} - 2 \, \int \frac {32 \, {\left (16 \, x - 17\right )} e^{x}}{4096 \, x^{4} + 34816 \, x^{3} + 110976 \, x^{2} + 157216 \, x + 83521}\,{d x} \end {gather*}

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")

((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 + 289) - 2*integrate(32*(16*x - 17)*e^x/(4096*x^4 + 34816*x^3 + 110976*x^2 +

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mupad [B] time = 0.12, size = 14, normalized size = 0.64 \begin {gather*} \frac {{\left ({\mathrm {e}}^x-4\right )}^2}{{\left (8\,x+17\right )}^2} \end {gather*}

int(-(exp(x)*(64*x + 8) - exp(2*x)*(16*x + 18) + 256)/(6936*x + 3264*x^2 + 512*x^3 + 4913),x)

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sympy [B] time = 0.15, size = 61, normalized size = 2.77 \begin {gather*} \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} \end {gather*}

((-512*x**2 - 2176*x - 2312)*exp(x) + (64*x**2 + 272*x + 289)*exp(2*x))/(4096*x**4 + 34816*x**3 + 110976*x**2

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3.67.82 \(\int \frac {-256+e^x (-8-64 x)+e^{2 x} (18+16 x)}{4913+6936 x+3264 x^2+512 x^3} \, dx\)