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

Optimal. Leaf size=19 \[ \frac {e^{1+6 x}}{5-4 e^{-4+x}} \]

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Rubi [A]  time = 0.10, antiderivative size = 21, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2282, 12, 74} \begin {gather*} \frac {e^{6 x+5}}{5 e^4-4 e^x} \end {gather*}

Antiderivative was successfully verified.

[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 74

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 2282

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 {gather*} \begin {aligned} \text {integral} &=\operatorname {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 ) \operatorname {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 {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 39, normalized size = 2.05 \begin {gather*} -\frac {15625 e^{29}-12500 e^{25+x}+4096 e^{5+6 x}}{4096 \left (-5 e^4+4 e^x\right )} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.83, size = 23, normalized size = 1.21 \begin {gather*} -\frac {e^{25}}{4 \, e^{\left (-5 \, x + 20\right )} - 5 \, e^{\left (-6 \, x + 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: NotImplementedError >> Unable to parse Giac output: 10*((-50000*exp(1)*exp(4)^6+80000*exp(1)
*exp(4)^4*exp(8)-31875*exp(1)*exp(4)^2*exp(8)^2+1875*exp(1)*exp(8)^3)/8192*ln(abs(16*exp(sageVARx)^2-40*exp(sa
geVARx)*exp(4)+25*exp

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maple [A]  time = 0.17, size = 24, normalized size = 1.26

method	result	size
norman	\(\frac {{\mathrm e}^{25} {\mathrm e}^{5 x -20}}{5 \,{\mathrm e}^{-x +4}-4}\)	\(24\)
risch	\(-\frac {625 \,{\mathrm e}^{x +21}}{1024}-\frac {125 \,{\mathrm e}^{17+2 x}}{256}-\frac {25 \,{\mathrm e}^{3 x +13}}{64}-\frac {5 \,{\mathrm e}^{4 x +9}}{16}-\frac {{\mathrm e}^{5 x +5}}{4}+\frac {3125 \,{\mathrm e}^{25}}{1024 \left (5 \,{\mathrm e}^{-x +4}-4\right )}\)	\(56\)
default	\(\frac {-\frac {15 \,{\mathrm e} \,{\mathrm e}^{8} {\mathrm e}^{5 x}}{8}-\frac {125 \,{\mathrm e} \,{\mathrm e}^{12} {\mathrm e}^{4 x}}{32}-\frac {625 \,{\mathrm e} \,{\mathrm e}^{16} {\mathrm e}^{3 x}}{64}-\frac {9375 \,{\mathrm e}^{20} {\mathrm e} \,{\mathrm e}^{2 x}}{256}+\frac {234375 \,{\mathrm e}^{28} {\mathrm e}}{2048}}{5 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{x}}+\frac {{\mathrm e} \,{\mathrm e}^{4} {\mathrm e}^{6 x}+\frac {15 \,{\mathrm e} \,{\mathrm e}^{8} {\mathrm e}^{5 x}}{8}+\frac {125 \,{\mathrm e} \,{\mathrm e}^{12} {\mathrm e}^{4 x}}{32}+\frac {625 \,{\mathrm e} \,{\mathrm e}^{16} {\mathrm e}^{3 x}}{64}+\frac {9375 \,{\mathrm e}^{20} {\mathrm e} \,{\mathrm e}^{2 x}}{256}-\frac {234375 \,{\mathrm e}^{28} {\mathrm e}}{2048}}{5 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{x}}\)	\(152\)

Verification of antiderivative is not currently implemented for this CAS.

[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(25)/exp(-x+4)^5/(5*exp(-x+4)-4)

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maxima [A]  time = 0.36, size = 23, normalized size = 1.21 \begin {gather*} -\frac {e^{25}}{4 \, e^{\left (-5 \, x + 20\right )} - 5 \, e^{\left (-6 \, x + 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.16, size = 55, normalized size = 2.89 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 0.29, size = 76, normalized size = 4.00 \begin {gather*} - \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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