3.41.54 \(\int \frac {e^{2 x} (8-20 x+24 x^2-14 x^3+2 x^4)+e^{3+x} (-5184 x^3+3888 x^4-972 x^5+81 x^6)}{-5184 x^3+3888 x^4-972 x^5+81 x^6} \, dx\)

Optimal. Leaf size=28 \[ e^{3+x}+\frac {e^{2 x} (-1+x)^2}{81 (-4+x)^2 x^2} \]

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Rubi [B]  time = 0.49, antiderivative size = 62, normalized size of antiderivative = 2.21, number of steps used = 17, number of rules used = 5, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {6688, 2194, 6742, 2177, 2178} \begin {gather*} \frac {e^{2 x}}{1296 x^2}+e^{x+3}-\frac {e^{2 x}}{864 x}-\frac {e^{2 x}}{864 (4-x)}+\frac {e^{2 x}}{144 (4-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*x)*(8 - 20*x + 24*x^2 - 14*x^3 + 2*x^4) + E^(3 + x)*(-5184*x^3 + 3888*x^4 - 972*x^5 + 81*x^6))/(-518
4*x^3 + 3888*x^4 - 972*x^5 + 81*x^6),x]

[Out]

E^(3 + x) + E^(2*x)/(144*(4 - x)^2) - E^(2*x)/(864*(4 - x)) + E^(2*x)/(1296*x^2) - E^(2*x)/(864*x)

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{3+x}+\frac {2 e^{2 x} \left (4-10 x+12 x^2-7 x^3+x^4\right )}{81 (-4+x)^3 x^3}\right ) \, dx\\ &=\frac {2}{81} \int \frac {e^{2 x} \left (4-10 x+12 x^2-7 x^3+x^4\right )}{(-4+x)^3 x^3} \, dx+\int e^{3+x} \, dx\\ &=e^{3+x}+\frac {2}{81} \int \left (-\frac {9 e^{2 x}}{16 (-4+x)^3}+\frac {33 e^{2 x}}{64 (-4+x)^2}+\frac {3 e^{2 x}}{32 (-4+x)}-\frac {e^{2 x}}{16 x^3}+\frac {7 e^{2 x}}{64 x^2}-\frac {3 e^{2 x}}{32 x}\right ) \, dx\\ &=e^{3+x}-\frac {1}{648} \int \frac {e^{2 x}}{x^3} \, dx+\frac {1}{432} \int \frac {e^{2 x}}{-4+x} \, dx-\frac {1}{432} \int \frac {e^{2 x}}{x} \, dx+\frac {7 \int \frac {e^{2 x}}{x^2} \, dx}{2592}+\frac {11}{864} \int \frac {e^{2 x}}{(-4+x)^2} \, dx-\frac {1}{72} \int \frac {e^{2 x}}{(-4+x)^3} \, dx\\ &=e^{3+x}+\frac {e^{2 x}}{144 (4-x)^2}+\frac {11 e^{2 x}}{864 (4-x)}+\frac {e^{2 x}}{1296 x^2}-\frac {7 e^{2 x}}{2592 x}+\frac {1}{432} e^8 \text {Ei}(-2 (4-x))-\frac {\text {Ei}(2 x)}{432}-\frac {1}{648} \int \frac {e^{2 x}}{x^2} \, dx+\frac {7 \int \frac {e^{2 x}}{x} \, dx}{1296}-\frac {1}{72} \int \frac {e^{2 x}}{(-4+x)^2} \, dx+\frac {11}{432} \int \frac {e^{2 x}}{-4+x} \, dx\\ &=e^{3+x}+\frac {e^{2 x}}{144 (4-x)^2}-\frac {e^{2 x}}{864 (4-x)}+\frac {e^{2 x}}{1296 x^2}-\frac {e^{2 x}}{864 x}+\frac {1}{36} e^8 \text {Ei}(-2 (4-x))+\frac {\text {Ei}(2 x)}{324}-\frac {1}{324} \int \frac {e^{2 x}}{x} \, dx-\frac {1}{36} \int \frac {e^{2 x}}{-4+x} \, dx\\ &=e^{3+x}+\frac {e^{2 x}}{144 (4-x)^2}-\frac {e^{2 x}}{864 (4-x)}+\frac {e^{2 x}}{1296 x^2}-\frac {e^{2 x}}{864 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 38, normalized size = 1.36 \begin {gather*} \frac {e^x \left (e^x (-1+x)^2+81 e^3 (-4+x)^2 x^2\right )}{81 (-4+x)^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(8 - 20*x + 24*x^2 - 14*x^3 + 2*x^4) + E^(3 + x)*(-5184*x^3 + 3888*x^4 - 972*x^5 + 81*x^6))
/(-5184*x^3 + 3888*x^4 - 972*x^5 + 81*x^6),x]

[Out]

(E^x*(E^x*(-1 + x)^2 + 81*E^3*(-4 + x)^2*x^2))/(81*(-4 + x)^2*x^2)

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fricas [B]  time = 0.58, size = 56, normalized size = 2.00 \begin {gather*} \frac {{\left ({\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x + 6\right )} + 81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (x + 9\right )}\right )} e^{\left (-6\right )}}{81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+
3888*x^4-5184*x^3),x, algorithm="fricas")

[Out]

1/81*((x^2 - 2*x + 1)*e^(2*x + 6) + 81*(x^4 - 8*x^3 + 16*x^2)*e^(x + 9))*e^(-6)/(x^4 - 8*x^3 + 16*x^2)

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giac [B]  time = 0.16, size = 65, normalized size = 2.32 \begin {gather*} \frac {81 \, x^{4} e^{\left (x + 3\right )} - 648 \, x^{3} e^{\left (x + 3\right )} + x^{2} e^{\left (2 \, x\right )} + 1296 \, x^{2} e^{\left (x + 3\right )} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )}}{81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+
3888*x^4-5184*x^3),x, algorithm="giac")

[Out]

1/81*(81*x^4*e^(x + 3) - 648*x^3*e^(x + 3) + x^2*e^(2*x) + 1296*x^2*e^(x + 3) - 2*x*e^(2*x) + e^(2*x))/(x^4 -
8*x^3 + 16*x^2)

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maple [A]  time = 0.09, size = 28, normalized size = 1.00

method	result	size
risch	\(\frac {\left (x^{2}-2 x +1\right ) {\mathrm e}^{2 x}}{81 \left (x -4\right )^{2} x^{2}}+{\mathrm e}^{3+x}\)	\(28\)
norman	\(\frac {{\mathrm e}^{x} {\mathrm e}^{3} x^{4}+\frac {{\mathrm e}^{2 x}}{81}-\frac {2 x \,{\mathrm e}^{2 x}}{81}+\frac {{\mathrm e}^{2 x} x^{2}}{81}+16 x^{2} {\mathrm e}^{3} {\mathrm e}^{x}-8 \,{\mathrm e}^{x} {\mathrm e}^{3} x^{3}}{\left (x -4\right )^{2} x^{2}}\)	\(59\)
default	\(\frac {{\mathrm e}^{2 x}}{144 \left (x -4\right )^{2}}+\frac {{\mathrm e}^{2 x}}{864 x -3456}-\frac {{\mathrm e}^{2 x}}{864 x}+\frac {{\mathrm e}^{2 x}}{1296 x^{2}}-64 \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{x}}{2 \left (x -4\right )^{2}}-\frac {{\mathrm e}^{x}}{2 \left (x -4\right )}-\frac {{\mathrm e}^{4} \expIntegralEi \left (1, -x +4\right )}{2}\right )+48 \,{\mathrm e}^{3} \left (-\frac {2 \,{\mathrm e}^{x}}{\left (x -4\right )^{2}}-\frac {3 \,{\mathrm e}^{x}}{x -4}-3 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x +4\right )\right )-12 \,{\mathrm e}^{3} \left (-\frac {8 \,{\mathrm e}^{x}}{\left (x -4\right )^{2}}-\frac {16 \,{\mathrm e}^{x}}{x -4}-17 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x +4\right )\right )+{\mathrm e}^{3} \left ({\mathrm e}^{x}-\frac {32 \,{\mathrm e}^{x}}{\left (x -4\right )^{2}}-\frac {80 \,{\mathrm e}^{x}}{x -4}-92 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x +4\right )\right )\)	\(179\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+3888*x
^4-5184*x^3),x,method=_RETURNVERBOSE)

[Out]

1/81*(x^2-2*x+1)/(x-4)^2/x^2*exp(2*x)+exp(3+x)

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maxima [B]  time = 0.40, size = 57, normalized size = 2.04 \begin {gather*} \frac {{\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + 81 \, {\left (x^{4} e^{3} - 8 \, x^{3} e^{3} + 16 \, x^{2} e^{3}\right )} e^{x}}{81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x^6-972*x^5+3888*x^4-5184*x^3)*exp(3+x)+(2*x^4-14*x^3+24*x^2-20*x+8)*exp(x)^2)/(81*x^6-972*x^5+
3888*x^4-5184*x^3),x, algorithm="maxima")

[Out]

1/81*((x^2 - 2*x + 1)*e^(2*x) + 81*(x^4*e^3 - 8*x^3*e^3 + 16*x^2*e^3)*e^x)/(x^4 - 8*x^3 + 16*x^2)

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mupad [B]  time = 3.21, size = 55, normalized size = 1.96 \begin {gather*} \frac {{\mathrm {e}}^{x-3}\,\left ({\mathrm {e}}^{x+3}-2\,x\,{\mathrm {e}}^{x+3}+x^2\,{\mathrm {e}}^{x+3}+1296\,x^2\,{\mathrm {e}}^6-648\,x^3\,{\mathrm {e}}^6+81\,x^4\,{\mathrm {e}}^6\right )}{81\,x^2\,{\left (x-4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(24*x^2 - 20*x - 14*x^3 + 2*x^4 + 8) - exp(x + 3)*(5184*x^3 - 3888*x^4 + 972*x^5 - 81*x^6))/(51
84*x^3 - 3888*x^4 + 972*x^5 - 81*x^6),x)

[Out]

(exp(x - 3)*(exp(x + 3) - 2*x*exp(x + 3) + x^2*exp(x + 3) + 1296*x^2*exp(6) - 648*x^3*exp(6) + 81*x^4*exp(6)))
/(81*x^2*(x - 4)^2)

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sympy [B]  time = 0.19, size = 61, normalized size = 2.18 \begin {gather*} \frac {\left (x^{2} - 2 x + 1\right ) e^{2 x} + \left (81 x^{4} e^{3} - 648 x^{3} e^{3} + 1296 x^{2} e^{3}\right ) \sqrt {e^{2 x}}}{81 x^{4} - 648 x^{3} + 1296 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((81*x**6-972*x**5+3888*x**4-5184*x**3)*exp(3+x)+(2*x**4-14*x**3+24*x**2-20*x+8)*exp(x)**2)/(81*x**6
-972*x**5+3888*x**4-5184*x**3),x)

[Out]

((x**2 - 2*x + 1)*exp(2*x) + (81*x**4*exp(3) - 648*x**3*exp(3) + 1296*x**2*exp(3))*sqrt(exp(2*x)))/(81*x**4 -
648*x**3 + 1296*x**2)

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