3.54.98 \(\int \frac {e^{-x} (-4 e^x+e^4 (-243+405 x-270 x^2+90 x^3-15 x^4+x^5)+e^{3 x} (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5))}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx\)

Optimal. Leaf size=20 \[ -e^{4-x}+e^{2 x}+\frac {1}{(-3+x)^4} \]

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Rubi [A]  time = 0.54, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6688, 6742, 2194} \begin {gather*} -e^{4-x}+e^{2 x}+\frac {1}{(x-3)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*E^x + E^4*(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5) + E^(3*x)*(-486 + 810*x - 540*x^2 + 180*x^3
- 30*x^4 + 2*x^5))/(E^x*(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5)),x]

[Out]

-E^(4 - x) + E^(2*x) + (-3 + x)^(-4)

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 \frac {e^{-x} \left (4 e^x-e^4 (-3+x)^5-2 e^{3 x} (-3+x)^5\right )}{(3-x)^5} \, dx\\ &=\int \left (e^{4-x}+2 e^{2 x}-\frac {4}{(-3+x)^5}\right ) \, dx\\ &=\frac {1}{(-3+x)^4}+2 \int e^{2 x} \, dx+\int e^{4-x} \, dx\\ &=-e^{4-x}+e^{2 x}+\frac {1}{(-3+x)^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 20, normalized size = 1.00 \begin {gather*} -e^{4-x}+e^{2 x}+\frac {1}{(-3+x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*E^x + E^4*(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5) + E^(3*x)*(-486 + 810*x - 540*x^2 + 1
80*x^3 - 30*x^4 + 2*x^5))/(E^x*(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5)),x]

[Out]

-E^(4 - x) + E^(2*x) + (-3 + x)^(-4)

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fricas [B]  time = 0.93, size = 76, normalized size = 3.80 \begin {gather*} -\frac {{\left ({\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )} e^{4} - {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )} e^{\left (3 \, x\right )} - e^{x}\right )} e^{\left (-x\right )}}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5-30*x^4+180*x^3-540*x^2+810*x-486)*exp(x)^3-4*exp(x)+(x^5-15*x^4+90*x^3-270*x^2+405*x-243)*ex
p(4))/(x^5-15*x^4+90*x^3-270*x^2+405*x-243)/exp(x),x, algorithm="fricas")

[Out]

-((x^4 - 12*x^3 + 54*x^2 - 108*x + 81)*e^4 - (x^4 - 12*x^3 + 54*x^2 - 108*x + 81)*e^(3*x) - e^x)*e^(-x)/(x^4 -
12*x^3 + 54*x^2 - 108*x + 81)

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giac [B]  time = 0.23, size = 112, normalized size = 5.60 \begin {gather*} \frac {x^{4} e^{\left (2 \, x\right )} - x^{4} e^{\left (-x + 4\right )} - 12 \, x^{3} e^{\left (2 \, x\right )} + 12 \, x^{3} e^{\left (-x + 4\right )} + 54 \, x^{2} e^{\left (2 \, x\right )} - 54 \, x^{2} e^{\left (-x + 4\right )} - 108 \, x e^{\left (2 \, x\right )} + 108 \, x e^{\left (-x + 4\right )} + 81 \, e^{\left (2 \, x\right )} - 81 \, e^{\left (-x + 4\right )} + 1}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5-30*x^4+180*x^3-540*x^2+810*x-486)*exp(x)^3-4*exp(x)+(x^5-15*x^4+90*x^3-270*x^2+405*x-243)*ex
p(4))/(x^5-15*x^4+90*x^3-270*x^2+405*x-243)/exp(x),x, algorithm="giac")

[Out]

(x^4*e^(2*x) - x^4*e^(-x + 4) - 12*x^3*e^(2*x) + 12*x^3*e^(-x + 4) + 54*x^2*e^(2*x) - 54*x^2*e^(-x + 4) - 108*
x*e^(2*x) + 108*x*e^(-x + 4) + 81*e^(2*x) - 81*e^(-x + 4) + 1)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81)

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maple [A]  time = 0.15, size = 34, normalized size = 1.70

method	result	size
risch	\(\frac {1}{x^{4}-12 x^{3}+54 x^{2}-108 x +81}+{\mathrm e}^{2 x}-{\mathrm e}^{-x +4}\)	\(34\)
norman	\(\frac {\left (x^{4} {\mathrm e}^{3 x}+{\mathrm e}^{x}+81 \,{\mathrm e}^{3 x}+108 x \,{\mathrm e}^{4}-108 x \,{\mathrm e}^{3 x}-54 x^{2} {\mathrm e}^{4}+54 x^{2} {\mathrm e}^{3 x}+12 x^{3} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3 x}-x^{4} {\mathrm e}^{4}-81 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-x}}{\left (x -3\right )^{4}}\)	\(83\)
default	\({\mathrm e}^{4} \left (-{\mathrm e}^{-x}-\frac {9 \,{\mathrm e}^{-x} \left (11 x^{3}-30 x^{2}-15 x +72\right )}{8 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}-\frac {21 \,{\mathrm e}^{-3} \expIntegralEi \left (1, x -3\right )}{8}\right )+\frac {1}{\left (x -3\right )^{4}}-243 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-x} \left (x^{3}-10 x^{2}+35 x -48\right )}{24 x^{4}-288 x^{3}+1296 x^{2}-2592 x +1944}-\frac {{\mathrm e}^{-3} \expIntegralEi \left (1, x -3\right )}{24}\right )+{\mathrm e}^{2 x}+405 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-x} \left (x^{3}-10 x^{2}+35 x -24\right )}{24 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}+\frac {{\mathrm e}^{-3} \expIntegralEi \left (1, x -3\right )}{24}\right )-270 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-x} \left (x^{3}-10 x^{2}+43 x -48\right )}{8 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}+\frac {{\mathrm e}^{-3} \expIntegralEi \left (1, x -3\right )}{8}\right )+90 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-x} x \left (x^{2}-18 x +27\right )}{8 x^{4}-96 x^{3}+432 x^{2}-864 x +648}-\frac {{\mathrm e}^{-3} \expIntegralEi \left (1, x -3\right )}{8}\right )-15 \,{\mathrm e}^{4} \left (\frac {3 \,{\mathrm e}^{-x} \left (x^{3}-42 x^{2}+147 x -144\right )}{8 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}-\frac {11 \,{\mathrm e}^{-3} \expIntegralEi \left (1, x -3\right )}{8}\right )\)	\(332\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^5-30*x^4+180*x^3-540*x^2+810*x-486)*exp(x)^3-4*exp(x)+(x^5-15*x^4+90*x^3-270*x^2+405*x-243)*exp(4))/
(x^5-15*x^4+90*x^3-270*x^2+405*x-243)/exp(x),x,method=_RETURNVERBOSE)

[Out]

1/(x^4-12*x^3+54*x^2-108*x+81)+exp(2*x)-exp(-x+4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {{\left (x^{5} - 15 \, x^{4} + 90 \, x^{3} - 270 \, x^{2} + 405 \, x - 243\right )} e^{\left (2 \, x\right )} - {\left (x^{5} e^{4} - 15 \, x^{4} e^{4} + 90 \, x^{3} e^{4} - 270 \, x^{2} e^{4} + 405 \, x e^{4}\right )} e^{\left (-x\right )} + x - 3}{x^{5} - 15 \, x^{4} + 90 \, x^{3} - 270 \, x^{2} + 405 \, x - 243} + \frac {243 \, e E_{5}\left (x - 3\right )}{{\left (x - 3\right )}^{4}} - 1215 \, \int \frac {e^{\left (-x + 4\right )}}{x^{6} - 18 \, x^{5} + 135 \, x^{4} - 540 \, x^{3} + 1215 \, x^{2} - 1458 \, x + 729}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^5-30*x^4+180*x^3-540*x^2+810*x-486)*exp(x)^3-4*exp(x)+(x^5-15*x^4+90*x^3-270*x^2+405*x-243)*ex
p(4))/(x^5-15*x^4+90*x^3-270*x^2+405*x-243)/exp(x),x, algorithm="maxima")

[Out]

((x^5 - 15*x^4 + 90*x^3 - 270*x^2 + 405*x - 243)*e^(2*x) - (x^5*e^4 - 15*x^4*e^4 + 90*x^3*e^4 - 270*x^2*e^4 +
405*x*e^4)*e^(-x) + x - 3)/(x^5 - 15*x^4 + 90*x^3 - 270*x^2 + 405*x - 243) + 243*e*exp_integral_e(5, x - 3)/(x
- 3)^4 - 1215*integrate(e^(-x + 4)/(x^6 - 18*x^5 + 135*x^4 - 540*x^3 + 1215*x^2 - 1458*x + 729), x)

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mupad [B]  time = 3.67, size = 55, normalized size = 2.75 \begin {gather*} \frac {{\mathrm {e}}^x}{81\,{\mathrm {e}}^x+54\,x^2\,{\mathrm {e}}^x-12\,x^3\,{\mathrm {e}}^x+x^4\,{\mathrm {e}}^x-108\,x\,{\mathrm {e}}^x}-{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^4-{\mathrm {e}}^{3\,x}+\frac {{\mathrm {e}}^x}{81}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(exp(4)*(405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5 - 243) - 4*exp(x) + exp(3*x)*(810*x - 540*x^2 +
180*x^3 - 30*x^4 + 2*x^5 - 486)))/(405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5 - 243),x)

[Out]

exp(x)/(81*exp(x) + 54*x^2*exp(x) - 12*x^3*exp(x) + x^4*exp(x) - 108*x*exp(x)) - exp(-x)*(exp(4) - exp(3*x) +
exp(x)/81)

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sympy [A]  time = 0.23, size = 32, normalized size = 1.60 \begin {gather*} e^{2 x} - e^{4} e^{- x} + \frac {4}{4 x^{4} - 48 x^{3} + 216 x^{2} - 432 x + 324} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**5-30*x**4+180*x**3-540*x**2+810*x-486)*exp(x)**3-4*exp(x)+(x**5-15*x**4+90*x**3-270*x**2+405*
x-243)*exp(4))/(x**5-15*x**4+90*x**3-270*x**2+405*x-243)/exp(x),x)

[Out]

exp(2*x) - exp(4)*exp(-x) + 4/(4*x**4 - 48*x**3 + 216*x**2 - 432*x + 324)

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