3.31.17 \(\int \frac {e^{-21+4 x} (4-22 x+8 x^2)}{-200 x^2+300 x^3-150 x^4+25 x^5} \, dx\)

Optimal. Leaf size=23 \[ \frac {2 e^{-21+4 x} x}{25 (-x+(-1+x) x)^2} \]

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Rubi [B]  time = 0.52, antiderivative size = 51, normalized size of antiderivative = 2.22, number of steps used = 13, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6688, 12, 6742, 2178, 2177} \begin {gather*} \frac {e^{4 x-21}}{50 x}+\frac {e^{4 x-21}}{50 (2-x)}+\frac {e^{4 x-21}}{25 (2-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-21 + 4*x)*(4 - 22*x + 8*x^2))/(-200*x^2 + 300*x^3 - 150*x^4 + 25*x^5),x]

[Out]

E^(-21 + 4*x)/(25*(2 - x)^2) + E^(-21 + 4*x)/(50*(2 - x)) + E^(-21 + 4*x)/(50*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 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 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 {2 e^{-21+4 x} \left (-2+11 x-4 x^2\right )}{25 (2-x)^3 x^2} \, dx\\ &=\frac {2}{25} \int \frac {e^{-21+4 x} \left (-2+11 x-4 x^2\right )}{(2-x)^3 x^2} \, dx\\ &=\frac {2}{25} \int \left (\frac {e^{-21+4 x}}{2-x}-\frac {e^{-21+4 x}}{(-2+x)^3}+\frac {9 e^{-21+4 x}}{4 (-2+x)^2}-\frac {e^{-21+4 x}}{4 x^2}+\frac {e^{-21+4 x}}{x}\right ) \, dx\\ &=-\left (\frac {1}{50} \int \frac {e^{-21+4 x}}{x^2} \, dx\right )+\frac {2}{25} \int \frac {e^{-21+4 x}}{2-x} \, dx-\frac {2}{25} \int \frac {e^{-21+4 x}}{(-2+x)^3} \, dx+\frac {2}{25} \int \frac {e^{-21+4 x}}{x} \, dx+\frac {9}{50} \int \frac {e^{-21+4 x}}{(-2+x)^2} \, dx\\ &=\frac {e^{-21+4 x}}{25 (2-x)^2}+\frac {9 e^{-21+4 x}}{50 (2-x)}+\frac {e^{-21+4 x}}{50 x}-\frac {2 \text {Ei}(-4 (2-x))}{25 e^{13}}+\frac {2 \text {Ei}(4 x)}{25 e^{21}}-\frac {2}{25} \int \frac {e^{-21+4 x}}{x} \, dx-\frac {4}{25} \int \frac {e^{-21+4 x}}{(-2+x)^2} \, dx+\frac {18}{25} \int \frac {e^{-21+4 x}}{-2+x} \, dx\\ &=\frac {e^{-21+4 x}}{25 (2-x)^2}+\frac {e^{-21+4 x}}{50 (2-x)}+\frac {e^{-21+4 x}}{50 x}+\frac {16 \text {Ei}(-4 (2-x))}{25 e^{13}}-\frac {16}{25} \int \frac {e^{-21+4 x}}{-2+x} \, dx\\ &=\frac {e^{-21+4 x}}{25 (2-x)^2}+\frac {e^{-21+4 x}}{50 (2-x)}+\frac {e^{-21+4 x}}{50 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 e^{-21+4 x}}{25 (-2+x)^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-21 + 4*x)*(4 - 22*x + 8*x^2))/(-200*x^2 + 300*x^3 - 150*x^4 + 25*x^5),x]

[Out]

(2*E^(-21 + 4*x))/(25*(-2 + x)^2*x)

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fricas [A]  time = 0.63, size = 22, normalized size = 0.96 \begin {gather*} \frac {2 \, e^{\left (4 \, x - 21\right )}}{25 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^2-22*x+4)/(25*x^5-150*x^4+300*x^3-200*x^2)/exp(-x+21/4)^4,x, algorithm="fricas")

[Out]

2/25*e^(4*x - 21)/(x^3 - 4*x^2 + 4*x)

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giac [A]  time = 0.21, size = 27, normalized size = 1.17 \begin {gather*} \frac {2 \, e^{\left (4 \, x\right )}}{25 \, {\left (x^{3} e^{21} - 4 \, x^{2} e^{21} + 4 \, x e^{21}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^2-22*x+4)/(25*x^5-150*x^4+300*x^3-200*x^2)/exp(-x+21/4)^4,x, algorithm="giac")

[Out]

2/25*e^(4*x)/(x^3*e^21 - 4*x^2*e^21 + 4*x*e^21)

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maple [A]  time = 0.10, size = 17, normalized size = 0.74

method	result	size
risch	\(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (x -2\right )^{2}}\)	\(17\)
norman	\(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (x -2\right )^{2}}\)	\(19\)
gosper	\(\frac {2 \,{\mathrm e}^{4 x -21}}{25 x \left (x^{2}-4 x +4\right )}\)	\(24\)
derivativedivides	\(\frac {109 \,{\mathrm e}^{4 x -21} \left (16 \left (-x +\frac {21}{4}\right )^{2}+184 x -505\right )}{50 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}+\frac {31 \,{\mathrm e}^{4 x -21} \left (48 \left (-x +\frac {21}{4}\right )^{2}-472 x +21\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}-\frac {{\mathrm e}^{4 x -21} \left (4976 \left (-x +\frac {21}{4}\right )^{2}+25480 x -108927\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}\)	\(143\)
default	\(\frac {109 \,{\mathrm e}^{4 x -21} \left (16 \left (-x +\frac {21}{4}\right )^{2}+184 x -505\right )}{50 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}+\frac {31 \,{\mathrm e}^{4 x -21} \left (48 \left (-x +\frac {21}{4}\right )^{2}-472 x +21\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}-\frac {{\mathrm e}^{4 x -21} \left (4976 \left (-x +\frac {21}{4}\right )^{2}+25480 x -108927\right )}{100 \left (64 \left (-x +\frac {21}{4}\right )^{3}-752 \left (-x +\frac {21}{4}\right )^{2}-2860 x +11466\right )}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^2-22*x+4)/(25*x^5-150*x^4+300*x^3-200*x^2)/exp(-x+21/4)^4,x,method=_RETURNVERBOSE)

[Out]

2/25/x/(x-2)^2*exp(4*x-21)

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maxima [A]  time = 0.80, size = 27, normalized size = 1.17 \begin {gather*} \frac {2 \, e^{\left (4 \, x\right )}}{25 \, {\left (x^{3} e^{21} - 4 \, x^{2} e^{21} + 4 \, x e^{21}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^2-22*x+4)/(25*x^5-150*x^4+300*x^3-200*x^2)/exp(-x+21/4)^4,x, algorithm="maxima")

[Out]

2/25*e^(4*x)/(x^3*e^21 - 4*x^2*e^21 + 4*x*e^21)

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mupad [B]  time = 0.11, size = 24, normalized size = 1.04 \begin {gather*} \frac {2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{-21}}{25\,\left (x^3-4\,x^2+4\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4*x - 21)*(8*x^2 - 22*x + 4))/(200*x^2 - 300*x^3 + 150*x^4 - 25*x^5),x)

[Out]

(2*exp(4*x)*exp(-21))/(25*(4*x - 4*x^2 + x^3))

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sympy [A]  time = 0.12, size = 20, normalized size = 0.87 \begin {gather*} \frac {2 e^{4 x - 21}}{25 x^{3} - 100 x^{2} + 100 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**2-22*x+4)/(25*x**5-150*x**4+300*x**3-200*x**2)/exp(-x+21/4)**4,x)

[Out]

2*exp(4*x - 21)/(25*x**3 - 100*x**2 + 100*x)

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