3.37.24 \(\int \frac {e^{-2 x} (-48-288 x-640 x^2-768 x^3-768 x^4-512 x^5)}{2401 x^4} \, dx\)

Optimal. Leaf size=19 \[ \frac {16 e^{-2 x} (1+2 x)^4}{2401 x^3} \]

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Rubi [B]  time = 0.19, antiderivative size = 56, normalized size of antiderivative = 2.95, number of steps used = 16, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {12, 2199, 2194, 2177, 2178, 2176} \begin {gather*} \frac {16 e^{-2 x}}{2401 x^3}+\frac {128 e^{-2 x}}{2401 x^2}+\frac {256 e^{-2 x} x}{2401}+\frac {512 e^{-2 x}}{2401}+\frac {384 e^{-2 x}}{2401 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-48 - 288*x - 640*x^2 - 768*x^3 - 768*x^4 - 512*x^5)/(2401*E^(2*x)*x^4),x]

[Out]

512/(2401*E^(2*x)) + 16/(2401*E^(2*x)*x^3) + 128/(2401*E^(2*x)*x^2) + 384/(2401*E^(2*x)*x) + (256*x)/(2401*E^(
2*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 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{-2 x} \left (-48-288 x-640 x^2-768 x^3-768 x^4-512 x^5\right )}{x^4} \, dx}{2401}\\ &=\frac {\int \left (-768 e^{-2 x}-\frac {48 e^{-2 x}}{x^4}-\frac {288 e^{-2 x}}{x^3}-\frac {640 e^{-2 x}}{x^2}-\frac {768 e^{-2 x}}{x}-512 e^{-2 x} x\right ) \, dx}{2401}\\ &=-\frac {48 \int \frac {e^{-2 x}}{x^4} \, dx}{2401}-\frac {288 \int \frac {e^{-2 x}}{x^3} \, dx}{2401}-\frac {512 \int e^{-2 x} x \, dx}{2401}-\frac {640 \int \frac {e^{-2 x}}{x^2} \, dx}{2401}-\frac {768 \int e^{-2 x} \, dx}{2401}-\frac {768 \int \frac {e^{-2 x}}{x} \, dx}{2401}\\ &=\frac {384 e^{-2 x}}{2401}+\frac {16 e^{-2 x}}{2401 x^3}+\frac {144 e^{-2 x}}{2401 x^2}+\frac {640 e^{-2 x}}{2401 x}+\frac {256 e^{-2 x} x}{2401}-\frac {768 \text {Ei}(-2 x)}{2401}+\frac {32 \int \frac {e^{-2 x}}{x^3} \, dx}{2401}-\frac {256 \int e^{-2 x} \, dx}{2401}+\frac {288 \int \frac {e^{-2 x}}{x^2} \, dx}{2401}+\frac {1280 \int \frac {e^{-2 x}}{x} \, dx}{2401}\\ &=\frac {512 e^{-2 x}}{2401}+\frac {16 e^{-2 x}}{2401 x^3}+\frac {128 e^{-2 x}}{2401 x^2}+\frac {352 e^{-2 x}}{2401 x}+\frac {256 e^{-2 x} x}{2401}+\frac {512 \text {Ei}(-2 x)}{2401}-\frac {32 \int \frac {e^{-2 x}}{x^2} \, dx}{2401}-\frac {576 \int \frac {e^{-2 x}}{x} \, dx}{2401}\\ &=\frac {512 e^{-2 x}}{2401}+\frac {16 e^{-2 x}}{2401 x^3}+\frac {128 e^{-2 x}}{2401 x^2}+\frac {384 e^{-2 x}}{2401 x}+\frac {256 e^{-2 x} x}{2401}-\frac {64 \text {Ei}(-2 x)}{2401}+\frac {64 \int \frac {e^{-2 x}}{x} \, dx}{2401}\\ &=\frac {512 e^{-2 x}}{2401}+\frac {16 e^{-2 x}}{2401 x^3}+\frac {128 e^{-2 x}}{2401 x^2}+\frac {384 e^{-2 x}}{2401 x}+\frac {256 e^{-2 x} x}{2401}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 19, normalized size = 1.00 \begin {gather*} \frac {16 e^{-2 x} (1+2 x)^4}{2401 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-48 - 288*x - 640*x^2 - 768*x^3 - 768*x^4 - 512*x^5)/(2401*E^(2*x)*x^4),x]

[Out]

(16*(1 + 2*x)^4)/(2401*E^(2*x)*x^3)

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fricas [A]  time = 0.47, size = 29, normalized size = 1.53 \begin {gather*} \frac {16 \, {\left (16 \, x^{4} + 32 \, x^{3} + 24 \, x^{2} + 8 \, x + 1\right )} e^{\left (-2 \, x\right )}}{2401 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2401*(-512*x^5-768*x^4-768*x^3-640*x^2-288*x-48)/exp(x)^2/x^4,x, algorithm="fricas")

[Out]

16/2401*(16*x^4 + 32*x^3 + 24*x^2 + 8*x + 1)*e^(-2*x)/x^3

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giac [B]  time = 0.14, size = 44, normalized size = 2.32 \begin {gather*} \frac {16 \, {\left (16 \, x^{4} e^{\left (-2 \, x\right )} + 32 \, x^{3} e^{\left (-2 \, x\right )} + 24 \, x^{2} e^{\left (-2 \, x\right )} + 8 \, x e^{\left (-2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}}{2401 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2401*(-512*x^5-768*x^4-768*x^3-640*x^2-288*x-48)/exp(x)^2/x^4,x, algorithm="giac")

[Out]

16/2401*(16*x^4*e^(-2*x) + 32*x^3*e^(-2*x) + 24*x^2*e^(-2*x) + 8*x*e^(-2*x) + e^(-2*x))/x^3

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maple [A]  time = 0.09, size = 29, normalized size = 1.53

method	result	size
norman	\(\frac {\left (\frac {16}{2401}+\frac {128}{2401} x +\frac {384}{2401} x^{2}+\frac {512}{2401} x^{3}+\frac {256}{2401} x^{4}\right ) {\mathrm e}^{-2 x}}{x^{3}}\)	\(29\)
gosper	\(\frac {16 \left (16 x^{4}+32 x^{3}+24 x^{2}+8 x +1\right ) {\mathrm e}^{-2 x}}{2401 x^{3}}\)	\(30\)
risch	\(\frac {16 \left (16 x^{4}+32 x^{3}+24 x^{2}+8 x +1\right ) {\mathrm e}^{-2 x}}{2401 x^{3}}\)	\(30\)
default	\(\frac {512 \,{\mathrm e}^{-2 x}}{2401}+\frac {256 x \,{\mathrm e}^{-2 x}}{2401}+\frac {16 \,{\mathrm e}^{-2 x}}{2401 x^{3}}+\frac {128 \,{\mathrm e}^{-2 x}}{2401 x^{2}}+\frac {384 \,{\mathrm e}^{-2 x}}{2401 x}\)	\(42\)
meijerg	\(\frac {48 \left (-6 x +3\right ) {\mathrm e}^{-2 x}}{2401 x^{2}}+\frac {2 \left (16 x^{2}-8 x +8\right ) {\mathrm e}^{-2 x}}{2401 x^{3}}+\frac {96}{2401 x^{2}}+\frac {160}{2401 x}+\frac {640 \,{\mathrm e}^{-2 x}}{2401 x}+\frac {64 \left (4 x +2\right ) {\mathrm e}^{-2 x}}{2401}-\frac {24 \left (36 x^{2}-24 x +6\right )}{2401 x^{2}}-\frac {320 \left (-4 x +2\right )}{2401 x}-\frac {2 \left (-176 x^{3}+144 x^{2}-72 x +24\right )}{7203 x^{3}}+\frac {384 \,{\mathrm e}^{-2 x}}{2401}+\frac {16}{2401 x^{3}}-\frac {64}{147}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2401*(-512*x^5-768*x^4-768*x^3-640*x^2-288*x-48)/exp(x)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

(16/2401+128/2401*x+384/2401*x^2+512/2401*x^3+256/2401*x^4)/exp(x)^2/x^3

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maxima [C]  time = 0.64, size = 45, normalized size = 2.37 \begin {gather*} \frac {128}{2401} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {768}{2401} \, {\rm Ei}\left (-2 \, x\right ) + \frac {384}{2401} \, e^{\left (-2 \, x\right )} + \frac {1280}{2401} \, \Gamma \left (-1, 2 \, x\right ) + \frac {1152}{2401} \, \Gamma \left (-2, 2 \, x\right ) + \frac {384}{2401} \, \Gamma \left (-3, 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2401*(-512*x^5-768*x^4-768*x^3-640*x^2-288*x-48)/exp(x)^2/x^4,x, algorithm="maxima")

[Out]

128/2401*(2*x + 1)*e^(-2*x) - 768/2401*Ei(-2*x) + 384/2401*e^(-2*x) + 1280/2401*gamma(-1, 2*x) + 1152/2401*gam
ma(-2, 2*x) + 384/2401*gamma(-3, 2*x)

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mupad [B]  time = 2.09, size = 16, normalized size = 0.84 \begin {gather*} \frac {16\,{\mathrm {e}}^{-2\,x}\,{\left (2\,x+1\right )}^4}{2401\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-2*x)*((288*x)/2401 + (640*x^2)/2401 + (768*x^3)/2401 + (768*x^4)/2401 + (512*x^5)/2401 + 48/2401))/
x^4,x)

[Out]

(16*exp(-2*x)*(2*x + 1)^4)/(2401*x^3)

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sympy [A]  time = 0.12, size = 29, normalized size = 1.53 \begin {gather*} \frac {\left (256 x^{4} + 512 x^{3} + 384 x^{2} + 128 x + 16\right ) e^{- 2 x}}{2401 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2401*(-512*x**5-768*x**4-768*x**3-640*x**2-288*x-48)/exp(x)**2/x**4,x)

[Out]

(256*x**4 + 512*x**3 + 384*x**2 + 128*x + 16)*exp(-2*x)/(2401*x**3)

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