3.56.23 \(\int \frac {e^{-4 x} (e^{4 x} (-18 x^2-2 x^3)+e^{6 x} (6 x+6 x^2+2 x^3)+e^{2 x} (12 x^3-4 x^4-2 x^5))}{675+675 x+225 x^2+25 x^3} \, dx\)

Optimal. Leaf size=28 \[ \frac {e^{2 x} x^2 \left (-1+e^{-2 x} x\right )^2}{25 (3+x)^2} \]

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Rubi [B]  time = 0.99, antiderivative size = 109, normalized size of antiderivative = 3.89, number of steps used = 28, number of rules used = 9, integrand size = 83, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6688, 12, 6742, 74, 2199, 2194, 2176, 2177, 2178} \begin {gather*} -\frac {2 x^3}{25 (x+3)^2}+\frac {1}{25} e^{-2 x} x^2-\frac {6}{25} e^{-2 x} x+\frac {27 e^{-2 x}}{25}+\frac {e^{2 x}}{25}-\frac {108 e^{-2 x}}{25 (x+3)}-\frac {6 e^{2 x}}{25 (x+3)}+\frac {81 e^{-2 x}}{25 (x+3)^2}+\frac {9 e^{2 x}}{25 (x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4*x)*(-18*x^2 - 2*x^3) + E^(6*x)*(6*x + 6*x^2 + 2*x^3) + E^(2*x)*(12*x^3 - 4*x^4 - 2*x^5))/(E^(4*x)*(6
75 + 675*x + 225*x^2 + 25*x^3)),x]

[Out]

27/(25*E^(2*x)) + E^(2*x)/25 - (6*x)/(25*E^(2*x)) + x^2/(25*E^(2*x)) + 81/(25*E^(2*x)*(3 + x)^2) + (9*E^(2*x))
/(25*(3 + x)^2) - (2*x^3)/(25*(3 + x)^2) - 108/(25*E^(2*x)*(3 + x)) - (6*E^(2*x))/(25*(3 + 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 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

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^{-2 x} x \left (-e^{2 x} x (9+x)-x^2 \left (-6+2 x+x^2\right )+e^{4 x} \left (3+3 x+x^2\right )\right )}{25 (3+x)^3} \, dx\\ &=\frac {2}{25} \int \frac {e^{-2 x} x \left (-e^{2 x} x (9+x)-x^2 \left (-6+2 x+x^2\right )+e^{4 x} \left (3+3 x+x^2\right )\right )}{(3+x)^3} \, dx\\ &=\frac {2}{25} \int \left (-\frac {x^2 (9+x)}{(3+x)^3}-\frac {e^{-2 x} x^3 \left (-6+2 x+x^2\right )}{(3+x)^3}+\frac {e^{2 x} x \left (3+3 x+x^2\right )}{(3+x)^3}\right ) \, dx\\ &=-\left (\frac {2}{25} \int \frac {x^2 (9+x)}{(3+x)^3} \, dx\right )-\frac {2}{25} \int \frac {e^{-2 x} x^3 \left (-6+2 x+x^2\right )}{(3+x)^3} \, dx+\frac {2}{25} \int \frac {e^{2 x} x \left (3+3 x+x^2\right )}{(3+x)^3} \, dx\\ &=-\frac {2 x^3}{25 (3+x)^2}-\frac {2}{25} \int \left (30 e^{-2 x}-7 e^{-2 x} x+e^{-2 x} x^2+\frac {81 e^{-2 x}}{(3+x)^3}+\frac {27 e^{-2 x}}{(3+x)^2}-\frac {108 e^{-2 x}}{3+x}\right ) \, dx+\frac {2}{25} \int \left (e^{2 x}-\frac {9 e^{2 x}}{(3+x)^3}+\frac {12 e^{2 x}}{(3+x)^2}-\frac {6 e^{2 x}}{3+x}\right ) \, dx\\ &=-\frac {2 x^3}{25 (3+x)^2}+\frac {2}{25} \int e^{2 x} \, dx-\frac {2}{25} \int e^{-2 x} x^2 \, dx-\frac {12}{25} \int \frac {e^{2 x}}{3+x} \, dx+\frac {14}{25} \int e^{-2 x} x \, dx-\frac {18}{25} \int \frac {e^{2 x}}{(3+x)^3} \, dx+\frac {24}{25} \int \frac {e^{2 x}}{(3+x)^2} \, dx-\frac {54}{25} \int \frac {e^{-2 x}}{(3+x)^2} \, dx-\frac {12}{5} \int e^{-2 x} \, dx-\frac {162}{25} \int \frac {e^{-2 x}}{(3+x)^3} \, dx+\frac {216}{25} \int \frac {e^{-2 x}}{3+x} \, dx\\ &=\frac {6 e^{-2 x}}{5}+\frac {e^{2 x}}{25}-\frac {7}{25} e^{-2 x} x+\frac {1}{25} e^{-2 x} x^2+\frac {81 e^{-2 x}}{25 (3+x)^2}+\frac {9 e^{2 x}}{25 (3+x)^2}-\frac {2 x^3}{25 (3+x)^2}+\frac {54 e^{-2 x}}{25 (3+x)}-\frac {24 e^{2 x}}{25 (3+x)}+\frac {216}{25} e^6 \text {Ei}(-2 (3+x))-\frac {12 \text {Ei}(2 (3+x))}{25 e^6}-\frac {2}{25} \int e^{-2 x} x \, dx+\frac {7}{25} \int e^{-2 x} \, dx-\frac {18}{25} \int \frac {e^{2 x}}{(3+x)^2} \, dx+\frac {48}{25} \int \frac {e^{2 x}}{3+x} \, dx+\frac {108}{25} \int \frac {e^{-2 x}}{3+x} \, dx+\frac {162}{25} \int \frac {e^{-2 x}}{(3+x)^2} \, dx\\ &=\frac {53 e^{-2 x}}{50}+\frac {e^{2 x}}{25}-\frac {6}{25} e^{-2 x} x+\frac {1}{25} e^{-2 x} x^2+\frac {81 e^{-2 x}}{25 (3+x)^2}+\frac {9 e^{2 x}}{25 (3+x)^2}-\frac {2 x^3}{25 (3+x)^2}-\frac {108 e^{-2 x}}{25 (3+x)}-\frac {6 e^{2 x}}{25 (3+x)}+\frac {324}{25} e^6 \text {Ei}(-2 (3+x))+\frac {36 \text {Ei}(2 (3+x))}{25 e^6}-\frac {1}{25} \int e^{-2 x} \, dx-\frac {36}{25} \int \frac {e^{2 x}}{3+x} \, dx-\frac {324}{25} \int \frac {e^{-2 x}}{3+x} \, dx\\ &=\frac {27 e^{-2 x}}{25}+\frac {e^{2 x}}{25}-\frac {6}{25} e^{-2 x} x+\frac {1}{25} e^{-2 x} x^2+\frac {81 e^{-2 x}}{25 (3+x)^2}+\frac {9 e^{2 x}}{25 (3+x)^2}-\frac {2 x^3}{25 (3+x)^2}-\frac {108 e^{-2 x}}{25 (3+x)}-\frac {6 e^{2 x}}{25 (3+x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 50, normalized size = 1.79 \begin {gather*} -\frac {e^{-2 x} \left (-e^{4 x} x^2-x^4+2 e^{2 x} \left (54+36 x+6 x^2+x^3\right )\right )}{25 (3+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*x)*(-18*x^2 - 2*x^3) + E^(6*x)*(6*x + 6*x^2 + 2*x^3) + E^(2*x)*(12*x^3 - 4*x^4 - 2*x^5))/(E^(4
*x)*(675 + 675*x + 225*x^2 + 25*x^3)),x]

[Out]

-1/25*(-(E^(4*x)*x^2) - x^4 + 2*E^(2*x)*(54 + 36*x + 6*x^2 + x^3))/(E^(2*x)*(3 + x)^2)

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fricas [A]  time = 0.79, size = 47, normalized size = 1.68 \begin {gather*} \frac {{\left (x^{4} + x^{2} e^{\left (4 \, x\right )} - 2 \, {\left (x^{3} + 6 \, x^{2} + 36 \, x + 54\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{25 \, {\left (x^{2} + 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+6*x^2+6*x)*exp(x)^2*exp(2*x)^2+(-2*x^3-18*x^2)*exp(x)^2*exp(2*x)+(-2*x^5-4*x^4+12*x^3)*exp(x
)^2)/(25*x^3+225*x^2+675*x+675)/exp(2*x)^2,x, algorithm="fricas")

[Out]

1/25*(x^4 + x^2*e^(4*x) - 2*(x^3 + 6*x^2 + 36*x + 54)*e^(2*x))*e^(-2*x)/(x^2 + 6*x + 9)

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giac [A]  time = 0.16, size = 43, normalized size = 1.54 \begin {gather*} \frac {x^{4} e^{\left (-2 \, x\right )} - 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 12 \, x^{2} - 72 \, x - 108}{25 \, {\left (x^{2} + 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+6*x^2+6*x)*exp(x)^2*exp(2*x)^2+(-2*x^3-18*x^2)*exp(x)^2*exp(2*x)+(-2*x^5-4*x^4+12*x^3)*exp(x
)^2)/(25*x^3+225*x^2+675*x+675)/exp(2*x)^2,x, algorithm="giac")

[Out]

1/25*(x^4*e^(-2*x) - 2*x^3 + x^2*e^(2*x) - 12*x^2 - 72*x - 108)/(x^2 + 6*x + 9)

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maple [A]  time = 0.08, size = 49, normalized size = 1.75

method	result	size
risch	\(-\frac {2 x}{25}+\frac {-\frac {54 x}{25}-\frac {108}{25}}{x^{2}+6 x +9}+\frac {x^{2} {\mathrm e}^{2 x}}{25 \left (3+x \right )^{2}}+\frac {x^{4} {\mathrm e}^{-2 x}}{25 \left (3+x \right )^{2}}\)	\(49\)
default	\(-\frac {54}{25 \left (3+x \right )}+\frac {54}{25 \left (3+x \right )^{2}}-\frac {2 x}{25}-\frac {6 \,{\mathrm e}^{-2 x}}{25}-\frac {162 \,{\mathrm e}^{-2 x} \left (4 x +11\right )}{25 \left (x^{2}+6 x +9\right )}+\frac {\left (2 x -17\right ) {\mathrm e}^{-2 x}}{25}-\frac {54 \,{\mathrm e}^{-2 x} \left (14 x +39\right )}{25 \left (x^{2}+6 x +9\right )}+\frac {\left (x^{2}-8 x +50\right ) {\mathrm e}^{-2 x}}{25}+\frac {81 \,{\mathrm e}^{-2 x} \left (16 x +45\right )}{25 \left (x^{2}+6 x +9\right )}-\frac {6 \,{\mathrm e}^{2 x}}{25 \left (3+x \right )}+\frac {9 \,{\mathrm e}^{2 x}}{25 \left (3+x \right )^{2}}+\frac {{\mathrm e}^{2 x}}{25}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3+6*x^2+6*x)*exp(x)^2*exp(2*x)^2+(-2*x^3-18*x^2)*exp(x)^2*exp(2*x)+(-2*x^5-4*x^4+12*x^3)*exp(x)^2)/(
25*x^3+225*x^2+675*x+675)/exp(2*x)^2,x,method=_RETURNVERBOSE)

[Out]

-2/25*x+(-54/25*x-108/25)/(x^2+6*x+9)+1/25*x^2/(3+x)^2*exp(2*x)+1/25*x^4/(3+x)^2*exp(-2*x)

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maxima [A]  time = 0.42, size = 43, normalized size = 1.54 \begin {gather*} \frac {x^{4} e^{\left (-2 \, x\right )} - 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 12 \, x^{2} - 72 \, x - 108}{25 \, {\left (x^{2} + 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+6*x^2+6*x)*exp(x)^2*exp(2*x)^2+(-2*x^3-18*x^2)*exp(x)^2*exp(2*x)+(-2*x^5-4*x^4+12*x^3)*exp(x
)^2)/(25*x^3+225*x^2+675*x+675)/exp(2*x)^2,x, algorithm="maxima")

[Out]

1/25*(x^4*e^(-2*x) - 2*x^3 + x^2*e^(2*x) - 12*x^2 - 72*x - 108)/(x^2 + 6*x + 9)

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mupad [B]  time = 0.17, size = 24, normalized size = 0.86 \begin {gather*} \frac {x^2\,{\mathrm {e}}^{-2\,x}\,{\left (x-{\mathrm {e}}^{2\,x}\right )}^2}{25\,{\left (x+3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4*x)*(exp(4*x)*(18*x^2 + 2*x^3) - exp(6*x)*(6*x + 6*x^2 + 2*x^3) + exp(2*x)*(4*x^4 - 12*x^3 + 2*x^5
)))/(675*x + 225*x^2 + 25*x^3 + 675),x)

[Out]

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

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sympy [B]  time = 0.20, size = 80, normalized size = 2.86 \begin {gather*} - \frac {2 x}{25} - \frac {54 x + 108}{25 x^{2} + 150 x + 225} + \frac {\left (25 x^{4} + 150 x^{3} + 225 x^{2}\right ) e^{2 x} + \left (25 x^{6} + 150 x^{5} + 225 x^{4}\right ) e^{- 2 x}}{625 x^{4} + 7500 x^{3} + 33750 x^{2} + 67500 x + 50625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+6*x**2+6*x)*exp(x)**2*exp(2*x)**2+(-2*x**3-18*x**2)*exp(x)**2*exp(2*x)+(-2*x**5-4*x**4+12*x
**3)*exp(x)**2)/(25*x**3+225*x**2+675*x+675)/exp(2*x)**2,x)

[Out]

-2*x/25 - (54*x + 108)/(25*x**2 + 150*x + 225) + ((25*x**4 + 150*x**3 + 225*x**2)*exp(2*x) + (25*x**6 + 150*x*
*5 + 225*x**4)*exp(-2*x))/(625*x**4 + 7500*x**3 + 33750*x**2 + 67500*x + 50625)

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