3.83.91 \(\int \frac {e^{20} (-52 x-4 x^2)}{-1113879+428415 x-65910 x^2+5070 x^3-195 x^4+3 x^5} \, dx\)

Optimal. Leaf size=19 \[ -2+\frac {2 e^{20} x^2}{3 (13-x)^4} \]

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Rubi [A]  time = 0.10, antiderivative size = 17, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 1593, 6688, 74} \begin {gather*} \frac {2 e^{20} x^2}{3 (13-x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^20*(-52*x - 4*x^2))/(-1113879 + 428415*x - 65910*x^2 + 5070*x^3 - 195*x^4 + 3*x^5),x]

[Out]

(2*E^20*x^2)/(3*(13 - x)^4)

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 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{20} \int \frac {-52 x-4 x^2}{-1113879+428415 x-65910 x^2+5070 x^3-195 x^4+3 x^5} \, dx\\ &=e^{20} \int \frac {(-52-4 x) x}{-1113879+428415 x-65910 x^2+5070 x^3-195 x^4+3 x^5} \, dx\\ &=e^{20} \int \frac {4 x (13+x)}{3 (13-x)^5} \, dx\\ &=\frac {1}{3} \left (4 e^{20}\right ) \int \frac {x (13+x)}{(13-x)^5} \, dx\\ &=\frac {2 e^{20} x^2}{3 (13-x)^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.79 \begin {gather*} \frac {2 e^{20} x^2}{3 (-13+x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^20*(-52*x - 4*x^2))/(-1113879 + 428415*x - 65910*x^2 + 5070*x^3 - 195*x^4 + 3*x^5),x]

[Out]

(2*E^20*x^2)/(3*(-13 + x)^4)

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fricas [A]  time = 0.68, size = 27, normalized size = 1.42 \begin {gather*} \frac {2 \, x^{2} e^{20}}{3 \, {\left (x^{4} - 52 \, x^{3} + 1014 \, x^{2} - 8788 \, x + 28561\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-52*x)*exp(5)^4/(3*x^5-195*x^4+5070*x^3-65910*x^2+428415*x-1113879),x, algorithm="fricas")

[Out]

2/3*x^2*e^20/(x^4 - 52*x^3 + 1014*x^2 - 8788*x + 28561)

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giac [A]  time = 0.12, size = 12, normalized size = 0.63 \begin {gather*} \frac {2 \, x^{2} e^{20}}{3 \, {\left (x - 13\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-52*x)*exp(5)^4/(3*x^5-195*x^4+5070*x^3-65910*x^2+428415*x-1113879),x, algorithm="giac")

[Out]

2/3*x^2*e^20/(x - 13)^4

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maple [A]  time = 0.04, size = 15, normalized size = 0.79

method	result	size
norman	\(\frac {2 x^{2} {\mathrm e}^{20}}{3 \left (x -13\right )^{4}}\)	\(15\)
risch	\(\frac {2 x^{2} {\mathrm e}^{20}}{3 \left (x^{4}-52 x^{3}+1014 x^{2}-8788 x +28561\right )}\)	\(28\)
default	\(\frac {4 \,{\mathrm e}^{20} \left (\frac {13}{\left (x -13\right )^{3}}+\frac {169}{2 \left (x -13\right )^{4}}+\frac {1}{2 \left (x -13\right )^{2}}\right )}{3}\)	\(29\)
gosper	\(\frac {2 x^{2} {\mathrm e}^{20}}{3 \left (x^{4}-52 x^{3}+1014 x^{2}-8788 x +28561\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^2-52*x)*exp(5)^4/(3*x^5-195*x^4+5070*x^3-65910*x^2+428415*x-1113879),x,method=_RETURNVERBOSE)

[Out]

2/3*x^2*exp(5)^4/(x-13)^4

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maxima [A]  time = 0.37, size = 27, normalized size = 1.42 \begin {gather*} \frac {2 \, x^{2} e^{20}}{3 \, {\left (x^{4} - 52 \, x^{3} + 1014 \, x^{2} - 8788 \, x + 28561\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^2-52*x)*exp(5)^4/(3*x^5-195*x^4+5070*x^3-65910*x^2+428415*x-1113879),x, algorithm="maxima")

[Out]

2/3*x^2*e^20/(x^4 - 52*x^3 + 1014*x^2 - 8788*x + 28561)

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mupad [B]  time = 5.23, size = 28, normalized size = 1.47 \begin {gather*} \frac {2\,{\mathrm {e}}^{20}}{3\,{\left (x-13\right )}^2}+\frac {52\,{\mathrm {e}}^{20}}{3\,{\left (x-13\right )}^3}+\frac {338\,{\mathrm {e}}^{20}}{3\,{\left (x-13\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(20)*(52*x + 4*x^2))/(428415*x - 65910*x^2 + 5070*x^3 - 195*x^4 + 3*x^5 - 1113879),x)

[Out]

(2*exp(20))/(3*(x - 13)^2) + (52*exp(20))/(3*(x - 13)^3) + (338*exp(20))/(3*(x - 13)^4)

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sympy [A]  time = 0.21, size = 27, normalized size = 1.42 \begin {gather*} \frac {2 x^{2} e^{20}}{3 x^{4} - 156 x^{3} + 3042 x^{2} - 26364 x + 85683} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**2-52*x)*exp(5)**4/(3*x**5-195*x**4+5070*x**3-65910*x**2+428415*x-1113879),x)

[Out]

2*x**2*exp(20)/(3*x**4 - 156*x**3 + 3042*x**2 - 26364*x + 85683)

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