∫ e3(1−34x)−3x2+6x3−105x4+102x5−867x6 ________________________________________________________________

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Rubi [A] time = 0.13, antiderivative size = 44, normalized size of antiderivative = 2.00, number of steps used = 5, number of rules used = 4, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1680, 12, 1814, 1586} \begin {gather*} -\left (x-\frac {1}{34}\right )^3-\frac {3}{34} \left (x-\frac {1}{34}\right )^2-\frac {3 x}{1156}+\frac {68 e^3}{1156 \left (x-\frac {1}{34}\right )^2+67} \end {gather*}

Int[(E^3*(1 - 34*x) - 3*x^2 + 6*x^3 - 105*x^4 + 102*x^5 - 867*x^6)/(1 - 2*x + 35*x^2 - 34*x^3 + 289*x^4),x]

(68*E^3)/(67 + 1156*(-1/34 + x)^2) - (3*(-1/34 + x)^2)/34 - (-1/34 + x)^3 - (3*x)/1156

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[

q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co

eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3

) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,

0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] && !IGtQ[p, 0]

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P

olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a

*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS

um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {-13467-915756 \left (1+\frac {2672672 e^3}{13467}\right ) x-16032564 x^2-31600416 x^3-541216080 x^4-272612544 x^5-4634413248 x^6}{1156 \left (67+1156 x^2\right )^2} \, dx,x,-\frac {1}{34}+x\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {-13467-915756 \left (1+\frac {2672672 e^3}{13467}\right ) x-16032564 x^2-31600416 x^3-541216080 x^4-272612544 x^5-4634413248 x^6}{\left (67+1156 x^2\right )^2} \, dx,x,-\frac {1}{34}+x\right )}{1156}\\ &=\frac {68 e^3}{67+(-1+34 x)^2}-\frac {\operatorname {Subst}\left (\int \frac {26934+1831512 x+31600416 x^2+31600416 x^3+537207072 x^4}{67+1156 x^2} \, dx,x,-\frac {1}{34}+x\right )}{154904}\\ &=\frac {68 e^3}{67+(-1+34 x)^2}-\frac {\operatorname {Subst}\left (\int \left (402+27336 x+464712 x^2\right ) \, dx,x,-\frac {1}{34}+x\right )}{154904}\\ &=-\frac {3 (1-34 x)^2}{39304}-\left (-\frac {1}{34}+x\right )^3-\frac {3 x}{1156}+\frac {68 e^3}{67+(-1+34 x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.02, size = 21, normalized size = 0.95 \begin {gather*} -x^3-\frac {e^3}{-1+x-17 x^2} \end {gather*}

Integrate[(E^3*(1 - 34*x) - 3*x^2 + 6*x^3 - 105*x^4 + 102*x^5 - 867*x^6)/(1 - 2*x + 35*x^2 - 34*x^3 + 289*x^4)

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fricas [A] time = 0.66, size = 32, normalized size = 1.45 \begin {gather*} -\frac {17 \, x^{5} - x^{4} + x^{3} - e^{3}}{17 \, x^{2} - x + 1} \end {gather*}

integrate(((-34*x+1)*exp(3)-867*x^6+102*x^5-105*x^4+6*x^3-3*x^2)/(289*x^4-34*x^3+35*x^2-2*x+1),x, algorithm="f

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giac [A] time = 0.13, size = 21, normalized size = 0.95 \begin {gather*} -x^{3} + \frac {e^{3}}{17 \, x^{2} - x + 1} \end {gather*}

integrate(((-34*x+1)*exp(3)-867*x^6+102*x^5-105*x^4+6*x^3-3*x^2)/(289*x^4-34*x^3+35*x^2-2*x+1),x, algorithm="g

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method	result	size

risch	\(\frac {{\mathrm e}^{3}}{17 x^{2}-x +1}-x^{3}\)	\(21\)
default	\(\frac {{\mathrm e}^{3}}{17 x^{2}-x +1}-x^{3}\)	\(22\)
gosper	\(\frac {-17 x^{5}+x^{4}-x^{3}+{\mathrm e}^{3}}{17 x^{2}-x +1}\)	\(30\)
norman	\(\frac {-17 x^{5}+x^{4}-x^{3}+{\mathrm e}^{3}}{17 x^{2}-x +1}\)	\(30\)

int(((-34*x+1)*exp(3)-867*x^6+102*x^5-105*x^4+6*x^3-3*x^2)/(289*x^4-34*x^3+35*x^2-2*x+1),x,method=_RETURNVERBO

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maxima [A] time = 0.35, size = 21, normalized size = 0.95 \begin {gather*} -x^{3} + \frac {e^{3}}{17 \, x^{2} - x + 1} \end {gather*}

integrate(((-34*x+1)*exp(3)-867*x^6+102*x^5-105*x^4+6*x^3-3*x^2)/(289*x^4-34*x^3+35*x^2-2*x+1),x, algorithm="m

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mupad [B] time = 0.07, size = 21, normalized size = 0.95 \begin {gather*} \frac {{\mathrm {e}}^3}{17\,x^2-x+1}-x^3 \end {gather*}

int(-(3*x^2 - 6*x^3 + 105*x^4 - 102*x^5 + 867*x^6 + exp(3)*(34*x - 1))/(35*x^2 - 2*x - 34*x^3 + 289*x^4 + 1),x

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sympy [A] time = 0.16, size = 14, normalized size = 0.64 \begin {gather*} - x^{3} + \frac {e^{3}}{17 x^{2} - x + 1} \end {gather*}

integrate(((-34*x+1)*exp(3)-867*x**6+102*x**5-105*x**4+6*x**3-3*x**2)/(289*x**4-34*x**3+35*x**2-2*x+1),x)

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3.86.71 \(\int \frac {e^3 (1-34 x)-3 x^2+6 x^3-105 x^4+102 x^5-867 x^6}{1-2 x+35 x^2-34 x^3+289 x^4} \, dx\)