∫ 3+16x−16x2+196x3−192x4+480x5−432x6+108x7+e2x(288x3−144x4−72x5+36x6)+ex(144x2−96x3+708x4−564x5+36x6+36x7)____________________________________________________________________________________________

3.37.14 $\int \frac {3+16 x-16 x^2+196 x^3-192 x^4+480 x^5-432 x^6+108 x^7+e^{2 x} (288 x^3-144 x^4-72 x^5+36 x^6)+e^x (144 x^2-96 x^3+708 x^4-564 x^5+36 x^6+36 x^7)}{8-8 x+2 x^2} \, dx$

Optimal. Leaf size=29 \[ -\frac {5-x}{-4+2 x}+\left (x+3 x^2 \left (e^x+x\right )\right )^2 \]

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Rubi [A] time = 0.79, antiderivative size = 51, normalized size of antiderivative = 1.76, number of steps used = 49, number of rules used = 7, integrand size = 110, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.064, Rules used = {27, 12, 6742, 43, 2196, 2176, 2194} \begin {gather*} 9 x^6+18 e^x x^5+9 e^{2 x} x^4+6 x^4+6 e^x x^3+x^2+\frac {3}{2 (2-x)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(3 + 16*x - 16*x^2 + 196*x^3 - 192*x^4 + 480*x^5 - 432*x^6 + 108*x^7 + E^(2*x)*(288*x^3 - 144*x^4 - 72*x^5

+ 36*x^6) + E^x*(144*x^2 - 96*x^3 + 708*x^4 - 564*x^5 + 36*x^6 + 36*x^7))/(8 - 8*x + 2*x^2),x]

[Out]

3/(2*(2 - x)) + x^2 + 6*E^x*x^3 + 6*x^4 + 9*E^(2*x)*x^4 + 18*E^x*x^5 + 9*x^6

Rule 12

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr

eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d

*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]

&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c

}, x] && PolynomialQ[u, x] && LinearQ[v, x] && !$UseGamma === True

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 {3+16 x-16 x^2+196 x^3-192 x^4+480 x^5-432 x^6+108 x^7+e^{2 x} \left (288 x^3-144 x^4-72 x^5+36 x^6\right )+e^x \left (144 x^2-96 x^3+708 x^4-564 x^5+36 x^6+36 x^7\right )}{2 (-2+x)^2} \, dx\\ &=\frac {1}{2} \int \frac {3+16 x-16 x^2+196 x^3-192 x^4+480 x^5-432 x^6+108 x^7+e^{2 x} \left (288 x^3-144 x^4-72 x^5+36 x^6\right )+e^x \left (144 x^2-96 x^3+708 x^4-564 x^5+36 x^6+36 x^7\right )}{(-2+x)^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {3}{(-2+x)^2}+\frac {16 x}{(-2+x)^2}-\frac {16 x^2}{(-2+x)^2}+\frac {196 x^3}{(-2+x)^2}-\frac {192 x^4}{(-2+x)^2}+\frac {480 x^5}{(-2+x)^2}-\frac {432 x^6}{(-2+x)^2}+\frac {108 x^7}{(-2+x)^2}+36 e^{2 x} x^3 (2+x)+12 e^x x^2 \left (3+x+15 x^2+3 x^3\right )\right ) \, dx\\ &=\frac {3}{2 (2-x)}+6 \int e^x x^2 \left (3+x+15 x^2+3 x^3\right ) \, dx+8 \int \frac {x}{(-2+x)^2} \, dx-8 \int \frac {x^2}{(-2+x)^2} \, dx+18 \int e^{2 x} x^3 (2+x) \, dx+54 \int \frac {x^7}{(-2+x)^2} \, dx-96 \int \frac {x^4}{(-2+x)^2} \, dx+98 \int \frac {x^3}{(-2+x)^2} \, dx-216 \int \frac {x^6}{(-2+x)^2} \, dx+240 \int \frac {x^5}{(-2+x)^2} \, dx\\ &=\frac {3}{2 (2-x)}+6 \int \left (3 e^x x^2+e^x x^3+15 e^x x^4+3 e^x x^5\right ) \, dx+8 \int \left (\frac {2}{(-2+x)^2}+\frac {1}{-2+x}\right ) \, dx-8 \int \left (1+\frac {4}{(-2+x)^2}+\frac {4}{-2+x}\right ) \, dx+18 \int \left (2 e^{2 x} x^3+e^{2 x} x^4\right ) \, dx+54 \int \left (192+\frac {128}{(-2+x)^2}+\frac {448}{-2+x}+80 x+32 x^2+12 x^3+4 x^4+x^5\right ) \, dx-96 \int \left (12+\frac {16}{(-2+x)^2}+\frac {32}{-2+x}+4 x+x^2\right ) \, dx+98 \int \left (4+\frac {8}{(-2+x)^2}+\frac {12}{-2+x}+x\right ) \, dx-216 \int \left (80+\frac {64}{(-2+x)^2}+\frac {192}{-2+x}+32 x+12 x^2+4 x^3+x^4\right ) \, dx+240 \int \left (32+\frac {32}{(-2+x)^2}+\frac {80}{-2+x}+12 x+4 x^2+x^3\right ) \, dx\\ &=\frac {3}{2 (2-x)}+x^2+6 x^4+9 x^6+6 \int e^x x^3 \, dx+18 \int e^x x^2 \, dx+18 \int e^{2 x} x^4 \, dx+18 \int e^x x^5 \, dx+36 \int e^{2 x} x^3 \, dx+90 \int e^x x^4 \, dx\\ &=\frac {3}{2 (2-x)}+x^2+18 e^x x^2+6 e^x x^3+18 e^{2 x} x^3+6 x^4+90 e^x x^4+9 e^{2 x} x^4+18 e^x x^5+9 x^6-18 \int e^x x^2 \, dx-36 \int e^x x \, dx-36 \int e^{2 x} x^3 \, dx-54 \int e^{2 x} x^2 \, dx-90 \int e^x x^4 \, dx-360 \int e^x x^3 \, dx\\ &=\frac {3}{2 (2-x)}-36 e^x x+x^2-27 e^{2 x} x^2-354 e^x x^3+6 x^4+9 e^{2 x} x^4+18 e^x x^5+9 x^6+36 \int e^x \, dx+36 \int e^x x \, dx+54 \int e^{2 x} x \, dx+54 \int e^{2 x} x^2 \, dx+360 \int e^x x^3 \, dx+1080 \int e^x x^2 \, dx\\ &=36 e^x+\frac {3}{2 (2-x)}+27 e^{2 x} x+x^2+1080 e^x x^2+6 e^x x^3+6 x^4+9 e^{2 x} x^4+18 e^x x^5+9 x^6-27 \int e^{2 x} \, dx-36 \int e^x \, dx-54 \int e^{2 x} x \, dx-1080 \int e^x x^2 \, dx-2160 \int e^x x \, dx\\ &=-\frac {27 e^{2 x}}{2}+\frac {3}{2 (2-x)}-2160 e^x x+x^2+6 e^x x^3+6 x^4+9 e^{2 x} x^4+18 e^x x^5+9 x^6+27 \int e^{2 x} \, dx+2160 \int e^x \, dx+2160 \int e^x x \, dx\\ &=2160 e^x+\frac {3}{2 (2-x)}+x^2+6 e^x x^3+6 x^4+9 e^{2 x} x^4+18 e^x x^5+9 x^6-2160 \int e^x \, dx\\ &=\frac {3}{2 (2-x)}+x^2+6 e^x x^3+6 x^4+9 e^{2 x} x^4+18 e^x x^5+9 x^6\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.07, size = 52, normalized size = 1.79 \begin {gather*} \frac {1}{2} \left (-\frac {3}{-2+x}+2 x^2+12 x^4+18 e^{2 x} x^4+18 x^6+e^x \left (12 x^3+36 x^5\right )\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 + 16*x - 16*x^2 + 196*x^3 - 192*x^4 + 480*x^5 - 432*x^6 + 108*x^7 + E^(2*x)*(288*x^3 - 144*x^4 -

72*x^5 + 36*x^6) + E^x*(144*x^2 - 96*x^3 + 708*x^4 - 564*x^5 + 36*x^6 + 36*x^7))/(8 - 8*x + 2*x^2),x]

[Out]

(-3/(-2 + x) + 2*x^2 + 12*x^4 + 18*E^(2*x)*x^4 + 18*x^6 + E^x*(12*x^3 + 36*x^5))/2

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fricas [B] time = 0.64, size = 77, normalized size = 2.66 \begin {gather*} \frac {18 \, x^{7} - 36 \, x^{6} + 12 \, x^{5} - 24 \, x^{4} + 2 \, x^{3} - 4 \, x^{2} + 18 \, {\left (x^{5} - 2 \, x^{4}\right )} e^{\left (2 \, x\right )} + 12 \, {\left (3 \, x^{6} - 6 \, x^{5} + x^{4} - 2 \, x^{3}\right )} e^{x} - 3}{2 \, {\left (x - 2\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x^6-72*x^5-144*x^4+288*x^3)*exp(x)^2+(36*x^7+36*x^6-564*x^5+708*x^4-96*x^3+144*x^2)*exp(x)+108*

x^7-432*x^6+480*x^5-192*x^4+196*x^3-16*x^2+16*x+3)/(2*x^2-8*x+8),x, algorithm="fricas")

[Out]

1/2*(18*x^7 - 36*x^6 + 12*x^5 - 24*x^4 + 2*x^3 - 4*x^2 + 18*(x^5 - 2*x^4)*e^(2*x) + 12*(3*x^6 - 6*x^5 + x^4 -

2*x^3)*e^x - 3)/(x - 2)

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giac [B] time = 0.14, size = 85, normalized size = 2.93 \begin {gather*} \frac {18 \, x^{7} + 36 \, x^{6} e^{x} - 36 \, x^{6} + 18 \, x^{5} e^{\left (2 \, x\right )} - 72 \, x^{5} e^{x} + 12 \, x^{5} - 36 \, x^{4} e^{\left (2 \, x\right )} + 12 \, x^{4} e^{x} - 24 \, x^{4} - 24 \, x^{3} e^{x} + 2 \, x^{3} - 4 \, x^{2} - 3}{2 \, {\left (x - 2\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x^6-72*x^5-144*x^4+288*x^3)*exp(x)^2+(36*x^7+36*x^6-564*x^5+708*x^4-96*x^3+144*x^2)*exp(x)+108*

x^7-432*x^6+480*x^5-192*x^4+196*x^3-16*x^2+16*x+3)/(2*x^2-8*x+8),x, algorithm="giac")

[Out]

1/2*(18*x^7 + 36*x^6*e^x - 36*x^6 + 18*x^5*e^(2*x) - 72*x^5*e^x + 12*x^5 - 36*x^4*e^(2*x) + 12*x^4*e^x - 24*x^

4 - 24*x^3*e^x + 2*x^3 - 4*x^2 - 3)/(x - 2)

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maple [A] time = 0.16, size = 45, normalized size = 1.55


method	result	size

default	$-\frac {3}{2 \left (x -2\right )}+x^{2}+6 x^{4}+9 x^{6}+6 \,{\mathrm e}^{x} x^{3}+9 \,{\mathrm e}^{2 x} x^{4}+18 x^{5} {\mathrm e}^{x}$	$45$
risch	$9 x^{6}+6 x^{4}+x^{2}-\frac {3}{2 \left (x -2\right )}+9 \,{\mathrm e}^{2 x} x^{4}+\left (18 x^{5}+6 x^{3}\right ) {\mathrm e}^{x}$	$45$
norman	$\frac {x^{3}-2 x^{2}-12 x^{4}+6 x^{5}-18 x^{6}+9 x^{7}-36 x^{5} {\mathrm e}^{x}+9 x^{5} {\mathrm e}^{2 x}+18 x^{6} {\mathrm e}^{x}-12 \,{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{x} x^{4}-18 \,{\mathrm e}^{2 x} x^{4}-\frac {3}{2}}{x -2}$	$83$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((36*x^6-72*x^5-144*x^4+288*x^3)*exp(x)^2+(36*x^7+36*x^6-564*x^5+708*x^4-96*x^3+144*x^2)*exp(x)+108*x^7-43

2*x^6+480*x^5-192*x^4+196*x^3-16*x^2+16*x+3)/(2*x^2-8*x+8),x,method=_RETURNVERBOSE)

[Out]

-3/2/(x-2)+x^2+6*x^4+9*x^6+6*exp(x)*x^3+9*exp(x)^2*x^4+18*x^5*exp(x)

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maxima [A] time = 0.60, size = 43, normalized size = 1.48 \begin {gather*} 9 \, x^{6} + 9 \, x^{4} e^{\left (2 \, x\right )} + 6 \, x^{4} + x^{2} + 6 \, {\left (3 \, x^{5} + x^{3}\right )} e^{x} - \frac {3}{2 \, {\left (x - 2\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x^6-72*x^5-144*x^4+288*x^3)*exp(x)^2+(36*x^7+36*x^6-564*x^5+708*x^4-96*x^3+144*x^2)*exp(x)+108*

x^7-432*x^6+480*x^5-192*x^4+196*x^3-16*x^2+16*x+3)/(2*x^2-8*x+8),x, algorithm="maxima")

[Out]

9*x^6 + 9*x^4*e^(2*x) + 6*x^4 + x^2 + 6*(3*x^5 + x^3)*e^x - 3/2/(x - 2)

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mupad [B] time = 2.13, size = 44, normalized size = 1.52 \begin {gather*} 6\,x^3\,{\mathrm {e}}^x+18\,x^5\,{\mathrm {e}}^x+x^4\,\left (9\,{\mathrm {e}}^{2\,x}+6\right )-\frac {3}{2\,x-4}+x^2+9\,x^6 \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((16*x + exp(2*x)*(288*x^3 - 144*x^4 - 72*x^5 + 36*x^6) - 16*x^2 + 196*x^3 - 192*x^4 + 480*x^5 - 432*x^6 +

108*x^7 + exp(x)*(144*x^2 - 96*x^3 + 708*x^4 - 564*x^5 + 36*x^6 + 36*x^7) + 3)/(2*x^2 - 8*x + 8),x)

[Out]

6*x^3*exp(x) + 18*x^5*exp(x) + x^4*(9*exp(2*x) + 6) - 3/(2*x - 4) + x^2 + 9*x^6

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sympy [B] time = 0.21, size = 42, normalized size = 1.45 \begin {gather*} 9 x^{6} + 9 x^{4} e^{2 x} + 6 x^{4} + x^{2} + \left (18 x^{5} + 6 x^{3}\right ) e^{x} - \frac {3}{2 x - 4} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x**6-72*x**5-144*x**4+288*x**3)*exp(x)**2+(36*x**7+36*x**6-564*x**5+708*x**4-96*x**3+144*x**2)*

exp(x)+108*x**7-432*x**6+480*x**5-192*x**4+196*x**3-16*x**2+16*x+3)/(2*x**2-8*x+8),x)

[Out]

9*x**6 + 9*x**4*exp(2*x) + 6*x**4 + x**2 + (18*x**5 + 6*x**3)*exp(x) - 3/(2*x - 4)

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3.37.14 \(\int \frac {3+16 x-16 x^2+196 x^3-192 x^4+480 x^5-432 x^6+108 x^7+e^{2 x} (288 x^3-144 x^4-72 x^5+36 x^6)+e^x (144 x^2-96 x^3+708 x^4-564 x^5+36 x^6+36 x^7)}{8-8 x+2 x^2} \, dx\)