3.13.82 \(\int \frac {396 x^2+285 x^3+72 x^4+262 x^5+192 x^6+48 x^7+4 x^8+e^{2 x} (-192+240 x+252 x^2+197 x^3+230 x^4+120 x^5+26 x^6+2 x^7)+e^x (384 x^2+288 x^3+456 x^4+422 x^5+168 x^6+30 x^7+2 x^8)}{192 x^2+144 x^3+36 x^4+3 x^5} \, dx\)

Optimal. Leaf size=29 \[ x+\frac {x}{(4+x)^2}+\frac {1}{3} \left (e^x+x\right )^2 \left (\frac {3}{x}+x^2\right ) \]

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Rubi [B]  time = 2.15, antiderivative size = 59, normalized size of antiderivative = 2.03, number of steps used = 25, number of rules used = 9, integrand size = 141, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {6688, 12, 2196, 2194, 2176, 2199, 2177, 2178, 1850} \begin {gather*} \frac {x^4}{3}+\frac {2 e^x x^3}{3}+\frac {1}{3} e^{2 x} x^2+2 x+2 e^x+\frac {1}{x+4}-\frac {4}{(x+4)^2}+\frac {e^{2 x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(396*x^2 + 285*x^3 + 72*x^4 + 262*x^5 + 192*x^6 + 48*x^7 + 4*x^8 + E^(2*x)*(-192 + 240*x + 252*x^2 + 197*x
^3 + 230*x^4 + 120*x^5 + 26*x^6 + 2*x^7) + E^x*(384*x^2 + 288*x^3 + 456*x^4 + 422*x^5 + 168*x^6 + 30*x^7 + 2*x
^8))/(192*x^2 + 144*x^3 + 36*x^4 + 3*x^5),x]

[Out]

2*E^x + E^(2*x)/x + 2*x + (E^(2*x)*x^2)/3 + (2*E^x*x^3)/3 + x^4/3 - 4/(4 + x)^2 + (4 + x)^(-1)

Rule 12

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

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

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 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 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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{3} \left (2 e^x \left (3+3 x^2+x^3\right )+\frac {e^{2 x} \left (-3+6 x+2 x^3+2 x^4\right )}{x^2}+\frac {396+285 x+72 x^2+262 x^3+192 x^4+48 x^5+4 x^6}{(4+x)^3}\right ) \, dx\\ &=\frac {1}{3} \int \left (2 e^x \left (3+3 x^2+x^3\right )+\frac {e^{2 x} \left (-3+6 x+2 x^3+2 x^4\right )}{x^2}+\frac {396+285 x+72 x^2+262 x^3+192 x^4+48 x^5+4 x^6}{(4+x)^3}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{2 x} \left (-3+6 x+2 x^3+2 x^4\right )}{x^2} \, dx+\frac {1}{3} \int \frac {396+285 x+72 x^2+262 x^3+192 x^4+48 x^5+4 x^6}{(4+x)^3} \, dx+\frac {2}{3} \int e^x \left (3+3 x^2+x^3\right ) \, dx\\ &=\frac {1}{3} \int \left (-\frac {3 e^{2 x}}{x^2}+\frac {6 e^{2 x}}{x}+2 e^{2 x} x+2 e^{2 x} x^2\right ) \, dx+\frac {1}{3} \int \left (6+4 x^3+\frac {24}{(4+x)^3}-\frac {3}{(4+x)^2}\right ) \, dx+\frac {2}{3} \int \left (3 e^x+3 e^x x^2+e^x x^3\right ) \, dx\\ &=2 x+\frac {x^4}{3}-\frac {4}{(4+x)^2}+\frac {1}{4+x}+\frac {2}{3} \int e^{2 x} x \, dx+\frac {2}{3} \int e^{2 x} x^2 \, dx+\frac {2}{3} \int e^x x^3 \, dx+2 \int e^x \, dx+2 \int \frac {e^{2 x}}{x} \, dx+2 \int e^x x^2 \, dx-\int \frac {e^{2 x}}{x^2} \, dx\\ &=2 e^x+\frac {e^{2 x}}{x}+2 x+\frac {1}{3} e^{2 x} x+2 e^x x^2+\frac {1}{3} e^{2 x} x^2+\frac {2 e^x x^3}{3}+\frac {x^4}{3}-\frac {4}{(4+x)^2}+\frac {1}{4+x}+2 \text {Ei}(2 x)-\frac {1}{3} \int e^{2 x} \, dx-\frac {2}{3} \int e^{2 x} x \, dx-2 \int \frac {e^{2 x}}{x} \, dx-2 \int e^x x^2 \, dx-4 \int e^x x \, dx\\ &=2 e^x-\frac {e^{2 x}}{6}+\frac {e^{2 x}}{x}+2 x-4 e^x x+\frac {1}{3} e^{2 x} x^2+\frac {2 e^x x^3}{3}+\frac {x^4}{3}-\frac {4}{(4+x)^2}+\frac {1}{4+x}+\frac {1}{3} \int e^{2 x} \, dx+4 \int e^x \, dx+4 \int e^x x \, dx\\ &=6 e^x+\frac {e^{2 x}}{x}+2 x+\frac {1}{3} e^{2 x} x^2+\frac {2 e^x x^3}{3}+\frac {x^4}{3}-\frac {4}{(4+x)^2}+\frac {1}{4+x}-4 \int e^x \, dx\\ &=2 e^x+\frac {e^{2 x}}{x}+2 x+\frac {1}{3} e^{2 x} x^2+\frac {2 e^x x^3}{3}+\frac {x^4}{3}-\frac {4}{(4+x)^2}+\frac {1}{4+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 50, normalized size = 1.72 \begin {gather*} \frac {1}{3} \left (-232+6 x+x^4-\frac {12}{(4+x)^2}+\frac {3}{4+x}+2 e^x \left (3+x^3\right )+\frac {e^{2 x} \left (3+x^3\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(396*x^2 + 285*x^3 + 72*x^4 + 262*x^5 + 192*x^6 + 48*x^7 + 4*x^8 + E^(2*x)*(-192 + 240*x + 252*x^2 +
197*x^3 + 230*x^4 + 120*x^5 + 26*x^6 + 2*x^7) + E^x*(384*x^2 + 288*x^3 + 456*x^4 + 422*x^5 + 168*x^6 + 30*x^7
+ 2*x^8))/(192*x^2 + 144*x^3 + 36*x^4 + 3*x^5),x]

[Out]

(-232 + 6*x + x^4 - 12/(4 + x)^2 + 3/(4 + x) + 2*E^x*(3 + x^3) + (E^(2*x)*(3 + x^3))/x)/3

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fricas [B]  time = 0.75, size = 104, normalized size = 3.59 \begin {gather*} \frac {x^{7} + 8 \, x^{6} + 16 \, x^{5} + 6 \, x^{4} + 48 \, x^{3} + 99 \, x^{2} + {\left (x^{5} + 8 \, x^{4} + 16 \, x^{3} + 3 \, x^{2} + 24 \, x + 48\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{6} + 8 \, x^{5} + 16 \, x^{4} + 3 \, x^{3} + 24 \, x^{2} + 48 \, x\right )} e^{x}}{3 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^7+26*x^6+120*x^5+230*x^4+197*x^3+252*x^2+240*x-192)*exp(x)^2+(2*x^8+30*x^7+168*x^6+422*x^5+456
*x^4+288*x^3+384*x^2)*exp(x)+4*x^8+48*x^7+192*x^6+262*x^5+72*x^4+285*x^3+396*x^2)/(3*x^5+36*x^4+144*x^3+192*x^
2),x, algorithm="fricas")

[Out]

1/3*(x^7 + 8*x^6 + 16*x^5 + 6*x^4 + 48*x^3 + 99*x^2 + (x^5 + 8*x^4 + 16*x^3 + 3*x^2 + 24*x + 48)*e^(2*x) + 2*(
x^6 + 8*x^5 + 16*x^4 + 3*x^3 + 24*x^2 + 48*x)*e^x)/(x^3 + 8*x^2 + 16*x)

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giac [B]  time = 0.19, size = 133, normalized size = 4.59 \begin {gather*} \frac {x^{7} + 2 \, x^{6} e^{x} + 8 \, x^{6} + x^{5} e^{\left (2 \, x\right )} + 16 \, x^{5} e^{x} + 16 \, x^{5} + 8 \, x^{4} e^{\left (2 \, x\right )} + 32 \, x^{4} e^{x} + 6 \, x^{4} + 16 \, x^{3} e^{\left (2 \, x\right )} + 6 \, x^{3} e^{x} + 48 \, x^{3} + 3 \, x^{2} e^{\left (2 \, x\right )} + 48 \, x^{2} e^{x} + 99 \, x^{2} + 24 \, x e^{\left (2 \, x\right )} + 96 \, x e^{x} + 48 \, e^{\left (2 \, x\right )}}{3 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^7+26*x^6+120*x^5+230*x^4+197*x^3+252*x^2+240*x-192)*exp(x)^2+(2*x^8+30*x^7+168*x^6+422*x^5+456
*x^4+288*x^3+384*x^2)*exp(x)+4*x^8+48*x^7+192*x^6+262*x^5+72*x^4+285*x^3+396*x^2)/(3*x^5+36*x^4+144*x^3+192*x^
2),x, algorithm="giac")

[Out]

1/3*(x^7 + 2*x^6*e^x + 8*x^6 + x^5*e^(2*x) + 16*x^5*e^x + 16*x^5 + 8*x^4*e^(2*x) + 32*x^4*e^x + 6*x^4 + 16*x^3
*e^(2*x) + 6*x^3*e^x + 48*x^3 + 3*x^2*e^(2*x) + 48*x^2*e^x + 99*x^2 + 24*x*e^(2*x) + 96*x*e^x + 48*e^(2*x))/(x
^3 + 8*x^2 + 16*x)

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maple [A]  time = 0.10, size = 46, normalized size = 1.59

method	result	size
risch	\(\frac {x^{4}}{3}+2 x +\frac {x}{x^{2}+8 x +16}+\frac {\left (x^{3}+3\right ) {\mathrm e}^{2 x}}{3 x}+\left (2+\frac {2 x^{3}}{3}\right ) {\mathrm e}^{x}\)	\(46\)
default	\(-\frac {4}{\left (4+x \right )^{2}}+\frac {1}{4+x}+2 x +\frac {x^{4}}{3}+\frac {{\mathrm e}^{2 x}}{x}+2 \,{\mathrm e}^{x}+\frac {2 \,{\mathrm e}^{x} x^{3}}{3}+\frac {{\mathrm e}^{2 x} x^{2}}{3}\)	\(50\)
norman	\(\frac {-256 x -95 x^{2}+{\mathrm e}^{2 x} x^{2}+2 x^{4}+\frac {16 x^{5}}{3}+\frac {8 x^{6}}{3}+\frac {x^{7}}{3}+16 \,{\mathrm e}^{2 x}+8 x \,{\mathrm e}^{2 x}+\frac {16 x^{5} {\mathrm e}^{x}}{3}+\frac {x^{5} {\mathrm e}^{2 x}}{3}+\frac {2 x^{6} {\mathrm e}^{x}}{3}+32 \,{\mathrm e}^{x} x +16 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} x^{3}+\frac {32 \,{\mathrm e}^{x} x^{4}}{3}+\frac {16 \,{\mathrm e}^{2 x} x^{3}}{3}+\frac {8 \,{\mathrm e}^{2 x} x^{4}}{3}}{\left (4+x \right )^{2} x}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^7+26*x^6+120*x^5+230*x^4+197*x^3+252*x^2+240*x-192)*exp(x)^2+(2*x^8+30*x^7+168*x^6+422*x^5+456*x^4+2
88*x^3+384*x^2)*exp(x)+4*x^8+48*x^7+192*x^6+262*x^5+72*x^4+285*x^3+396*x^2)/(3*x^5+36*x^4+144*x^3+192*x^2),x,m
ethod=_RETURNVERBOSE)

[Out]

1/3*x^4+2*x+x/(x^2+8*x+16)+1/3*(x^3+3)/x*exp(2*x)+(2+2/3*x^3)*exp(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{3} \, x^{4} + 2 \, x + \frac {{\left (x^{6} + 12 \, x^{5} + 48 \, x^{4} + 67 \, x^{3} + 36 \, x^{2} + 144 \, x + 192\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{7} + 12 \, x^{6} + 48 \, x^{5} + 67 \, x^{4} + 36 \, x^{3} + 144 \, x^{2}\right )} e^{x}}{3 \, {\left (x^{4} + 12 \, x^{3} + 48 \, x^{2} + 64 \, x\right )}} - \frac {4096 \, {\left (5 \, x + 18\right )}}{x^{2} + 8 \, x + 16} + \frac {8192 \, {\left (3 \, x + 11\right )}}{3 \, {\left (x^{2} + 8 \, x + 16\right )}} - \frac {4192 \, {\left (3 \, x + 10\right )}}{3 \, {\left (x^{2} + 8 \, x + 16\right )}} + \frac {8192 \, {\left (2 \, x + 7\right )}}{x^{2} + 8 \, x + 16} + \frac {192 \, {\left (x + 3\right )}}{x^{2} + 8 \, x + 16} - \frac {95 \, {\left (x + 2\right )}}{x^{2} + 8 \, x + 16} - \frac {128 \, e^{\left (-4\right )} E_{3}\left (-x - 4\right )}{{\left (x + 4\right )}^{2}} - \frac {66}{x^{2} + 8 \, x + 16} - 384 \, \int \frac {e^{x}}{x^{4} + 16 \, x^{3} + 96 \, x^{2} + 256 \, x + 256}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^7+26*x^6+120*x^5+230*x^4+197*x^3+252*x^2+240*x-192)*exp(x)^2+(2*x^8+30*x^7+168*x^6+422*x^5+456
*x^4+288*x^3+384*x^2)*exp(x)+4*x^8+48*x^7+192*x^6+262*x^5+72*x^4+285*x^3+396*x^2)/(3*x^5+36*x^4+144*x^3+192*x^
2),x, algorithm="maxima")

[Out]

1/3*x^4 + 2*x + 1/3*((x^6 + 12*x^5 + 48*x^4 + 67*x^3 + 36*x^2 + 144*x + 192)*e^(2*x) + 2*(x^7 + 12*x^6 + 48*x^
5 + 67*x^4 + 36*x^3 + 144*x^2)*e^x)/(x^4 + 12*x^3 + 48*x^2 + 64*x) - 4096*(5*x + 18)/(x^2 + 8*x + 16) + 8192/3
*(3*x + 11)/(x^2 + 8*x + 16) - 4192/3*(3*x + 10)/(x^2 + 8*x + 16) + 8192*(2*x + 7)/(x^2 + 8*x + 16) + 192*(x +
3)/(x^2 + 8*x + 16) - 95*(x + 2)/(x^2 + 8*x + 16) - 128*e^(-4)*exp_integral_e(3, -x - 4)/(x + 4)^2 - 66/(x^2
+ 8*x + 16) - 384*integrate(e^x/(x^4 + 16*x^3 + 96*x^2 + 256*x + 256), x)

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mupad [B]  time = 1.04, size = 72, normalized size = 2.48 \begin {gather*} 2\,x+2\,{\mathrm {e}}^x+\frac {2\,x^3\,{\mathrm {e}}^x}{3}+\frac {x^2\,{\mathrm {e}}^{2\,x}}{3}+\frac {48\,{\mathrm {e}}^{2\,x}+24\,x\,{\mathrm {e}}^{2\,x}+x^2\,\left (3\,{\mathrm {e}}^{2\,x}+3\right )}{3\,x^3+24\,x^2+48\,x}+\frac {x^4}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(240*x + 252*x^2 + 197*x^3 + 230*x^4 + 120*x^5 + 26*x^6 + 2*x^7 - 192) + exp(x)*(384*x^2 + 288*x
^3 + 456*x^4 + 422*x^5 + 168*x^6 + 30*x^7 + 2*x^8) + 396*x^2 + 285*x^3 + 72*x^4 + 262*x^5 + 192*x^6 + 48*x^7 +
4*x^8)/(192*x^2 + 144*x^3 + 36*x^4 + 3*x^5),x)

[Out]

2*x + 2*exp(x) + (2*x^3*exp(x))/3 + (x^2*exp(2*x))/3 + (48*exp(2*x) + 24*x*exp(2*x) + x^2*(3*exp(2*x) + 3))/(4
8*x + 24*x^2 + 3*x^3) + x^4/3

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sympy [A]  time = 0.22, size = 44, normalized size = 1.52 \begin {gather*} \frac {x^{4}}{3} + 2 x + \frac {x}{x^{2} + 8 x + 16} + \frac {\left (3 x^{3} + 9\right ) e^{2 x} + \left (6 x^{4} + 18 x\right ) e^{x}}{9 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**7+26*x**6+120*x**5+230*x**4+197*x**3+252*x**2+240*x-192)*exp(x)**2+(2*x**8+30*x**7+168*x**6+4
22*x**5+456*x**4+288*x**3+384*x**2)*exp(x)+4*x**8+48*x**7+192*x**6+262*x**5+72*x**4+285*x**3+396*x**2)/(3*x**5
+36*x**4+144*x**3+192*x**2),x)

[Out]

x**4/3 + 2*x + x/(x**2 + 8*x + 16) + ((3*x**3 + 9)*exp(2*x) + (6*x**4 + 18*x)*exp(x))/(9*x)

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