3.103.58 \(\int \frac {-95-60 x+e^2 (15+10 x)+e^x (-19 x-12 x^2+e^2 (3 x+2 x^2))+e^x (-2-21 x-25 x^2-6 x^3+e^2 (3 x+4 x^2+x^3)) \log (2+19 x+6 x^2+e^2 (-3 x-x^2))}{-6-57 x-18 x^2+e^2 (9 x+3 x^2)} \, dx\)

Optimal. Leaf size=29 \[ -2+\frac {1}{3} \left (5+e^x x\right ) \log \left (2+x+\left (6-e^2\right ) x (3+x)\right ) \]

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Rubi [B]  time = 2.65, antiderivative size = 91, normalized size of antiderivative = 3.14, number of steps used = 22, number of rules used = 10, integrand size = 125, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6741, 6728, 628, 6688, 6742, 2194, 2270, 2178, 2176, 2554} \begin {gather*} -\frac {1}{3} e^x \log \left (\left (6-e^2\right ) x^2+\left (19-3 e^2\right ) x+2\right )+\frac {1}{3} e^x (x+1) \log \left (\left (6-e^2\right ) x^2+\left (19-3 e^2\right ) x+2\right )+\frac {5}{3} \log \left (\left (6-e^2\right ) x^2+\left (19-3 e^2\right ) x+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-95 - 60*x + E^2*(15 + 10*x) + E^x*(-19*x - 12*x^2 + E^2*(3*x + 2*x^2)) + E^x*(-2 - 21*x - 25*x^2 - 6*x^3
+ E^2*(3*x + 4*x^2 + x^3))*Log[2 + 19*x + 6*x^2 + E^2*(-3*x - x^2)])/(-6 - 57*x - 18*x^2 + E^2*(9*x + 3*x^2))
,x]

[Out]

(5*Log[2 + (19 - 3*E^2)*x + (6 - E^2)*x^2])/3 - (E^x*Log[2 + (19 - 3*E^2)*x + (6 - E^2)*x^2])/3 + (E^x*(1 + x)
*Log[2 + (19 - 3*E^2)*x + (6 - E^2)*x^2])/3

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 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 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 2270

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6688

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

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6741

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

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 {95+60 x-e^2 (15+10 x)-e^x \left (-19 x-12 x^2+e^2 \left (3 x+2 x^2\right )\right )-e^x \left (-2-21 x-25 x^2-6 x^3+e^2 \left (3 x+4 x^2+x^3\right )\right ) \log \left (2+19 x+6 x^2+e^2 \left (-3 x-x^2\right )\right )}{6+3 \left (19-3 e^2\right ) x+3 \left (6-e^2\right ) x^2} \, dx\\ &=\int \left (\frac {5 \left (19-3 e^2+2 \left (6-e^2\right ) x\right )}{3 \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )}+\frac {e^x \left (19 \left (1-\frac {3 e^2}{19}\right ) x+12 \left (1-\frac {e^2}{6}\right ) x^2+2 \log \left (2+19 x+6 x^2-e^2 x (3+x)\right )+21 \left (1-\frac {e^2}{7}\right ) x \log \left (2+19 x+6 x^2-e^2 x (3+x)\right )+25 \left (1-\frac {4 e^2}{25}\right ) x^2 \log \left (2+19 x+6 x^2-e^2 x (3+x)\right )+6 \left (1-\frac {e^2}{6}\right ) x^3 \log \left (2+19 x+6 x^2-e^2 x (3+x)\right )\right )}{3 \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^x \left (19 \left (1-\frac {3 e^2}{19}\right ) x+12 \left (1-\frac {e^2}{6}\right ) x^2+2 \log \left (2+19 x+6 x^2-e^2 x (3+x)\right )+21 \left (1-\frac {e^2}{7}\right ) x \log \left (2+19 x+6 x^2-e^2 x (3+x)\right )+25 \left (1-\frac {4 e^2}{25}\right ) x^2 \log \left (2+19 x+6 x^2-e^2 x (3+x)\right )+6 \left (1-\frac {e^2}{6}\right ) x^3 \log \left (2+19 x+6 x^2-e^2 x (3+x)\right )\right )}{2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2} \, dx+\frac {5}{3} \int \frac {19-3 e^2+2 \left (6-e^2\right ) x}{2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2} \, dx\\ &=\frac {5}{3} \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} \int e^x \left (\frac {x \left (-19-12 x+e^2 (3+2 x)\right )}{-2+\left (-19+3 e^2\right ) x+\left (-6+e^2\right ) x^2}+(1+x) \log \left (2+19 x+6 x^2-e^2 x (3+x)\right )\right ) \, dx\\ &=\frac {5}{3} \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} \int \left (\frac {e^x x \left (19-3 e^2+2 \left (6-e^2\right ) x\right )}{2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2}+e^x (1+x) \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )\right ) \, dx\\ &=\frac {5}{3} \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} \int \frac {e^x x \left (19-3 e^2+2 \left (6-e^2\right ) x\right )}{2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2} \, dx+\frac {1}{3} \int e^x (1+x) \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right ) \, dx\\ &=\frac {5}{3} \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )-\frac {1}{3} e^x \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} e^x (1+x) \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )-\frac {1}{3} \int \frac {e^x x \left (19-3 e^2+2 \left (6-e^2\right ) x\right )}{2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2} \, dx+\frac {1}{3} \int \left (2 e^x-\frac {e^x \left (4+\left (19-3 e^2\right ) x\right )}{2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2}\right ) \, dx\\ &=\frac {5}{3} \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )-\frac {1}{3} e^x \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} e^x (1+x) \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )-\frac {1}{3} \int \frac {e^x \left (4+\left (19-3 e^2\right ) x\right )}{2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2} \, dx-\frac {1}{3} \int \left (2 e^x-\frac {e^x \left (4+\left (19-3 e^2\right ) x\right )}{2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2}\right ) \, dx+\frac {2 \int e^x \, dx}{3}\\ &=\frac {2 e^x}{3}+\frac {5}{3} \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )-\frac {1}{3} e^x \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} e^x (1+x) \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} \int \frac {e^x \left (4+\left (19-3 e^2\right ) x\right )}{2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2} \, dx-\frac {1}{3} \int \left (\frac {e^x \left (19-3 e^2-\sqrt {313-106 e^2+9 e^4}\right )}{19-3 e^2-\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x}+\frac {e^x \left (19-3 e^2+\sqrt {313-106 e^2+9 e^4}\right )}{19-3 e^2+\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x}\right ) \, dx-\frac {2 \int e^x \, dx}{3}\\ &=\frac {5}{3} \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )-\frac {1}{3} e^x \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} e^x (1+x) \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} \int \left (\frac {e^x \left (19-3 e^2-\sqrt {313-106 e^2+9 e^4}\right )}{19-3 e^2-\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x}+\frac {e^x \left (19-3 e^2+\sqrt {313-106 e^2+9 e^4}\right )}{19-3 e^2+\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x}\right ) \, dx-\frac {1}{3} \left (19-3 e^2-\sqrt {313-106 e^2+9 e^4}\right ) \int \frac {e^x}{19-3 e^2-\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x} \, dx-\frac {1}{3} \left (19-3 e^2+\sqrt {313-106 e^2+9 e^4}\right ) \int \frac {e^x}{19-3 e^2+\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x} \, dx\\ &=-\frac {\exp \left (-\frac {19-3 e^2-\sqrt {313-106 e^2+9 e^4}}{2 \left (6-e^2\right )}\right ) \left (19-3 e^2-\sqrt {313-106 e^2+9 e^4}\right ) \text {Ei}\left (\frac {19-3 e^2-\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x}{2 \left (6-e^2\right )}\right )}{6 \left (6-e^2\right )}-\frac {\exp \left (-\frac {19-3 e^2+\sqrt {313-106 e^2+9 e^4}}{2 \left (6-e^2\right )}\right ) \left (19-3 e^2+\sqrt {313-106 e^2+9 e^4}\right ) \text {Ei}\left (\frac {19-3 e^2+\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x}{2 \left (6-e^2\right )}\right )}{6 \left (6-e^2\right )}+\frac {5}{3} \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )-\frac {1}{3} e^x \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} e^x (1+x) \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} \left (19-3 e^2-\sqrt {313-106 e^2+9 e^4}\right ) \int \frac {e^x}{19-3 e^2-\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x} \, dx+\frac {1}{3} \left (19-3 e^2+\sqrt {313-106 e^2+9 e^4}\right ) \int \frac {e^x}{19-3 e^2+\sqrt {313-106 e^2+9 e^4}+2 \left (6-e^2\right ) x} \, dx\\ &=\frac {5}{3} \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )-\frac {1}{3} e^x \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )+\frac {1}{3} e^x (1+x) \log \left (2+\left (19-3 e^2\right ) x+\left (6-e^2\right ) x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 33, normalized size = 1.14 \begin {gather*} \frac {1}{3} \left (5+e^x x\right ) \log \left (2+\left (19-3 e^2\right ) x-\left (-6+e^2\right ) x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-95 - 60*x + E^2*(15 + 10*x) + E^x*(-19*x - 12*x^2 + E^2*(3*x + 2*x^2)) + E^x*(-2 - 21*x - 25*x^2 -
6*x^3 + E^2*(3*x + 4*x^2 + x^3))*Log[2 + 19*x + 6*x^2 + E^2*(-3*x - x^2)])/(-6 - 57*x - 18*x^2 + E^2*(9*x + 3
*x^2)),x]

[Out]

((5 + E^x*x)*Log[2 + (19 - 3*E^2)*x - (-6 + E^2)*x^2])/3

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fricas [A]  time = 0.65, size = 30, normalized size = 1.03 \begin {gather*} \frac {1}{3} \, {\left (x e^{x} + 5\right )} \log \left (6 \, x^{2} - {\left (x^{2} + 3 \, x\right )} e^{2} + 19 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3+4*x^2+3*x)*exp(2)-6*x^3-25*x^2-21*x-2)*exp(x)*log((-x^2-3*x)*exp(2)+6*x^2+19*x+2)+((2*x^2+3*x
)*exp(2)-12*x^2-19*x)*exp(x)+(10*x+15)*exp(2)-60*x-95)/((3*x^2+9*x)*exp(2)-18*x^2-57*x-6),x, algorithm="fricas
")

[Out]

1/3*(x*e^x + 5)*log(6*x^2 - (x^2 + 3*x)*e^2 + 19*x + 2)

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giac [B]  time = 0.27, size = 53, normalized size = 1.83 \begin {gather*} \frac {1}{3} \, x e^{x} \log \left (-x^{2} e^{2} + 6 \, x^{2} - 3 \, x e^{2} + 19 \, x + 2\right ) + \frac {5}{3} \, \log \left (x^{2} e^{2} - 6 \, x^{2} + 3 \, x e^{2} - 19 \, x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3+4*x^2+3*x)*exp(2)-6*x^3-25*x^2-21*x-2)*exp(x)*log((-x^2-3*x)*exp(2)+6*x^2+19*x+2)+((2*x^2+3*x
)*exp(2)-12*x^2-19*x)*exp(x)+(10*x+15)*exp(2)-60*x-95)/((3*x^2+9*x)*exp(2)-18*x^2-57*x-6),x, algorithm="giac")

[Out]

1/3*x*e^x*log(-x^2*e^2 + 6*x^2 - 3*x*e^2 + 19*x + 2) + 5/3*log(x^2*e^2 - 6*x^2 + 3*x*e^2 - 19*x - 2)

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maple [A]  time = 0.42, size = 51, normalized size = 1.76

method	result	size
risch	\(\frac {{\mathrm e}^{x} x \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3}+\frac {5 \ln \left (\left ({\mathrm e}^{2}-6\right ) x^{2}+\left (3 \,{\mathrm e}^{2}-19\right ) x -2\right )}{3}\)	\(51\)
default	\(\frac {{\mathrm e}^{x} x \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3}+\frac {5 \ln \left (x^{2} {\mathrm e}^{2}+3 \,{\mathrm e}^{2} x -6 x^{2}-19 x -2\right )}{3}\)	\(54\)
norman	\(\frac {5 \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3}+\frac {{\mathrm e}^{x} x \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^3+4*x^2+3*x)*exp(2)-6*x^3-25*x^2-21*x-2)*exp(x)*ln((-x^2-3*x)*exp(2)+6*x^2+19*x+2)+((2*x^2+3*x)*exp(2
)-12*x^2-19*x)*exp(x)+(10*x+15)*exp(2)-60*x-95)/((3*x^2+9*x)*exp(2)-18*x^2-57*x-6),x,method=_RETURNVERBOSE)

[Out]

1/3*exp(x)*x*ln((-x^2-3*x)*exp(2)+6*x^2+19*x+2)+5/3*ln((exp(2)-6)*x^2+(3*exp(2)-19)*x-2)

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maxima [B]  time = 0.42, size = 391, normalized size = 13.48 \begin {gather*} \frac {1}{3} \, x e^{x} \log \left (-x^{2} {\left (e^{2} - 6\right )} - x {\left (3 \, e^{2} - 19\right )} + 2\right ) - \frac {5}{3} \, {\left (\frac {{\left (3 \, e^{2} - 19\right )} \log \left (\frac {2 \, x {\left (e^{2} - 6\right )} - \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}{2 \, x {\left (e^{2} - 6\right )} + \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}\right )}{\sqrt {9 \, e^{4} - 106 \, e^{2} + 313} {\left (e^{2} - 6\right )}} - \frac {\log \left (x^{2} {\left (e^{2} - 6\right )} + x {\left (3 \, e^{2} - 19\right )} - 2\right )}{e^{2} - 6}\right )} e^{2} + \frac {5 \, e^{2} \log \left (\frac {2 \, x {\left (e^{2} - 6\right )} - \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}{2 \, x {\left (e^{2} - 6\right )} + \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}\right )}{\sqrt {9 \, e^{4} - 106 \, e^{2} + 313}} - \frac {95 \, \log \left (\frac {2 \, x {\left (e^{2} - 6\right )} - \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}{2 \, x {\left (e^{2} - 6\right )} + \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}\right )}{3 \, \sqrt {9 \, e^{4} - 106 \, e^{2} + 313}} + \frac {10 \, {\left (3 \, e^{2} - 19\right )} \log \left (\frac {2 \, x {\left (e^{2} - 6\right )} - \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}{2 \, x {\left (e^{2} - 6\right )} + \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}\right )}{\sqrt {9 \, e^{4} - 106 \, e^{2} + 313} {\left (e^{2} - 6\right )}} - \frac {10 \, \log \left (x^{2} {\left (e^{2} - 6\right )} + x {\left (3 \, e^{2} - 19\right )} - 2\right )}{e^{2} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^3+4*x^2+3*x)*exp(2)-6*x^3-25*x^2-21*x-2)*exp(x)*log((-x^2-3*x)*exp(2)+6*x^2+19*x+2)+((2*x^2+3*x
)*exp(2)-12*x^2-19*x)*exp(x)+(10*x+15)*exp(2)-60*x-95)/((3*x^2+9*x)*exp(2)-18*x^2-57*x-6),x, algorithm="maxima
")

[Out]

1/3*x*e^x*log(-x^2*(e^2 - 6) - x*(3*e^2 - 19) + 2) - 5/3*((3*e^2 - 19)*log((2*x*(e^2 - 6) - sqrt(9*e^4 - 106*e
^2 + 313) + 3*e^2 - 19)/(2*x*(e^2 - 6) + sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19))/(sqrt(9*e^4 - 106*e^2 + 31
3)*(e^2 - 6)) - log(x^2*(e^2 - 6) + x*(3*e^2 - 19) - 2)/(e^2 - 6))*e^2 + 5*e^2*log((2*x*(e^2 - 6) - sqrt(9*e^4
- 106*e^2 + 313) + 3*e^2 - 19)/(2*x*(e^2 - 6) + sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19))/sqrt(9*e^4 - 106*e
^2 + 313) - 95/3*log((2*x*(e^2 - 6) - sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19)/(2*x*(e^2 - 6) + sqrt(9*e^4 -
106*e^2 + 313) + 3*e^2 - 19))/sqrt(9*e^4 - 106*e^2 + 313) + 10*(3*e^2 - 19)*log((2*x*(e^2 - 6) - sqrt(9*e^4 -
106*e^2 + 313) + 3*e^2 - 19)/(2*x*(e^2 - 6) + sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19))/(sqrt(9*e^4 - 106*e^2
+ 313)*(e^2 - 6)) - 10*log(x^2*(e^2 - 6) + x*(3*e^2 - 19) - 2)/(e^2 - 6)

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mupad [B]  time = 7.81, size = 49, normalized size = 1.69 \begin {gather*} \frac {5\,\ln \left (\left ({\mathrm {e}}^2-6\right )\,x^2+\left (3\,{\mathrm {e}}^2-19\right )\,x-2\right )}{3}+\frac {x\,{\mathrm {e}}^x\,\ln \left (19\,x-{\mathrm {e}}^2\,\left (x^2+3\,x\right )+6\,x^2+2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((60*x + exp(x)*(19*x - exp(2)*(3*x + 2*x^2) + 12*x^2) - exp(2)*(10*x + 15) + exp(x)*log(19*x - exp(2)*(3*x
+ x^2) + 6*x^2 + 2)*(21*x - exp(2)*(3*x + 4*x^2 + x^3) + 25*x^2 + 6*x^3 + 2) + 95)/(57*x - exp(2)*(9*x + 3*x^
2) + 18*x^2 + 6),x)

[Out]

(5*log(x*(3*exp(2) - 19) + x^2*(exp(2) - 6) - 2))/3 + (x*exp(x)*log(19*x - exp(2)*(3*x + x^2) + 6*x^2 + 2))/3

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sympy [B]  time = 5.05, size = 53, normalized size = 1.83 \begin {gather*} \frac {x e^{x} \log {\left (6 x^{2} + 19 x + \left (- x^{2} - 3 x\right ) e^{2} + 2 \right )}}{3} + \frac {5 \log {\left (x^{2} \left (-6 + e^{2}\right ) + x \left (-19 + 3 e^{2}\right ) - 2 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**3+4*x**2+3*x)*exp(2)-6*x**3-25*x**2-21*x-2)*exp(x)*ln((-x**2-3*x)*exp(2)+6*x**2+19*x+2)+((2*x*
*2+3*x)*exp(2)-12*x**2-19*x)*exp(x)+(10*x+15)*exp(2)-60*x-95)/((3*x**2+9*x)*exp(2)-18*x**2-57*x-6),x)

[Out]

x*exp(x)*log(6*x**2 + 19*x + (-x**2 - 3*x)*exp(2) + 2)/3 + 5*log(x**2*(-6 + exp(2)) + x*(-19 + 3*exp(2)) - 2)/
3

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