3.8.10 \(\int \frac {-15+29 x+4 x^2-4 x^3+(-3+7 x-2 x^2) \log (5)+e^{2 x^2} (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+(-5 x+12 x^2-28 x^3+8 x^4) \log (5))+(25 x+5 x^2+5 x \log (5)) \log (5 x+x^2+x \log (5))}{45 x-21 x^2-x^3+x^4+(9 x-6 x^2+x^3) \log (5)} \, dx\)

Optimal. Leaf size=31 \[ \frac {(-1+2 x) \left (-e^{2 x^2}+\log (x (5+x+\log (5)))\right )}{3-x} \]

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Rubi [B]  time = 7.01, antiderivative size = 1255, normalized size of antiderivative = 40.48, number of steps used = 39, number of rules used = 22, integrand size = 144, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.153, Rules used = {6741, 6742, 2067, 2063, 44, 2081, 2076, 77, 1586, 27, 2288, 800, 634, 618, 206, 628, 72, 88, 2495, 36, 31, 29}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(-15 + 29*x + 4*x^2 - 4*x^3 + (-3 + 7*x - 2*x^2)*Log[5] + E^(2*x^2)*(-25*x + 55*x^2 - 128*x^3 + 12*x^4 + 8
*x^5 + (-5*x + 12*x^2 - 28*x^3 + 8*x^4)*Log[5]) + (25*x + 5*x^2 + 5*x*Log[5])*Log[5*x + x^2 + x*Log[5]])/(45*x
- 21*x^2 - x^3 + x^4 + (9*x - 6*x^2 + x^3)*Log[5]),x]

[Out]

(E^(2*x^2)*(3*x - 7*x^2 + 2*x^3))/((3 - x)^2*x) - (29*(1024 - 6*Log[5]^2 + 2*Log[5]^3 - 27*Log[25] + 3*Log[5]*
(146 + 9*Log[25])))/((64 - 2*Log[5] + Log[5]^2 + 9*Log[25])*(2*x*(64 - 2*Log[5] + Log[5]^2 + 9*Log[25]) - 3*(1
28 - Log[25] + Log[5]*(34 + Log[25])))) + (10*(1024 - 6*Log[5]^2 + 2*Log[5]^3 - 27*Log[25] + 3*Log[5]*(146 + 9
*Log[25])))/((128 - Log[25] + Log[5]*(34 + Log[25]))*(2*x*(64 - 2*Log[5] + Log[5]^2 + 9*Log[25]) - 3*(128 - Lo
g[25] + Log[5]*(34 + Log[25])))) - (6*(128 - Log[25] + Log[5]*(34 + Log[25]))*(1024 - 6*Log[5]^2 + 2*Log[5]^3
- 27*Log[25] + 3*Log[5]*(146 + 9*Log[25])))/((64 - 2*Log[5] + Log[5]^2 + 9*Log[25])^2*(2*x*(64 - 2*Log[5] + Lo
g[5]^2 + 9*Log[25]) - 3*(128 - Log[25] + Log[5]*(34 + Log[25])))) + (9*(128 - Log[25] + Log[5]*(34 + Log[25]))
^2*(1024 - 6*Log[5]^2 + 2*Log[5]^3 - 27*Log[25] + 3*Log[5]*(146 + 9*Log[25])))/((64 - 2*Log[5] + Log[5]^2 + 9*
Log[25])^3*(2*x*(64 - 2*Log[5] + Log[5]^2 + 9*Log[25]) - 3*(128 - Log[25] + Log[5]*(34 + Log[25])))) + (ArcTan
h[(2 + 2*x + Log[5])/Sqrt[64 + Log[5]^2 + 4*Log[125] + Log[625]]]*Log[5]*(58 + Log[48828125]))/(3*(5 + Log[5])
*Sqrt[64 + Log[5]^2 + 4*Log[125] + Log[625]]) + (5*Log[3 - x])/3 + (5*Log[3 - x])/(8 + Log[5]) - (5*Log[x])/3
- (Log[5]*Log[x])/(3*(5 + Log[5])) - (5*(2*(1 - Log[5])^3 - 243*(5 + Log[5]) + 27*(1 - Log[5])*(7 + Log[25]))^
2*Log[x])/(3*(320 - 3*Log[5]^2 + Log[5]^3 - 12*Log[25] + 12*Log[5]*(14 + Log[25]))*(128 - Log[25] + Log[5]*(34
+ Log[25]))^2) - (5*Log[5 + x + Log[5]])/(8 + Log[5]) + (5*Log[x*(5 + x + Log[5])])/(3 - x) + (29*Log[320 - 3
*Log[5]^2 + Log[5]^3 - 12*Log[25] + 12*Log[5]*(14 + Log[25]) + x*(64 - 2*Log[5] + Log[5]^2 + 9*Log[25])])/(64
- 2*Log[5] + Log[5]^2 + 9*Log[25]) + (15*Log[320 - 3*Log[5]^2 + Log[5]^3 - 12*Log[25] + 12*Log[5]*(14 + Log[25
]) + x*(64 - 2*Log[5] + Log[5]^2 + 9*Log[25])])/(320 - 3*Log[5]^2 + Log[5]^3 - 12*Log[25] + 12*Log[5]*(14 + Lo
g[25])) - (4*(320 - 3*Log[5]^2 + Log[5]^3 - 12*Log[25] + 12*Log[5]*(14 + Log[25]))*Log[320 - 3*Log[5]^2 + Log[
5]^3 - 12*Log[25] + 12*Log[5]*(14 + Log[25]) + x*(64 - 2*Log[5] + Log[5]^2 + 9*Log[25])])/(64 - 2*Log[5] + Log
[5]^2 + 9*Log[25])^2 - (4*(320 - 3*Log[5]^2 + Log[5]^3 - 12*Log[25] + 12*Log[5]*(14 + Log[25]))^2*Log[320 - 3*
Log[5]^2 + Log[5]^3 - 12*Log[25] + 12*Log[5]*(14 + Log[25]) + x*(64 - 2*Log[5] + Log[5]^2 + 9*Log[25])])/(64 -
2*Log[5] + Log[5]^2 + 9*Log[25])^3 - (29*Log[-128*x + 4*x*Log[5] - 2*x*Log[5]^2 - 18*x*Log[25] + 3*(128 - Log
[25] + Log[5]*(34 + Log[25]))])/(64 - 2*Log[5] + Log[5]^2 + 9*Log[25]) + (4*(320 - 3*Log[5]^2 + Log[5]^3 - 12*
Log[25] + 12*Log[5]*(14 + Log[25]))*Log[-128*x + 4*x*Log[5] - 2*x*Log[5]^2 - 18*x*Log[25] + 3*(128 - Log[25] +
Log[5]*(34 + Log[25]))])/(64 - 2*Log[5] + Log[5]^2 + 9*Log[25])^2 + (20*(704 - 3*Log[5]^2 + Log[5]^3 - 15*Log
[25] + 15*Log[5]*(18 + Log[25]))*Log[-128*x + 4*x*Log[5] - 2*x*Log[5]^2 - 18*x*Log[25] + 3*(128 - Log[25] + Lo
g[5]*(34 + Log[25]))])/(3*(128 - Log[25] + Log[5]*(34 + Log[25]))^2) - (3*(212992 - 8192*Log[25] + 51*Log[25]^
2 + 4*Log[5]^4*(34 + Log[25]) + Log[5]*(155648 + 5684*Log[25] - 102*Log[25]^2) + 3*Log[5]^2*(8260 + 840*Log[25
] + 17*Log[25]^2) + 8*Log[5]^3*(13 - Log[625]))*Log[-128*x + 4*x*Log[5] - 2*x*Log[5]^2 - 18*x*Log[25] + 3*(128
- Log[25] + Log[5]*(34 + Log[25]))])/(64 - 2*Log[5] + Log[5]^2 + 9*Log[25])^3 + (Log[5]*Log[15 - x^2 - x*(2 +
Log[5]) + Log[125]])/(6*(5 + Log[5]))

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1586

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]

Rule 2063

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> Dist[1/(3^(3*p)*a^(2*p)), Int[(3*a - b*x)^p*(3*a +
2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d}, x] && EqQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]

Rule 2067

Int[(P3_)^(p_), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3
, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x,
x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]

Rule 2076

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_.), x_Symbol] :> Dist[1/(3^(3*p)*a^(2*p))
, Int[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[4*b^3 + 27*
a^2*d, 0] && IntegerQ[p]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 2495

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(
h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]

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 {-15+29 x+4 x^2-4 x^3+\left (-3+7 x-2 x^2\right ) \log (5)+e^{2 x^2} \left (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+\left (-5 x+12 x^2-28 x^3+8 x^4\right ) \log (5)\right )+\left (25 x+5 x^2+5 x \log (5)\right ) \log \left (5 x+x^2+x \log (5)\right )}{x \left (x^3+x^2 (-1+\log (5))+9 (5+\log (5))-3 x (7+\log (25))\right )} \, dx\\ &=\int \frac {-15+29 x+4 x^2-4 x^3+\left (-3+7 x-2 x^2\right ) \log (5)+e^{2 x^2} \left (-25 x+55 x^2-128 x^3+12 x^4+8 x^5+\left (-5 x+12 x^2-28 x^3+8 x^4\right ) \log (5)\right )+\left (25 x+5 x^2+5 x \log (5)\right ) \log \left (5 x+x^2+x \log (5)\right )}{x \left (x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))\right )} \, dx\\ &=\int \left (\frac {29}{x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))}+\frac {4 x}{x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))}+\frac {e^{2 x^2} \left (5-12 x+28 x^2-8 x^3\right ) (-5-x-\log (5))}{x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))}+\frac {(1-2 x) (-3+x) \log (5)}{x \left (x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))\right )}+\frac {15}{x \left (-x^3+x^2 (1-\log (5))-9 (5+\log (5))+3 x (7+\log (25))\right )}+\frac {4 x^2}{-x^3+x^2 (1-\log (5))-9 (5+\log (5))+3 x (7+\log (25))}+\frac {5 (5+x+\log (5)) \log (x (5+x+\log (5)))}{x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))}\right ) \, dx\\ &=4 \int \frac {x}{x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))} \, dx+4 \int \frac {x^2}{-x^3+x^2 (1-\log (5))-9 (5+\log (5))+3 x (7+\log (25))} \, dx+5 \int \frac {(5+x+\log (5)) \log (x (5+x+\log (5)))}{x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))} \, dx+15 \int \frac {1}{x \left (-x^3+x^2 (1-\log (5))-9 (5+\log (5))+3 x (7+\log (25))\right )} \, dx+29 \int \frac {1}{x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))} \, dx+\log (5) \int \frac {(1-2 x) (-3+x)}{x \left (x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))\right )} \, dx+\int \frac {e^{2 x^2} \left (5-12 x+28 x^2-8 x^3\right ) (-5-x-\log (5))}{x^3-x^2 (1-\log (5))+9 (5+\log (5))-3 x (7+\log (25))} \, dx\\ &=4 \operatorname {Subst}\left (\int \frac {x+\frac {1}{3} (1-\log (5))}{x^3-\frac {1}{3} x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )+\frac {1}{27} \left (-2 (1-\log (5))^3+243 (5+\log (5))-27 (1-\log (5)) (7+\log (25))\right )} \, dx,x,x+\frac {1}{3} (-1+\log (5))\right )+4 \operatorname {Subst}\left (\int \frac {\left (x+\frac {1}{3} (1-\log (5))\right )^2}{-x^3+\frac {1}{3} x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )+\frac {1}{27} \left (2 (1-\log (5))^3-243 (5+\log (5))+27 (1-\log (5)) (7+\log (25))\right )} \, dx,x,x+\frac {1}{3} (-1+\log (5))\right )+5 \int \frac {\log (x (5+x+\log (5)))}{9-6 x+x^2+6 \log (5)-3 \log (25)} \, dx+15 \operatorname {Subst}\left (\int \frac {1}{\left (x+\frac {1}{3} (1-\log (5))\right ) \left (-x^3+\frac {1}{3} x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )+\frac {1}{27} \left (2 (1-\log (5))^3-243 (5+\log (5))+27 (1-\log (5)) (7+\log (25))\right )\right )} \, dx,x,x+\frac {1}{3} (-1+\log (5))\right )+29 \operatorname {Subst}\left (\int \frac {1}{x^3-\frac {1}{3} x \left (64-2 \log (5)+\log ^2(5)+9 \log (25)\right )+\frac {1}{27} \left (-2 (1-\log (5))^3+243 (5+\log (5))-27 (1-\log (5)) (7+\log (25))\right )} \, dx,x,x+\frac {1}{3} (-1+\log (5))\right )+\log (5) \int \frac {1-2 x}{x \left (-15+x^2+3 \log (5)+x (2+\log (5))-3 \log (25)\right )} \, dx+\int \frac {e^{2 x^2} \left (5-12 x+28 x^2-8 x^3\right )}{-9+6 x-x^2-6 \log (5)+3 \log (25)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.35, size = 76, normalized size = 2.45 \begin {gather*} \frac {e^{2 x^2} \left (-1+2 x+8 \log ^3(5)-4 \log ^2(5) (-14+\log (25))-76 \log (25)-4 \log (5) (-38+7 \log (25))\right )}{-3+x}-2 \log (x)-2 \log (5+x+\log (5))-\frac {5 \log (x (5+x+\log (5)))}{-3+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15 + 29*x + 4*x^2 - 4*x^3 + (-3 + 7*x - 2*x^2)*Log[5] + E^(2*x^2)*(-25*x + 55*x^2 - 128*x^3 + 12*x
^4 + 8*x^5 + (-5*x + 12*x^2 - 28*x^3 + 8*x^4)*Log[5]) + (25*x + 5*x^2 + 5*x*Log[5])*Log[5*x + x^2 + x*Log[5]])
/(45*x - 21*x^2 - x^3 + x^4 + (9*x - 6*x^2 + x^3)*Log[5]),x]

[Out]

(E^(2*x^2)*(-1 + 2*x + 8*Log[5]^3 - 4*Log[5]^2*(-14 + Log[25]) - 76*Log[25] - 4*Log[5]*(-38 + 7*Log[25])))/(-3
+ x) - 2*Log[x] - 2*Log[5 + x + Log[5]] - (5*Log[x*(5 + x + Log[5])])/(-3 + x)

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fricas [A]  time = 1.10, size = 38, normalized size = 1.23 \begin {gather*} \frac {{\left (2 \, x - 1\right )} e^{\left (2 \, x^{2}\right )} - {\left (2 \, x - 1\right )} \log \left (x^{2} + x \log \relax (5) + 5 \, x\right )}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x*log(5)+5*x^2+25*x)*log(x*log(5)+x^2+5*x)+((8*x^4-28*x^3+12*x^2-5*x)*log(5)+8*x^5+12*x^4-128*x^
3+55*x^2-25*x)*exp(x^2)^2+(-2*x^2+7*x-3)*log(5)-4*x^3+4*x^2+29*x-15)/((x^3-6*x^2+9*x)*log(5)+x^4-x^3-21*x^2+45
*x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.45, size = 64, normalized size = 2.06 \begin {gather*} \frac {2 \, x e^{\left (2 \, x^{2}\right )} - 2 \, x \log \left (x + \log \relax (5) + 5\right ) - 2 \, x \log \relax (x) - e^{\left (2 \, x^{2}\right )} - 5 \, \log \left (x^{2} + x \log \relax (5) + 5 \, x\right ) + 6 \, \log \left (x + \log \relax (5) + 5\right ) + 6 \, \log \relax (x)}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x*log(5)+5*x^2+25*x)*log(x*log(5)+x^2+5*x)+((8*x^4-28*x^3+12*x^2-5*x)*log(5)+8*x^5+12*x^4-128*x^
3+55*x^2-25*x)*exp(x^2)^2+(-2*x^2+7*x-3)*log(5)-4*x^3+4*x^2+29*x-15)/((x^3-6*x^2+9*x)*log(5)+x^4-x^3-21*x^2+45
*x),x, algorithm="giac")

[Out]

(2*x*e^(2*x^2) - 2*x*log(x + log(5) + 5) - 2*x*log(x) - e^(2*x^2) - 5*log(x^2 + x*log(5) + 5*x) + 6*log(x + lo
g(5) + 5) + 6*log(x))/(x - 3)

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maple [C]  time = 0.28, size = 161, normalized size = 5.19

method	result	size
risch	\(-\frac {5 \ln \left (x +\ln \relax (5)+5\right )}{x -3}-\frac {-5 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x +\ln \relax (5)+5\right )\right ) \mathrm {csgn}\left (i x \left (x +\ln \relax (5)+5\right )\right )+5 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x +\ln \relax (5)+5\right )\right )^{2}+5 i \pi \,\mathrm {csgn}\left (i \left (x +\ln \relax (5)+5\right )\right ) \mathrm {csgn}\left (i x \left (x +\ln \relax (5)+5\right )\right )^{2}-5 i \pi \mathrm {csgn}\left (i x \left (x +\ln \relax (5)+5\right )\right )^{3}-4 x \,{\mathrm e}^{2 x^{2}}+4 \ln \left (x^{2}+\left (5+\ln \relax (5)\right ) x \right ) x +2 \,{\mathrm e}^{2 x^{2}}-12 \ln \left (x^{2}+\left (5+\ln \relax (5)\right ) x \right )+10 \ln \relax (x )}{2 \left (x -3\right )}\)	\(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x*ln(5)+5*x^2+25*x)*ln(x*ln(5)+x^2+5*x)+((8*x^4-28*x^3+12*x^2-5*x)*ln(5)+8*x^5+12*x^4-128*x^3+55*x^2-2
5*x)*exp(x^2)^2+(-2*x^2+7*x-3)*ln(5)-4*x^3+4*x^2+29*x-15)/((x^3-6*x^2+9*x)*ln(5)+x^4-x^3-21*x^2+45*x),x,method
=_RETURNVERBOSE)

[Out]

-5/(x-3)*ln(x+ln(5)+5)-1/2*(-5*I*Pi*csgn(I*x)*csgn(I*(x+ln(5)+5))*csgn(I*x*(x+ln(5)+5))+5*I*Pi*csgn(I*x)*csgn(
I*x*(x+ln(5)+5))^2+5*I*Pi*csgn(I*(x+ln(5)+5))*csgn(I*x*(x+ln(5)+5))^2-5*I*Pi*csgn(I*x*(x+ln(5)+5))^3-4*x*exp(2
*x^2)+4*ln(x^2+(5+ln(5))*x)*x+2*exp(2*x^2)-12*ln(x^2+(5+ln(5))*x)+10*ln(x))/(x-3)

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maxima [B]  time = 0.85, size = 464, normalized size = 14.97 \begin {gather*} 2 \, {\left (\frac {{\left (\log \relax (5) + 5\right )} \log \left (x + \log \relax (5) + 5\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} - \frac {{\left (\log \relax (5) + 5\right )} \log \left (x - 3\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} + \frac {3}{x {\left (\log \relax (5) + 8\right )} - 3 \, \log \relax (5) - 24}\right )} \log \relax (5) + \frac {1}{3} \, {\left (\frac {{\left (\log \relax (5) + 11\right )} \log \left (x - 3\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} + \frac {9 \, \log \left (x + \log \relax (5) + 5\right )}{\log \relax (5)^{3} + 21 \, \log \relax (5)^{2} + 144 \, \log \relax (5) + 320} - \frac {\log \relax (x)}{\log \relax (5) + 5} + \frac {3}{x {\left (\log \relax (5) + 8\right )} - 3 \, \log \relax (5) - 24}\right )} \log \relax (5) + 7 \, {\left (\frac {\log \left (x + \log \relax (5) + 5\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} - \frac {\log \left (x - 3\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} - \frac {1}{x {\left (\log \relax (5) + 8\right )} - 3 \, \log \relax (5) - 24}\right )} \log \relax (5) - \frac {4 \, {\left (\log \relax (5)^{2} + 10 \, \log \relax (5) + 25\right )} \log \left (x + \log \relax (5) + 5\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} - \frac {4 \, {\left (\log \relax (5) + 5\right )} \log \left (x + \log \relax (5) + 5\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} - \frac {12 \, {\left (2 \, \log \relax (5) + 13\right )} \log \left (x - 3\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} + \frac {5 \, {\left (\log \relax (5) + 11\right )} \log \left (x - 3\right )}{3 \, {\left (\log \relax (5)^{2} + 16 \, \log \relax (5) + 64\right )}} + \frac {5 \, {\left (\log \relax (5) + 11\right )} \log \left (x - 3\right )}{3 \, {\left (\log \relax (5) + 8\right )}} + \frac {4 \, {\left (\log \relax (5) + 5\right )} \log \left (x - 3\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} + \frac {{\left (2 \, x - 1\right )} e^{\left (2 \, x^{2}\right )} - 5 \, \log \left (x + \log \relax (5) + 5\right ) - 5 \, \log \relax (x)}{x - 3} + \frac {15 \, \log \left (x + \log \relax (5) + 5\right )}{\log \relax (5)^{3} + 21 \, \log \relax (5)^{2} + 144 \, \log \relax (5) + 320} + \frac {29 \, \log \left (x + \log \relax (5) + 5\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} - \frac {5 \, \log \left (x + \log \relax (5) + 5\right )}{\log \relax (5) + 8} - \frac {29 \, \log \left (x - 3\right )}{\log \relax (5)^{2} + 16 \, \log \relax (5) + 64} - \frac {5 \, \log \relax (x)}{3 \, {\left (\log \relax (5) + 5\right )}} - \frac {5}{3} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x*log(5)+5*x^2+25*x)*log(x*log(5)+x^2+5*x)+((8*x^4-28*x^3+12*x^2-5*x)*log(5)+8*x^5+12*x^4-128*x^
3+55*x^2-25*x)*exp(x^2)^2+(-2*x^2+7*x-3)*log(5)-4*x^3+4*x^2+29*x-15)/((x^3-6*x^2+9*x)*log(5)+x^4-x^3-21*x^2+45
*x),x, algorithm="maxima")

[Out]

2*((log(5) + 5)*log(x + log(5) + 5)/(log(5)^2 + 16*log(5) + 64) - (log(5) + 5)*log(x - 3)/(log(5)^2 + 16*log(5
) + 64) + 3/(x*(log(5) + 8) - 3*log(5) - 24))*log(5) + 1/3*((log(5) + 11)*log(x - 3)/(log(5)^2 + 16*log(5) + 6
4) + 9*log(x + log(5) + 5)/(log(5)^3 + 21*log(5)^2 + 144*log(5) + 320) - log(x)/(log(5) + 5) + 3/(x*(log(5) +
8) - 3*log(5) - 24))*log(5) + 7*(log(x + log(5) + 5)/(log(5)^2 + 16*log(5) + 64) - log(x - 3)/(log(5)^2 + 16*l
og(5) + 64) - 1/(x*(log(5) + 8) - 3*log(5) - 24))*log(5) - 4*(log(5)^2 + 10*log(5) + 25)*log(x + log(5) + 5)/(
log(5)^2 + 16*log(5) + 64) - 4*(log(5) + 5)*log(x + log(5) + 5)/(log(5)^2 + 16*log(5) + 64) - 12*(2*log(5) + 1
3)*log(x - 3)/(log(5)^2 + 16*log(5) + 64) + 5/3*(log(5) + 11)*log(x - 3)/(log(5)^2 + 16*log(5) + 64) + 5/3*(lo
g(5) + 11)*log(x - 3)/(log(5) + 8) + 4*(log(5) + 5)*log(x - 3)/(log(5)^2 + 16*log(5) + 64) + ((2*x - 1)*e^(2*x
^2) - 5*log(x + log(5) + 5) - 5*log(x))/(x - 3) + 15*log(x + log(5) + 5)/(log(5)^3 + 21*log(5)^2 + 144*log(5)
+ 320) + 29*log(x + log(5) + 5)/(log(5)^2 + 16*log(5) + 64) - 5*log(x + log(5) + 5)/(log(5) + 8) - 29*log(x -
3)/(log(5)^2 + 16*log(5) + 64) - 5/3*log(x)/(log(5) + 5) - 5/3*log(x)

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mupad [B]  time = 1.31, size = 71, normalized size = 2.29 \begin {gather*} \ln \left (x+\ln \relax (5)+5\right )\,\left (\frac {\ln \left (25\right )+10}{\ln \relax (5)+5}-4\right )-\frac {5\,\ln \left (5\,x+x\,\ln \relax (5)+x^2\right )}{x-3}-\frac {\ln \relax (x)\,\left (\ln \left (25\right )+10\right )}{\ln \relax (5)+5}+\frac {{\mathrm {e}}^{2\,x^2}\,\left (2\,x-1\right )}{x-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5)*(2*x^2 - 7*x + 3) - 29*x + exp(2*x^2)*(25*x + log(5)*(5*x - 12*x^2 + 28*x^3 - 8*x^4) - 55*x^2 + 1
28*x^3 - 12*x^4 - 8*x^5) - log(5*x + x*log(5) + x^2)*(25*x + 5*x*log(5) + 5*x^2) - 4*x^2 + 4*x^3 + 15)/(45*x +
log(5)*(9*x - 6*x^2 + x^3) - 21*x^2 - x^3 + x^4),x)

[Out]

log(x + log(5) + 5)*((log(25) + 10)/(log(5) + 5) - 4) - (5*log(5*x + x*log(5) + x^2))/(x - 3) - (log(x)*(log(2
5) + 10))/(log(5) + 5) + (exp(2*x^2)*(2*x - 1))/(x - 3)

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sympy [A]  time = 0.70, size = 46, normalized size = 1.48 \begin {gather*} - 2 \log {\left (x^{2} + x \left (\log {\relax (5 )} + 5\right ) \right )} + \frac {\left (2 x - 1\right ) e^{2 x^{2}}}{x - 3} - \frac {5 \log {\left (x^{2} + x \log {\relax (5 )} + 5 x \right )}}{x - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x*ln(5)+5*x**2+25*x)*ln(x*ln(5)+x**2+5*x)+((8*x**4-28*x**3+12*x**2-5*x)*ln(5)+8*x**5+12*x**4-128
*x**3+55*x**2-25*x)*exp(x**2)**2+(-2*x**2+7*x-3)*ln(5)-4*x**3+4*x**2+29*x-15)/((x**3-6*x**2+9*x)*ln(5)+x**4-x*
*3-21*x**2+45*x),x)

[Out]

-2*log(x**2 + x*(log(5) + 5)) + (2*x - 1)*exp(2*x**2)/(x - 3) - 5*log(x**2 + x*log(5) + 5*x)/(x - 3)

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