3.74.19 \(\int \frac {-60000 x^3-50300 x^4+6904 x^5-292 x^6+4 x^7+e (-30000 x+32450 x^2-3748 x^3+150 x^4-2 x^5)+e^2 (-28800 x^3+3600 x^4-148 x^5+2 x^6)+(-62500 x^2-52500 x^3+7100 x^4-296 x^5+4 x^6+e (-31250+33750 x-3850 x^2+152 x^3-2 x^4)+e^2 (-30000 x^2+3700 x^3-150 x^4+2 x^5)) \log (\frac {e-e^2 x-2 x^2}{x})}{-31250 x^3+3750 x^4-150 x^5+2 x^6+e (15625 x-1875 x^2+75 x^3-x^4)+e^2 (-15625 x^2+1875 x^3-75 x^4+x^5)} \, dx\)

Optimal. Leaf size=32 \[ \left (-x+\frac {x}{25-x}-\log \left (-e^2+\frac {e}{x}-2 x\right )\right )^2 \]

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Rubi [C]  time = 6.76, antiderivative size = 791, normalized size of antiderivative = 24.72, number of steps used = 62, number of rules used = 24, integrand size = 237, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {6688, 12, 6728, 1628, 634, 618, 206, 628, 2528, 2523, 1657, 2525, 2524, 2357, 2301, 2316, 2315, 2317, 2391, 2418, 2394, 2390, 2393, 2392} \begin {gather*} -2 \text {Li}_2\left (-\frac {4 x-\sqrt {e \left (8+e^3\right )}+e^2}{2 \sqrt {e \left (8+e^3\right )}}\right )-2 \text {Li}_2\left (\frac {4 x+\sqrt {e \left (8+e^3\right )}+e^2}{2 \sqrt {e \left (8+e^3\right )}}\right )+x^2-\frac {e \left (4+1200 e-e^2+25 e^3\right ) \log \left (-2 x^2-e^2 x+e\right )}{2 \left (1250-e+25 e^2\right )}+\frac {25 \left (100+e^2\right ) \log \left (-2 x^2-e^2 x+e\right )}{1250-e+25 e^2}+\frac {1}{2} e^2 \log \left (-2 x^2-e^2 x+e\right )+2 x-\frac {1300}{25-x}+\frac {625}{(25-x)^2}-\log ^2(x)-\log ^2\left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right )-\log ^2\left (4 x+\sqrt {e \left (8+e^3\right )}+e^2\right )+2 x \log \left (-2 x+\frac {e}{x}-e^2\right )-\frac {50 \log \left (-2 x+\frac {e}{x}-e^2\right )}{25-x}-2 \log \left (-2 x+\frac {e}{x}-e^2\right ) \log (x)+2 \log \left (e^2+\sqrt {e \left (8+e^3\right )}\right ) \log (x)-2 \log (x)+2 \log \left (-2 x+\frac {e}{x}-e^2\right ) \log \left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right )+2 \log \left (-\frac {4 x}{e^2-\sqrt {e \left (8+e^3\right )}}\right ) \log \left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right )+2 \log \left (\frac {1}{4} \left (\sqrt {e \left (8+e^3\right )}-e^2\right )\right ) \log \left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right )+2 \log \left (-2 x+\frac {e}{x}-e^2\right ) \log \left (4 x+\sqrt {e \left (8+e^3\right )}+e^2\right )-2 \log \left (-\frac {4 x-\sqrt {e \left (8+e^3\right )}+e^2}{2 \sqrt {e \left (8+e^3\right )}}\right ) \log \left (4 x+\sqrt {e \left (8+e^3\right )}+e^2\right )-2 \log \left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right ) \log \left (\frac {4 x+\sqrt {e \left (8+e^3\right )}+e^2}{2 \sqrt {e \left (8+e^3\right )}}\right )+2 \log (x) \log \left (\frac {4 x}{e^2+\sqrt {e \left (8+e^3\right )}}+1\right )-\frac {50 \sqrt {e \left (8+e^3\right )} \tanh ^{-1}\left (\frac {4 x+e^2}{\sqrt {e \left (8+e^3\right )}}\right )}{1250-e+25 e^2}+\sqrt {e \left (8+e^3\right )} \tanh ^{-1}\left (\frac {4 x+e^2}{\sqrt {e \left (8+e^3\right )}}\right )-\frac {(2+e) \left (4-2 e+e^2\right ) \left (1200-e+25 e^2\right ) \sqrt {\frac {e}{8+e^3}} \tanh ^{-1}\left (\frac {4 x+e^2}{\sqrt {e \left (8+e^3\right )}}\right )}{1250-e+25 e^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-60000*x^3 - 50300*x^4 + 6904*x^5 - 292*x^6 + 4*x^7 + E*(-30000*x + 32450*x^2 - 3748*x^3 + 150*x^4 - 2*x^
5) + E^2*(-28800*x^3 + 3600*x^4 - 148*x^5 + 2*x^6) + (-62500*x^2 - 52500*x^3 + 7100*x^4 - 296*x^5 + 4*x^6 + E*
(-31250 + 33750*x - 3850*x^2 + 152*x^3 - 2*x^4) + E^2*(-30000*x^2 + 3700*x^3 - 150*x^4 + 2*x^5))*Log[(E - E^2*
x - 2*x^2)/x])/(-31250*x^3 + 3750*x^4 - 150*x^5 + 2*x^6 + E*(15625*x - 1875*x^2 + 75*x^3 - x^4) + E^2*(-15625*
x^2 + 1875*x^3 - 75*x^4 + x^5)),x]

[Out]

625/(25 - x)^2 - 1300/(25 - x) + 2*x + x^2 - ((2 + E)*(4 - 2*E + E^2)*(1200 - E + 25*E^2)*Sqrt[E/(8 + E^3)]*Ar
cTanh[(E^2 + 4*x)/Sqrt[E*(8 + E^3)]])/(1250 - E + 25*E^2) + Sqrt[E*(8 + E^3)]*ArcTanh[(E^2 + 4*x)/Sqrt[E*(8 +
E^3)]] - (50*Sqrt[E*(8 + E^3)]*ArcTanh[(E^2 + 4*x)/Sqrt[E*(8 + E^3)]])/(1250 - E + 25*E^2) - (50*Log[-E^2 + E/
x - 2*x])/(25 - x) + 2*x*Log[-E^2 + E/x - 2*x] - 2*Log[x] + 2*Log[E^2 + Sqrt[E*(8 + E^3)]]*Log[x] - 2*Log[-E^2
+ E/x - 2*x]*Log[x] - Log[x]^2 + 2*Log[(-E^2 + Sqrt[E*(8 + E^3)])/4]*Log[E^2 - Sqrt[E*(8 + E^3)] + 4*x] + 2*L
og[-E^2 + E/x - 2*x]*Log[E^2 - Sqrt[E*(8 + E^3)] + 4*x] + 2*Log[(-4*x)/(E^2 - Sqrt[E*(8 + E^3)])]*Log[E^2 - Sq
rt[E*(8 + E^3)] + 4*x] - Log[E^2 - Sqrt[E*(8 + E^3)] + 4*x]^2 + 2*Log[-E^2 + E/x - 2*x]*Log[E^2 + Sqrt[E*(8 +
E^3)] + 4*x] - 2*Log[-1/2*(E^2 - Sqrt[E*(8 + E^3)] + 4*x)/Sqrt[E*(8 + E^3)]]*Log[E^2 + Sqrt[E*(8 + E^3)] + 4*x
] - Log[E^2 + Sqrt[E*(8 + E^3)] + 4*x]^2 - 2*Log[E^2 - Sqrt[E*(8 + E^3)] + 4*x]*Log[(E^2 + Sqrt[E*(8 + E^3)] +
4*x)/(2*Sqrt[E*(8 + E^3)])] + 2*Log[x]*Log[1 + (4*x)/(E^2 + Sqrt[E*(8 + E^3)])] + (E^2*Log[E - E^2*x - 2*x^2]
)/2 + (25*(100 + E^2)*Log[E - E^2*x - 2*x^2])/(1250 - E + 25*E^2) - (E*(4 + 1200*E - E^2 + 25*E^3)*Log[E - E^2
*x - 2*x^2])/(2*(1250 - E + 25*E^2)) - 2*PolyLog[2, -1/2*(E^2 - Sqrt[E*(8 + E^3)] + 4*x)/Sqrt[E*(8 + E^3)]] -
2*PolyLog[2, (E^2 + Sqrt[E*(8 + E^3)] + 4*x)/(2*Sqrt[E*(8 + E^3)])]

Rule 12

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

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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2316

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

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2392

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (e^2 x^2 \left (600-50 x+x^2\right )-e \left (-625+650 x-51 x^2+x^3\right )+2 x^2 \left (625+550 x-49 x^2+x^3\right )\right ) \left ((-24+x) x+(-25+x) \log \left (-e^2+\frac {e}{x}-2 x\right )\right )}{(25-x)^3 x \left (e-e^2 x-2 x^2\right )} \, dx\\ &=2 \int \frac {\left (e^2 x^2 \left (600-50 x+x^2\right )-e \left (-625+650 x-51 x^2+x^3\right )+2 x^2 \left (625+550 x-49 x^2+x^3\right )\right ) \left ((-24+x) x+(-25+x) \log \left (-e^2+\frac {e}{x}-2 x\right )\right )}{(25-x)^3 x \left (e-e^2 x-2 x^2\right )} \, dx\\ &=2 \int \left (\frac {(24-x) \left (-625 e+650 e x-1250 \left (1+\frac {3 e (17+200 e)}{1250}\right ) x^2-1100 \left (1-\frac {e (1+50 e)}{1100}\right ) x^3+98 \left (1-\frac {e^2}{98}\right ) x^4-2 x^5\right )}{(25-x)^3 \left (e-e^2 x-2 x^2\right )}+\frac {\left (-625 e+650 e x-1250 \left (1+\frac {3 e (17+200 e)}{1250}\right ) x^2-1100 \left (1-\frac {e (1+50 e)}{1100}\right ) x^3+98 \left (1-\frac {e^2}{98}\right ) x^4-2 x^5\right ) \log \left (-e^2+\frac {e}{x}-2 x\right )}{(25-x)^2 x \left (e-e^2 x-2 x^2\right )}\right ) \, dx\\ &=2 \int \frac {(24-x) \left (-625 e+650 e x-1250 \left (1+\frac {3 e (17+200 e)}{1250}\right ) x^2-1100 \left (1-\frac {e (1+50 e)}{1100}\right ) x^3+98 \left (1-\frac {e^2}{98}\right ) x^4-2 x^5\right )}{(25-x)^3 \left (e-e^2 x-2 x^2\right )} \, dx+2 \int \frac {\left (-625 e+650 e x-1250 \left (1+\frac {3 e (17+200 e)}{1250}\right ) x^2-1100 \left (1-\frac {e (1+50 e)}{1100}\right ) x^3+98 \left (1-\frac {e^2}{98}\right ) x^4-2 x^5\right ) \log \left (-e^2+\frac {e}{x}-2 x\right )}{(25-x)^2 x \left (e-e^2 x-2 x^2\right )} \, dx\\ &=2 \int \left (2-\frac {625}{(-25+x)^3}-\frac {650}{(-25+x)^2}+\frac {1250+e}{\left (1250-e+25 e^2\right ) (-25+x)}+x+\frac {e \left (-2400+2 e-49 e^2+\left (4+1200 e-e^2+25 e^3\right ) x\right )}{\left (1250-e+25 e^2\right ) \left (e-e^2 x-2 x^2\right )}\right ) \, dx+2 \int \left (\log \left (-e^2+\frac {e}{x}-2 x\right )-\frac {25 \log \left (-e^2+\frac {e}{x}-2 x\right )}{(-25+x)^2}-\frac {\log \left (-e^2+\frac {e}{x}-2 x\right )}{x}+\frac {\left (e^2+4 x\right ) \log \left (-e^2+\frac {e}{x}-2 x\right )}{-e+e^2 x+2 x^2}\right ) \, dx\\ &=\frac {625}{(25-x)^2}-\frac {1300}{25-x}+4 x+x^2+\frac {2 (1250+e) \log (25-x)}{1250-e+25 e^2}+2 \int \log \left (-e^2+\frac {e}{x}-2 x\right ) \, dx-2 \int \frac {\log \left (-e^2+\frac {e}{x}-2 x\right )}{x} \, dx+2 \int \frac {\left (e^2+4 x\right ) \log \left (-e^2+\frac {e}{x}-2 x\right )}{-e+e^2 x+2 x^2} \, dx-50 \int \frac {\log \left (-e^2+\frac {e}{x}-2 x\right )}{(-25+x)^2} \, dx+\frac {(2 e) \int \frac {-2400+2 e-49 e^2+\left (4+1200 e-e^2+25 e^3\right ) x}{e-e^2 x-2 x^2} \, dx}{1250-e+25 e^2}\\ &=\frac {625}{(25-x)^2}-\frac {1300}{25-x}+4 x+x^2-\frac {50 \log \left (-e^2+\frac {e}{x}-2 x\right )}{25-x}+2 x \log \left (-e^2+\frac {e}{x}-2 x\right )+\frac {2 (1250+e) \log (25-x)}{1250-e+25 e^2}-2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log (x)-2 \int \frac {-e-2 x^2}{e-e^2 x-2 x^2} \, dx+2 \int \left (\frac {4 \log \left (-e^2+\frac {e}{x}-2 x\right )}{e^2-\sqrt {e \left (8+e^3\right )}+4 x}+\frac {4 \log \left (-e^2+\frac {e}{x}-2 x\right )}{e^2+\sqrt {e \left (8+e^3\right )}+4 x}\right ) \, dx+2 \int \frac {\left (-2-\frac {e}{x^2}\right ) \log (x)}{-e^2+\frac {e}{x}-2 x} \, dx-50 \int \frac {e+2 x^2}{(25-x) x \left (e-e^2 x-2 x^2\right )} \, dx-\frac {\left (e (2+e) \left (4-2 e+e^2\right ) \left (1200-e+25 e^2\right )\right ) \int \frac {1}{e-e^2 x-2 x^2} \, dx}{2 \left (1250-e+25 e^2\right )}-\frac {\left (e \left (4+1200 e-e^2+25 e^3\right )\right ) \int \frac {-e^2-4 x}{e-e^2 x-2 x^2} \, dx}{2 \left (1250-e+25 e^2\right )}\\ &=\frac {625}{(25-x)^2}-\frac {1300}{25-x}+4 x+x^2-\frac {50 \log \left (-e^2+\frac {e}{x}-2 x\right )}{25-x}+2 x \log \left (-e^2+\frac {e}{x}-2 x\right )+\frac {2 (1250+e) \log (25-x)}{1250-e+25 e^2}-2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log (x)-\frac {e \left (4+1200 e-e^2+25 e^3\right ) \log \left (e-e^2 x-2 x^2\right )}{2 \left (1250-e+25 e^2\right )}-2 \int \left (1-\frac {2 e-e^2 x}{e-e^2 x-2 x^2}\right ) \, dx+2 \int \left (-\frac {\log (x)}{x}+\frac {\left (e^2+4 x\right ) \log (x)}{-e+e^2 x+2 x^2}\right ) \, dx+8 \int \frac {\log \left (-e^2+\frac {e}{x}-2 x\right )}{e^2-\sqrt {e \left (8+e^3\right )}+4 x} \, dx+8 \int \frac {\log \left (-e^2+\frac {e}{x}-2 x\right )}{e^2+\sqrt {e \left (8+e^3\right )}+4 x} \, dx-50 \int \left (\frac {1250+e}{25 \left (1250-e+25 e^2\right ) (-25+x)}+\frac {1}{25 x}+\frac {e \left (4+50 e+e^3\right )+2 \left (100+e^2\right ) x}{\left (1250-e+25 e^2\right ) \left (e-e^2 x-2 x^2\right )}\right ) \, dx+\frac {\left (e (2+e) \left (4-2 e+e^2\right ) \left (1200-e+25 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{e \left (8+e^3\right )-x^2} \, dx,x,-e^2-4 x\right )}{1250-e+25 e^2}\\ &=\frac {625}{(25-x)^2}-\frac {1300}{25-x}+2 x+x^2-\frac {(2+e) \left (4-2 e+e^2\right ) \left (1200-e+25 e^2\right ) \sqrt {\frac {e}{8+e^3}} \tanh ^{-1}\left (\frac {e^2+4 x}{\sqrt {e \left (8+e^3\right )}}\right )}{1250-e+25 e^2}-\frac {50 \log \left (-e^2+\frac {e}{x}-2 x\right )}{25-x}+2 x \log \left (-e^2+\frac {e}{x}-2 x\right )-2 \log (x)-2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log (x)+2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log \left (e^2-\sqrt {e \left (8+e^3\right )}+4 x\right )+2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log \left (e^2+\sqrt {e \left (8+e^3\right )}+4 x\right )-\frac {e \left (4+1200 e-e^2+25 e^3\right ) \log \left (e-e^2 x-2 x^2\right )}{2 \left (1250-e+25 e^2\right )}+2 \int \frac {2 e-e^2 x}{e-e^2 x-2 x^2} \, dx-2 \int \frac {\log (x)}{x} \, dx+2 \int \frac {\left (e^2+4 x\right ) \log (x)}{-e+e^2 x+2 x^2} \, dx-2 \int \frac {\left (-2-\frac {e}{x^2}\right ) \log \left (e^2-\sqrt {e \left (8+e^3\right )}+4 x\right )}{-e^2+\frac {e}{x}-2 x} \, dx-2 \int \frac {\left (-2-\frac {e}{x^2}\right ) \log \left (e^2+\sqrt {e \left (8+e^3\right )}+4 x\right )}{-e^2+\frac {e}{x}-2 x} \, dx-\frac {50 \int \frac {e \left (4+50 e+e^3\right )+2 \left (100+e^2\right ) x}{e-e^2 x-2 x^2} \, dx}{1250-e+25 e^2}\\ &=\frac {625}{(25-x)^2}-\frac {1300}{25-x}+2 x+x^2-\frac {(2+e) \left (4-2 e+e^2\right ) \left (1200-e+25 e^2\right ) \sqrt {\frac {e}{8+e^3}} \tanh ^{-1}\left (\frac {e^2+4 x}{\sqrt {e \left (8+e^3\right )}}\right )}{1250-e+25 e^2}-\frac {50 \log \left (-e^2+\frac {e}{x}-2 x\right )}{25-x}+2 x \log \left (-e^2+\frac {e}{x}-2 x\right )-2 \log (x)-2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log (x)-\log ^2(x)+2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log \left (e^2-\sqrt {e \left (8+e^3\right )}+4 x\right )+2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log \left (e^2+\sqrt {e \left (8+e^3\right )}+4 x\right )-\frac {e \left (4+1200 e-e^2+25 e^3\right ) \log \left (e-e^2 x-2 x^2\right )}{2 \left (1250-e+25 e^2\right )}+2 \int \left (\frac {4 \log (x)}{e^2-\sqrt {e \left (8+e^3\right )}+4 x}+\frac {4 \log (x)}{e^2+\sqrt {e \left (8+e^3\right )}+4 x}\right ) \, dx-2 \int \left (-\frac {\log \left (e^2-\sqrt {e \left (8+e^3\right )}+4 x\right )}{x}+\frac {\left (e^2+4 x\right ) \log \left (e^2-\sqrt {e \left (8+e^3\right )}+4 x\right )}{-e+e^2 x+2 x^2}\right ) \, dx-2 \int \left (-\frac {\log \left (e^2+\sqrt {e \left (8+e^3\right )}+4 x\right )}{x}+\frac {\left (e^2+4 x\right ) \log \left (e^2+\sqrt {e \left (8+e^3\right )}+4 x\right )}{-e+e^2 x+2 x^2}\right ) \, dx+\frac {1}{2} e^2 \int \frac {-e^2-4 x}{e-e^2 x-2 x^2} \, dx+\frac {\left (25 \left (100+e^2\right )\right ) \int \frac {-e^2-4 x}{e-e^2 x-2 x^2} \, dx}{1250-e+25 e^2}+\frac {1}{2} \left (e \left (8+e^3\right )\right ) \int \frac {1}{e-e^2 x-2 x^2} \, dx-\frac {\left (25 e \left (8+e^3\right )\right ) \int \frac {1}{e-e^2 x-2 x^2} \, dx}{1250-e+25 e^2}\\ &=\frac {625}{(25-x)^2}-\frac {1300}{25-x}+2 x+x^2-\frac {(2+e) \left (4-2 e+e^2\right ) \left (1200-e+25 e^2\right ) \sqrt {\frac {e}{8+e^3}} \tanh ^{-1}\left (\frac {e^2+4 x}{\sqrt {e \left (8+e^3\right )}}\right )}{1250-e+25 e^2}-\frac {50 \log \left (-e^2+\frac {e}{x}-2 x\right )}{25-x}+2 x \log \left (-e^2+\frac {e}{x}-2 x\right )-2 \log (x)-2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log (x)-\log ^2(x)+2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log \left (e^2-\sqrt {e \left (8+e^3\right )}+4 x\right )+2 \log \left (-e^2+\frac {e}{x}-2 x\right ) \log \left (e^2+\sqrt {e \left (8+e^3\right )}+4 x\right )+\frac {1}{2} e^2 \log \left (e-e^2 x-2 x^2\right )+\frac {25 \left (100+e^2\right ) \log \left (e-e^2 x-2 x^2\right )}{1250-e+25 e^2}-\frac {e \left (4+1200 e-e^2+25 e^3\right ) \log \left (e-e^2 x-2 x^2\right )}{2 \left (1250-e+25 e^2\right )}+2 \int \frac {\log \left (e^2-\sqrt {e \left (8+e^3\right )}+4 x\right )}{x} \, dx-2 \int \frac {\left (e^2+4 x\right ) \log \left (e^2-\sqrt {e \left (8+e^3\right )}+4 x\right )}{-e+e^2 x+2 x^2} \, dx+2 \int \frac {\log \left (e^2+\sqrt {e \left (8+e^3\right )}+4 x\right )}{x} \, dx-2 \int \frac {\left (e^2+4 x\right ) \log \left (e^2+\sqrt {e \left (8+e^3\right )}+4 x\right )}{-e+e^2 x+2 x^2} \, dx+8 \int \frac {\log (x)}{e^2-\sqrt {e \left (8+e^3\right )}+4 x} \, dx+8 \int \frac {\log (x)}{e^2+\sqrt {e \left (8+e^3\right )}+4 x} \, dx-\left (e \left (8+e^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{e \left (8+e^3\right )-x^2} \, dx,x,-e^2-4 x\right )+\frac {\left (50 e \left (8+e^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{e \left (8+e^3\right )-x^2} \, dx,x,-e^2-4 x\right )}{1250-e+25 e^2}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.13, size = 96, normalized size = 3.00 \begin {gather*} 2 \left (\frac {625}{2 (-25+x)^2}+\frac {650}{-25+x}+x+\frac {x^2}{2}+\frac {\left (25-25 x+x^2\right ) \log \left (-e^2+\frac {e}{x}-2 x\right )}{-25+x}+\frac {1}{2} \log ^2\left (-e^2+\frac {e}{x}-2 x\right )-\log (x)+\log \left (-e+e^2 x+2 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-60000*x^3 - 50300*x^4 + 6904*x^5 - 292*x^6 + 4*x^7 + E*(-30000*x + 32450*x^2 - 3748*x^3 + 150*x^4
- 2*x^5) + E^2*(-28800*x^3 + 3600*x^4 - 148*x^5 + 2*x^6) + (-62500*x^2 - 52500*x^3 + 7100*x^4 - 296*x^5 + 4*x^
6 + E*(-31250 + 33750*x - 3850*x^2 + 152*x^3 - 2*x^4) + E^2*(-30000*x^2 + 3700*x^3 - 150*x^4 + 2*x^5))*Log[(E
- E^2*x - 2*x^2)/x])/(-31250*x^3 + 3750*x^4 - 150*x^5 + 2*x^6 + E*(15625*x - 1875*x^2 + 75*x^3 - x^4) + E^2*(-
15625*x^2 + 1875*x^3 - 75*x^4 + x^5)),x]

[Out]

2*(625/(2*(-25 + x)^2) + 650/(-25 + x) + x + x^2/2 + ((25 - 25*x + x^2)*Log[-E^2 + E/x - 2*x])/(-25 + x) + Log
[-E^2 + E/x - 2*x]^2/2 - Log[x] + Log[-E + E^2*x + 2*x^2])

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fricas [B]  time = 0.64, size = 94, normalized size = 2.94 \begin {gather*} \frac {x^{4} - 48 \, x^{3} + {\left (x^{2} - 50 \, x + 625\right )} \log \left (-\frac {2 \, x^{2} + x e^{2} - e}{x}\right )^{2} + 525 \, x^{2} + 2 \, {\left (x^{3} - 49 \, x^{2} + 600 \, x\right )} \log \left (-\frac {2 \, x^{2} + x e^{2} - e}{x}\right ) + 2550 \, x - 31875}{x^{2} - 50 \, x + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5-150*x^4+3700*x^3-30000*x^2)*exp(2)+(-2*x^4+152*x^3-3850*x^2+33750*x-31250)*exp(1)+4*x^6-296
*x^5+7100*x^4-52500*x^3-62500*x^2)*log((-exp(2)*x+exp(1)-2*x^2)/x)+(2*x^6-148*x^5+3600*x^4-28800*x^3)*exp(2)+(
-2*x^5+150*x^4-3748*x^3+32450*x^2-30000*x)*exp(1)+4*x^7-292*x^6+6904*x^5-50300*x^4-60000*x^3)/((x^5-75*x^4+187
5*x^3-15625*x^2)*exp(2)+(-x^4+75*x^3-1875*x^2+15625*x)*exp(1)+2*x^6-150*x^5+3750*x^4-31250*x^3),x, algorithm="
fricas")

[Out]

(x^4 - 48*x^3 + (x^2 - 50*x + 625)*log(-(2*x^2 + x*e^2 - e)/x)^2 + 525*x^2 + 2*(x^3 - 49*x^2 + 600*x)*log(-(2*
x^2 + x*e^2 - e)/x) + 2550*x - 31875)/(x^2 - 50*x + 625)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (2 \, x^{7} - 146 \, x^{6} + 3452 \, x^{5} - 25150 \, x^{4} - 30000 \, x^{3} + {\left (x^{6} - 74 \, x^{5} + 1800 \, x^{4} - 14400 \, x^{3}\right )} e^{2} - {\left (x^{5} - 75 \, x^{4} + 1874 \, x^{3} - 16225 \, x^{2} + 15000 \, x\right )} e + {\left (2 \, x^{6} - 148 \, x^{5} + 3550 \, x^{4} - 26250 \, x^{3} - 31250 \, x^{2} + {\left (x^{5} - 75 \, x^{4} + 1850 \, x^{3} - 15000 \, x^{2}\right )} e^{2} - {\left (x^{4} - 76 \, x^{3} + 1925 \, x^{2} - 16875 \, x + 15625\right )} e\right )} \log \left (-\frac {2 \, x^{2} + x e^{2} - e}{x}\right )\right )}}{2 \, x^{6} - 150 \, x^{5} + 3750 \, x^{4} - 31250 \, x^{3} + {\left (x^{5} - 75 \, x^{4} + 1875 \, x^{3} - 15625 \, x^{2}\right )} e^{2} - {\left (x^{4} - 75 \, x^{3} + 1875 \, x^{2} - 15625 \, x\right )} e}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5-150*x^4+3700*x^3-30000*x^2)*exp(2)+(-2*x^4+152*x^3-3850*x^2+33750*x-31250)*exp(1)+4*x^6-296
*x^5+7100*x^4-52500*x^3-62500*x^2)*log((-exp(2)*x+exp(1)-2*x^2)/x)+(2*x^6-148*x^5+3600*x^4-28800*x^3)*exp(2)+(
-2*x^5+150*x^4-3748*x^3+32450*x^2-30000*x)*exp(1)+4*x^7-292*x^6+6904*x^5-50300*x^4-60000*x^3)/((x^5-75*x^4+187
5*x^3-15625*x^2)*exp(2)+(-x^4+75*x^3-1875*x^2+15625*x)*exp(1)+2*x^6-150*x^5+3750*x^4-31250*x^3),x, algorithm="
giac")

[Out]

integrate(2*(2*x^7 - 146*x^6 + 3452*x^5 - 25150*x^4 - 30000*x^3 + (x^6 - 74*x^5 + 1800*x^4 - 14400*x^3)*e^2 -
(x^5 - 75*x^4 + 1874*x^3 - 16225*x^2 + 15000*x)*e + (2*x^6 - 148*x^5 + 3550*x^4 - 26250*x^3 - 31250*x^2 + (x^5
- 75*x^4 + 1850*x^3 - 15000*x^2)*e^2 - (x^4 - 76*x^3 + 1925*x^2 - 16875*x + 15625)*e)*log(-(2*x^2 + x*e^2 - e
)/x))/(2*x^6 - 150*x^5 + 3750*x^4 - 31250*x^3 + (x^5 - 75*x^4 + 1875*x^3 - 15625*x^2)*e^2 - (x^4 - 75*x^3 + 18
75*x^2 - 15625*x)*e), x)

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maple [B]  time = 0.36, size = 156, normalized size = 4.88

method	result	size
norman	\(\frac {x^{4}+x^{2} \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right )^{2}+28800 x -98 \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right ) x^{2}+1200 x \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right )-48 x^{3}+625 \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right )^{2}-50 x \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right )^{2}-360000+2 \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right ) x^{3}}{\left (x -25\right )^{2}}\)	\(156\)
default	error in gcdex: invalid arguments\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^5-150*x^4+3700*x^3-30000*x^2)*exp(2)+(-2*x^4+152*x^3-3850*x^2+33750*x-31250)*exp(1)+4*x^6-296*x^5+7
100*x^4-52500*x^3-62500*x^2)*ln((-exp(2)*x+exp(1)-2*x^2)/x)+(2*x^6-148*x^5+3600*x^4-28800*x^3)*exp(2)+(-2*x^5+
150*x^4-3748*x^3+32450*x^2-30000*x)*exp(1)+4*x^7-292*x^6+6904*x^5-50300*x^4-60000*x^3)/((x^5-75*x^4+1875*x^3-1
5625*x^2)*exp(2)+(-x^4+75*x^3-1875*x^2+15625*x)*exp(1)+2*x^6-150*x^5+3750*x^4-31250*x^3),x,method=_RETURNVERBO
SE)

[Out]

(x^4+x^2*ln((-exp(2)*x+exp(1)-2*x^2)/x)^2+28800*x-98*ln((-exp(2)*x+exp(1)-2*x^2)/x)*x^2+1200*x*ln((-exp(2)*x+e
xp(1)-2*x^2)/x)-48*x^3+625*ln((-exp(2)*x+exp(1)-2*x^2)/x)^2-50*x*ln((-exp(2)*x+exp(1)-2*x^2)/x)^2-360000+2*ln(
(-exp(2)*x+exp(1)-2*x^2)/x)*x^3)/(x-25)^2

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maxima [B]  time = 0.67, size = 123, normalized size = 3.84 \begin {gather*} \frac {x^{4} - 48 \, x^{3} + {\left (x^{2} - 50 \, x + 625\right )} \log \left (-2 \, x^{2} - x e^{2} + e\right )^{2} + {\left (x^{2} - 50 \, x + 625\right )} \log \relax (x)^{2} + 525 \, x^{2} + 2 \, {\left (x^{3} - 49 \, x^{2} - {\left (x^{2} - 50 \, x + 625\right )} \log \relax (x) + 600 \, x\right )} \log \left (-2 \, x^{2} - x e^{2} + e\right ) - 2 \, {\left (x^{3} - 49 \, x^{2} + 600 \, x\right )} \log \relax (x) + 2550 \, x - 31875}{x^{2} - 50 \, x + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^5-150*x^4+3700*x^3-30000*x^2)*exp(2)+(-2*x^4+152*x^3-3850*x^2+33750*x-31250)*exp(1)+4*x^6-296
*x^5+7100*x^4-52500*x^3-62500*x^2)*log((-exp(2)*x+exp(1)-2*x^2)/x)+(2*x^6-148*x^5+3600*x^4-28800*x^3)*exp(2)+(
-2*x^5+150*x^4-3748*x^3+32450*x^2-30000*x)*exp(1)+4*x^7-292*x^6+6904*x^5-50300*x^4-60000*x^3)/((x^5-75*x^4+187
5*x^3-15625*x^2)*exp(2)+(-x^4+75*x^3-1875*x^2+15625*x)*exp(1)+2*x^6-150*x^5+3750*x^4-31250*x^3),x, algorithm="
maxima")

[Out]

(x^4 - 48*x^3 + (x^2 - 50*x + 625)*log(-2*x^2 - x*e^2 + e)^2 + (x^2 - 50*x + 625)*log(x)^2 + 525*x^2 + 2*(x^3
- 49*x^2 - (x^2 - 50*x + 625)*log(x) + 600*x)*log(-2*x^2 - x*e^2 + e) - 2*(x^3 - 49*x^2 + 600*x)*log(x) + 2550
*x - 31875)/(x^2 - 50*x + 625)

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mupad [B]  time = 5.38, size = 98, normalized size = 3.06 \begin {gather*} 2\,x-48\,\ln \left (x^2+\frac {{\mathrm {e}}^2\,x}{2}-\frac {\mathrm {e}}{2}\right )+48\,\ln \relax (x)+\frac {1300\,x-31875}{x^2-50\,x+625}+{\ln \left (-\frac {2\,x^2+{\mathrm {e}}^2\,x-\mathrm {e}}{x}\right )}^2+x^2+\frac {\ln \left (-\frac {2\,x^2+{\mathrm {e}}^2\,x-\mathrm {e}}{x}\right )\,\left (x^2-600\right )}{\frac {x}{2}-\frac {25}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(-(x*exp(2) - exp(1) + 2*x^2)/x)*(exp(1)*(3850*x^2 - 33750*x - 152*x^3 + 2*x^4 + 31250) + 62500*x^2 +
52500*x^3 - 7100*x^4 + 296*x^5 - 4*x^6 + exp(2)*(30000*x^2 - 3700*x^3 + 150*x^4 - 2*x^5)) + exp(1)*(30000*x -
32450*x^2 + 3748*x^3 - 150*x^4 + 2*x^5) + 60000*x^3 + 50300*x^4 - 6904*x^5 + 292*x^6 - 4*x^7 + exp(2)*(28800*
x^3 - 3600*x^4 + 148*x^5 - 2*x^6))/(exp(1)*(15625*x - 1875*x^2 + 75*x^3 - x^4) - 31250*x^3 + 3750*x^4 - 150*x^
5 + 2*x^6 - exp(2)*(15625*x^2 - 1875*x^3 + 75*x^4 - x^5)),x)

[Out]

2*x - 48*log((x*exp(2))/2 - exp(1)/2 + x^2) + 48*log(x) + (1300*x - 31875)/(x^2 - 50*x + 625) + log(-(x*exp(2)
- exp(1) + 2*x^2)/x)^2 + x^2 + (log(-(x*exp(2) - exp(1) + 2*x^2)/x)*(x^2 - 600))/(x/2 - 25/2)

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sympy [B]  time = 4.87, size = 92, normalized size = 2.88 \begin {gather*} x^{2} + 2 x + \frac {1300 x - 31875}{x^{2} - 50 x + 625} - 2 \log {\relax (x )} + \log {\left (\frac {- 2 x^{2} - x e^{2} + e}{x} \right )}^{2} + 2 \log {\left (x^{2} + \frac {x e^{2}}{2} - \frac {e}{2} \right )} + \frac {\left (2 x^{2} - 50 x + 50\right ) \log {\left (\frac {- 2 x^{2} - x e^{2} + e}{x} \right )}}{x - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**5-150*x**4+3700*x**3-30000*x**2)*exp(2)+(-2*x**4+152*x**3-3850*x**2+33750*x-31250)*exp(1)+4*
x**6-296*x**5+7100*x**4-52500*x**3-62500*x**2)*ln((-exp(2)*x+exp(1)-2*x**2)/x)+(2*x**6-148*x**5+3600*x**4-2880
0*x**3)*exp(2)+(-2*x**5+150*x**4-3748*x**3+32450*x**2-30000*x)*exp(1)+4*x**7-292*x**6+6904*x**5-50300*x**4-600
00*x**3)/((x**5-75*x**4+1875*x**3-15625*x**2)*exp(2)+(-x**4+75*x**3-1875*x**2+15625*x)*exp(1)+2*x**6-150*x**5+
3750*x**4-31250*x**3),x)

[Out]

x**2 + 2*x + (1300*x - 31875)/(x**2 - 50*x + 625) - 2*log(x) + log((-2*x**2 - x*exp(2) + E)/x)**2 + 2*log(x**2
+ x*exp(2)/2 - E/2) + (2*x**2 - 50*x + 50)*log((-2*x**2 - x*exp(2) + E)/x)/(x - 25)

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