3.13.68 \(\int \frac {20+45 x+18 x^2+18 x^3+4 x^4+(-20-20 x-24 x^2-28 x^3-4 x^4) \log (x+x^2)}{125 x^2+125 x^3+100 x^4+100 x^5+20 x^6+20 x^7} \, dx\)

Optimal. Leaf size=25 \[ \frac {(4+x) \log \left (x+x^2\right )}{5 x \left (5+2 x^2\right )} \]

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Rubi [C]  time = 2.08, antiderivative size = 322, normalized size of antiderivative = 12.88, number of steps used = 99, number of rules used = 33, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6741, 12, 6742, 741, 801, 635, 203, 260, 894, 639, 823, 1647, 2513, 2357, 2304, 2323, 2324, 4848, 2391, 2335, 2418, 2395, 36, 29, 31, 2409, 2394, 2393, 2413, 706, 1805, 21, 30} \begin {gather*} -\frac {2 x^2 \log (x)}{25 \left (2 x^2+5\right )}-\frac {8 x \log (x)}{25 \left (2 x^2+5\right )}+\frac {\log (x+1)}{5 \left (2 x^2+5\right )}-\frac {(5-8 x) (\log (x)+\log (x+1)-\log (x (x+1)))}{25 \left (2 x^2+5\right )}-\frac {4}{175} \log \left (2 x^2+5\right )+\frac {4 \log \left (\sqrt {10}+2 i x\right )}{25 \left (2+i \sqrt {10}\right )}+\frac {\log (x)}{25}+\frac {4 \log (x+1)}{25 \left (-2 x+i \sqrt {10}\right )}-\frac {4 \log (x+1)}{25 \left (2 x+i \sqrt {10}\right )}-\frac {4 \log (x+1)}{25 \left (2+i \sqrt {10}\right )}-\frac {4 \log (x+1)}{25 \left (2-i \sqrt {10}\right )}+\frac {8}{175} \log (x+1)+\frac {4 \log \left (2 x+i \sqrt {10}\right )}{25 \left (2-i \sqrt {10}\right )}+\frac {4 \log (x)}{25 x}+\frac {4 \log (x+1)}{25 x}-\frac {4 (\log (x)+\log (x+1)-\log (x (x+1)))}{25 x}-\frac {4}{35} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20 + 45*x + 18*x^2 + 18*x^3 + 4*x^4 + (-20 - 20*x - 24*x^2 - 28*x^3 - 4*x^4)*Log[x + x^2])/(125*x^2 + 125
*x^3 + 100*x^4 + 100*x^5 + 20*x^6 + 20*x^7),x]

[Out]

(-4*Sqrt[2/5]*ArcTan[Sqrt[2/5]*x])/35 + (4*Log[Sqrt[10] + (2*I)*x])/(25*(2 + I*Sqrt[10])) + Log[x]/25 + (4*Log
[x])/(25*x) - (8*x*Log[x])/(25*(5 + 2*x^2)) - (2*x^2*Log[x])/(25*(5 + 2*x^2)) + (8*Log[1 + x])/175 - (4*Log[1
+ x])/(25*(2 - I*Sqrt[10])) - (4*Log[1 + x])/(25*(2 + I*Sqrt[10])) + (4*Log[1 + x])/(25*(I*Sqrt[10] - 2*x)) +
(4*Log[1 + x])/(25*x) - (4*Log[1 + x])/(25*(I*Sqrt[10] + 2*x)) + Log[1 + x]/(5*(5 + 2*x^2)) - (4*(Log[x] + Log
[1 + x] - Log[x*(1 + x)]))/(25*x) - ((5 - 8*x)*(Log[x] + Log[1 + x] - Log[x*(1 + x)]))/(25*(5 + 2*x^2)) + (4*L
og[I*Sqrt[10] + 2*x])/(25*(2 - I*Sqrt[10])) - (4*Log[5 + 2*x^2])/175

Rule 12

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 29

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 203

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

Rule 260

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

Rule 635

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

Rule 639

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)*(a + c*x^2)^(p + 1))/(2*a
*c*(p + 1)), x] + Dist[(d*(2*p + 3))/(2*a*(p + 1)), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 706

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

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2323

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(q + 1
)*(a + b*Log[c*x^n]))/(2*d*(q + 1)), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*Log[c*
x^n]), x], x] + Dist[(b*n)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] &&
LtQ[q, -1]

Rule 2324

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

Rule 2335

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

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 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 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 2395

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

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 2413

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(x_)^(m_.)*((f_.) + (g_.)*(x_)^(r_.))^(q_.), x_
Symbol] :> Simp[((f + g*x^r)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*r*(q + 1)), x] - Dist[(b*e*n*p)/(g*r*(q
+ 1)), Int[((f + g*x^r)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, n, q, r}, x] && EqQ[m, r - 1] && NeQ[q, -1] && IGtQ[p, 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 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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 {20+45 x+18 x^2+18 x^3+4 x^4+\left (-20-20 x-24 x^2-28 x^3-4 x^4\right ) \log \left (x+x^2\right )}{5 x^2 (1+x) \left (5+2 x^2\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {20+45 x+18 x^2+18 x^3+4 x^4+\left (-20-20 x-24 x^2-28 x^3-4 x^4\right ) \log \left (x+x^2\right )}{x^2 (1+x) \left (5+2 x^2\right )^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {18}{(1+x) \left (5+2 x^2\right )^2}+\frac {20}{x^2 (1+x) \left (5+2 x^2\right )^2}+\frac {45}{x (1+x) \left (5+2 x^2\right )^2}+\frac {18 x}{(1+x) \left (5+2 x^2\right )^2}+\frac {4 x^2}{(1+x) \left (5+2 x^2\right )^2}-\frac {4 \left (5+6 x^2+x^3\right ) \log (x (1+x))}{x^2 \left (5+2 x^2\right )^2}\right ) \, dx\\ &=\frac {4}{5} \int \frac {x^2}{(1+x) \left (5+2 x^2\right )^2} \, dx-\frac {4}{5} \int \frac {\left (5+6 x^2+x^3\right ) \log (x (1+x))}{x^2 \left (5+2 x^2\right )^2} \, dx+\frac {18}{5} \int \frac {1}{(1+x) \left (5+2 x^2\right )^2} \, dx+\frac {18}{5} \int \frac {x}{(1+x) \left (5+2 x^2\right )^2} \, dx+4 \int \frac {1}{x^2 (1+x) \left (5+2 x^2\right )^2} \, dx+9 \int \frac {1}{x (1+x) \left (5+2 x^2\right )^2} \, dx\\ &=-\frac {9 (1-x)}{35 \left (5+2 x^2\right )}+\frac {4 (5+2 x)}{175 \left (5+2 x^2\right )}-\frac {9}{350} \int \frac {10-10 x}{(1+x) \left (5+2 x^2\right )} \, dx-\frac {1}{25} \int \frac {-\frac {10}{7}+\frac {10 x}{7}}{(1+x) \left (5+2 x^2\right )} \, dx-\frac {9}{175} \int \frac {-12-2 x}{(1+x) \left (5+2 x^2\right )} \, dx-\frac {4}{5} \int \frac {\left (5+6 x^2+x^3\right ) \log (x)}{x^2 \left (5+2 x^2\right )^2} \, dx-\frac {4}{5} \int \frac {\left (5+6 x^2+x^3\right ) \log (1+x)}{x^2 \left (5+2 x^2\right )^2} \, dx+4 \int \left (\frac {1}{25 x^2}-\frac {1}{25 x}+\frac {1}{49 (1+x)}+\frac {4 (-1+x)}{35 \left (5+2 x^2\right )^2}+\frac {48 (-1+x)}{1225 \left (5+2 x^2\right )}\right ) \, dx+9 \int \left (\frac {1}{25 x}-\frac {1}{49 (1+x)}-\frac {2 (5+2 x)}{35 \left (5+2 x^2\right )^2}-\frac {2 (25+24 x)}{1225 \left (5+2 x^2\right )}\right ) \, dx+\frac {1}{5} (4 (\log (x)+\log (1+x)-\log (x (1+x)))) \int \frac {5+6 x^2+x^3}{x^2 \left (5+2 x^2\right )^2} \, dx\\ &=-\frac {4}{25 x}-\frac {9 (1-x)}{35 \left (5+2 x^2\right )}+\frac {4 (5+2 x)}{175 \left (5+2 x^2\right )}+\frac {\log (x)}{5}-\frac {5}{49} \log (1+x)-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}-\frac {18 \int \frac {25+24 x}{5+2 x^2} \, dx}{1225}-\frac {9}{350} \int \left (\frac {20}{7 (1+x)}-\frac {10 (3+4 x)}{7 \left (5+2 x^2\right )}\right ) \, dx-\frac {1}{25} \int \left (-\frac {20}{49 (1+x)}+\frac {10 (3+4 x)}{49 \left (5+2 x^2\right )}\right ) \, dx-\frac {9}{175} \int \left (-\frac {10}{7 (1+x)}+\frac {2 (-17+10 x)}{7 \left (5+2 x^2\right )}\right ) \, dx+\frac {192 \int \frac {-1+x}{5+2 x^2} \, dx}{1225}+\frac {16}{35} \int \frac {-1+x}{\left (5+2 x^2\right )^2} \, dx-\frac {18}{35} \int \frac {5+2 x}{\left (5+2 x^2\right )^2} \, dx-\frac {4}{5} \int \left (\frac {\log (x)}{5 x^2}+\frac {(4+x) \log (x)}{\left (5+2 x^2\right )^2}-\frac {2 \log (x)}{5 \left (5+2 x^2\right )}\right ) \, dx-\frac {4}{5} \int \left (\frac {\log (1+x)}{5 x^2}+\frac {(4+x) \log (1+x)}{\left (5+2 x^2\right )^2}-\frac {2 \log (1+x)}{5 \left (5+2 x^2\right )}\right ) \, dx-\frac {1}{25} (2 (\log (x)+\log (1+x)-\log (x (1+x)))) \int \frac {-10-4 x^2}{x^2 \left (5+2 x^2\right )} \, dx\\ &=-\frac {4}{25 x}+\frac {\log (x)}{5}-\frac {3}{35} \log (1+x)-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}-\frac {2}{245} \int \frac {3+4 x}{5+2 x^2} \, dx-\frac {18 \int \frac {-17+10 x}{5+2 x^2} \, dx}{1225}+\frac {9}{245} \int \frac {3+4 x}{5+2 x^2} \, dx-\frac {8}{175} \int \frac {1}{5+2 x^2} \, dx-\frac {192 \int \frac {1}{5+2 x^2} \, dx}{1225}+\frac {192 \int \frac {x}{5+2 x^2} \, dx}{1225}-\frac {4}{25} \int \frac {\log (x)}{x^2} \, dx-\frac {4}{25} \int \frac {\log (1+x)}{x^2} \, dx-\frac {9}{35} \int \frac {1}{5+2 x^2} \, dx+\frac {8}{25} \int \frac {\log (x)}{5+2 x^2} \, dx+\frac {8}{25} \int \frac {\log (1+x)}{5+2 x^2} \, dx-\frac {432 \int \frac {x}{5+2 x^2} \, dx}{1225}-\frac {18}{49} \int \frac {1}{5+2 x^2} \, dx-\frac {4}{5} \int \frac {(4+x) \log (x)}{\left (5+2 x^2\right )^2} \, dx-\frac {4}{5} \int \frac {(4+x) \log (1+x)}{\left (5+2 x^2\right )^2} \, dx+\frac {1}{25} (4 (\log (x)+\log (1+x)-\log (x (1+x)))) \int \frac {1}{x^2} \, dx\\ &=-\frac {349 \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )}{1225}-\frac {9 \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )}{35 \sqrt {10}}+\frac {\log (x)}{5}+\frac {4 \log (x)}{25 x}+\frac {4}{25} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right ) \log (x)-\frac {3}{35} \log (1+x)+\frac {4 \log (1+x)}{25 x}-\frac {4 (\log (x)+\log (1+x)-\log (x (1+x)))}{25 x}-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}-\frac {12}{245} \log \left (5+2 x^2\right )-\frac {6}{245} \int \frac {1}{5+2 x^2} \, dx-\frac {8}{245} \int \frac {x}{5+2 x^2} \, dx+\frac {27}{245} \int \frac {1}{5+2 x^2} \, dx-\frac {4}{25} \int \frac {1}{x (1+x)} \, dx+\frac {306 \int \frac {1}{5+2 x^2} \, dx}{1225}-\frac {8}{25} \int \frac {\tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )}{\sqrt {10} x} \, dx+\frac {8}{25} \int \left (\frac {i \log (1+x)}{2 \sqrt {5} \left (i \sqrt {5}-\sqrt {2} x\right )}+\frac {i \log (1+x)}{2 \sqrt {5} \left (i \sqrt {5}+\sqrt {2} x\right )}\right ) \, dx-\frac {4}{5} \int \left (\frac {4 \log (x)}{\left (5+2 x^2\right )^2}+\frac {x \log (x)}{\left (5+2 x^2\right )^2}\right ) \, dx-\frac {4}{5} \int \left (\frac {4 \log (1+x)}{\left (5+2 x^2\right )^2}+\frac {x \log (1+x)}{\left (5+2 x^2\right )^2}\right ) \, dx\\ &=-\frac {43}{175} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )+\frac {\log (x)}{5}+\frac {4 \log (x)}{25 x}+\frac {4}{25} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right ) \log (x)-\frac {3}{35} \log (1+x)+\frac {4 \log (1+x)}{25 x}-\frac {4 (\log (x)+\log (1+x)-\log (x (1+x)))}{25 x}-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}-\frac {2}{35} \log \left (5+2 x^2\right )-\frac {4}{25} \int \frac {1}{x} \, dx+\frac {4}{25} \int \frac {1}{1+x} \, dx-\frac {4}{5} \int \frac {x \log (x)}{\left (5+2 x^2\right )^2} \, dx-\frac {4}{5} \int \frac {x \log (1+x)}{\left (5+2 x^2\right )^2} \, dx-\frac {16}{5} \int \frac {\log (x)}{\left (5+2 x^2\right )^2} \, dx-\frac {16}{5} \int \frac {\log (1+x)}{\left (5+2 x^2\right )^2} \, dx-\frac {1}{25} \left (4 \sqrt {\frac {2}{5}}\right ) \int \frac {\tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )}{x} \, dx+\frac {(4 i) \int \frac {\log (1+x)}{i \sqrt {5}-\sqrt {2} x} \, dx}{25 \sqrt {5}}+\frac {(4 i) \int \frac {\log (1+x)}{i \sqrt {5}+\sqrt {2} x} \, dx}{25 \sqrt {5}}\\ &=-\frac {43}{175} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )+\frac {\log (x)}{25}+\frac {4 \log (x)}{25 x}-\frac {8 x \log (x)}{25 \left (5+2 x^2\right )}-\frac {2 x^2 \log (x)}{25 \left (5+2 x^2\right )}+\frac {4}{25} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right ) \log (x)+\frac {13}{175} \log (1+x)+\frac {4 \log (1+x)}{25 x}+\frac {\log (1+x)}{5 \left (5+2 x^2\right )}-\frac {4 (\log (x)+\log (1+x)-\log (x (1+x)))}{25 x}-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}-\frac {2}{25} i \sqrt {\frac {2}{5}} \log (1+x) \log \left (\frac {i \sqrt {5}-\sqrt {2} x}{\sqrt {2}+i \sqrt {5}}\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \log (1+x) \log \left (-\frac {i \sqrt {5}+\sqrt {2} x}{\sqrt {2}-i \sqrt {5}}\right )-\frac {2}{35} \log \left (5+2 x^2\right )+\frac {2}{25} \int \frac {x}{5+2 x^2} \, dx-\frac {1}{5} \int \frac {1}{(1+x) \left (5+2 x^2\right )} \, dx+\frac {8}{25} \int \frac {1}{5+2 x^2} \, dx-\frac {8}{25} \int \frac {\log (x)}{5+2 x^2} \, dx-\frac {16}{5} \int \left (-\frac {\log (1+x)}{10 \left (i \sqrt {10}-2 x\right )^2}-\frac {\log (1+x)}{10 \left (i \sqrt {10}+2 x\right )^2}-\frac {\log (1+x)}{5 \left (-10-4 x^2\right )}\right ) \, dx-\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \int \frac {\log \left (1-i \sqrt {\frac {2}{5}} x\right )}{x} \, dx+\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \int \frac {\log \left (1+i \sqrt {\frac {2}{5}} x\right )}{x} \, dx+\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \int \frac {\log \left (\frac {i \sqrt {5}-\sqrt {2} x}{\sqrt {2}+i \sqrt {5}}\right )}{1+x} \, dx-\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \int \frac {\log \left (\frac {i \sqrt {5}+\sqrt {2} x}{-\sqrt {2}+i \sqrt {5}}\right )}{1+x} \, dx\\ &=-\frac {3}{35} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )+\frac {\log (x)}{25}+\frac {4 \log (x)}{25 x}-\frac {8 x \log (x)}{25 \left (5+2 x^2\right )}-\frac {2 x^2 \log (x)}{25 \left (5+2 x^2\right )}+\frac {13}{175} \log (1+x)+\frac {4 \log (1+x)}{25 x}+\frac {\log (1+x)}{5 \left (5+2 x^2\right )}-\frac {4 (\log (x)+\log (1+x)-\log (x (1+x)))}{25 x}-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}-\frac {2}{25} i \sqrt {\frac {2}{5}} \log (1+x) \log \left (\frac {i \sqrt {5}-\sqrt {2} x}{\sqrt {2}+i \sqrt {5}}\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \log (1+x) \log \left (-\frac {i \sqrt {5}+\sqrt {2} x}{\sqrt {2}-i \sqrt {5}}\right )-\frac {13}{350} \log \left (5+2 x^2\right )-\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (-i \sqrt {\frac {2}{5}} x\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (i \sqrt {\frac {2}{5}} x\right )-\frac {1}{35} \int \frac {1}{1+x} \, dx-\frac {1}{35} \int \frac {2-2 x}{5+2 x^2} \, dx+\frac {8}{25} \int \frac {\tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )}{\sqrt {10} x} \, dx+\frac {8}{25} \int \frac {\log (1+x)}{\left (i \sqrt {10}-2 x\right )^2} \, dx+\frac {8}{25} \int \frac {\log (1+x)}{\left (i \sqrt {10}+2 x\right )^2} \, dx+\frac {16}{25} \int \frac {\log (1+x)}{-10-4 x^2} \, dx-\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {2} x}{-\sqrt {2}+i \sqrt {5}}\right )}{x} \, dx,x,1+x\right )+\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {2} x}{\sqrt {2}+i \sqrt {5}}\right )}{x} \, dx,x,1+x\right )\\ &=-\frac {3}{35} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )+\frac {\log (x)}{25}+\frac {4 \log (x)}{25 x}-\frac {8 x \log (x)}{25 \left (5+2 x^2\right )}-\frac {2 x^2 \log (x)}{25 \left (5+2 x^2\right )}+\frac {8}{175} \log (1+x)+\frac {4 \log (1+x)}{25 \left (i \sqrt {10}-2 x\right )}+\frac {4 \log (1+x)}{25 x}-\frac {4 \log (1+x)}{25 \left (i \sqrt {10}+2 x\right )}+\frac {\log (1+x)}{5 \left (5+2 x^2\right )}-\frac {4 (\log (x)+\log (1+x)-\log (x (1+x)))}{25 x}-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}-\frac {2}{25} i \sqrt {\frac {2}{5}} \log (1+x) \log \left (\frac {i \sqrt {5}-\sqrt {2} x}{\sqrt {2}+i \sqrt {5}}\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \log (1+x) \log \left (-\frac {i \sqrt {5}+\sqrt {2} x}{\sqrt {2}-i \sqrt {5}}\right )-\frac {13}{350} \log \left (5+2 x^2\right )-\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (-i \sqrt {\frac {2}{5}} x\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (i \sqrt {\frac {2}{5}} x\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (\frac {\sqrt {2} (1+x)}{\sqrt {2}-i \sqrt {5}}\right )-\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (\frac {\sqrt {2} (1+x)}{\sqrt {2}+i \sqrt {5}}\right )-\frac {2}{35} \int \frac {1}{5+2 x^2} \, dx+\frac {2}{35} \int \frac {x}{5+2 x^2} \, dx-\frac {4}{25} \int \frac {1}{\left (i \sqrt {10}-2 x\right ) (1+x)} \, dx+\frac {4}{25} \int \frac {1}{(1+x) \left (i \sqrt {10}+2 x\right )} \, dx+\frac {16}{25} \int \left (-\frac {i \log (1+x)}{4 \sqrt {5} \left (i \sqrt {5}-\sqrt {2} x\right )}-\frac {i \log (1+x)}{4 \sqrt {5} \left (i \sqrt {5}+\sqrt {2} x\right )}\right ) \, dx+\frac {1}{25} \left (4 \sqrt {\frac {2}{5}}\right ) \int \frac {\tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )}{x} \, dx\\ &=-\frac {4}{35} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )+\frac {\log (x)}{25}+\frac {4 \log (x)}{25 x}-\frac {8 x \log (x)}{25 \left (5+2 x^2\right )}-\frac {2 x^2 \log (x)}{25 \left (5+2 x^2\right )}+\frac {8}{175} \log (1+x)+\frac {4 \log (1+x)}{25 \left (i \sqrt {10}-2 x\right )}+\frac {4 \log (1+x)}{25 x}-\frac {4 \log (1+x)}{25 \left (i \sqrt {10}+2 x\right )}+\frac {\log (1+x)}{5 \left (5+2 x^2\right )}-\frac {4 (\log (x)+\log (1+x)-\log (x (1+x)))}{25 x}-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}-\frac {2}{25} i \sqrt {\frac {2}{5}} \log (1+x) \log \left (\frac {i \sqrt {5}-\sqrt {2} x}{\sqrt {2}+i \sqrt {5}}\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \log (1+x) \log \left (-\frac {i \sqrt {5}+\sqrt {2} x}{\sqrt {2}-i \sqrt {5}}\right )-\frac {4}{175} \log \left (5+2 x^2\right )-\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (-i \sqrt {\frac {2}{5}} x\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (i \sqrt {\frac {2}{5}} x\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (\frac {\sqrt {2} (1+x)}{\sqrt {2}-i \sqrt {5}}\right )-\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (\frac {\sqrt {2} (1+x)}{\sqrt {2}+i \sqrt {5}}\right )+\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \int \frac {\log \left (1-i \sqrt {\frac {2}{5}} x\right )}{x} \, dx-\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \int \frac {\log \left (1+i \sqrt {\frac {2}{5}} x\right )}{x} \, dx-\frac {(4 i) \int \frac {\log (1+x)}{i \sqrt {5}-\sqrt {2} x} \, dx}{25 \sqrt {5}}-\frac {(4 i) \int \frac {\log (1+x)}{i \sqrt {5}+\sqrt {2} x} \, dx}{25 \sqrt {5}}-\frac {4 \int \frac {1}{1+x} \, dx}{25 \left (2-i \sqrt {10}\right )}+\frac {8 \int \frac {1}{i \sqrt {10}+2 x} \, dx}{25 \left (2-i \sqrt {10}\right )}-\frac {4 \int \frac {1}{1+x} \, dx}{25 \left (2+i \sqrt {10}\right )}-\frac {8 \int \frac {1}{i \sqrt {10}-2 x} \, dx}{25 \left (2+i \sqrt {10}\right )}\\ &=-\frac {4}{35} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )+\frac {4 \log \left (\sqrt {10}+2 i x\right )}{25 \left (2+i \sqrt {10}\right )}+\frac {\log (x)}{25}+\frac {4 \log (x)}{25 x}-\frac {8 x \log (x)}{25 \left (5+2 x^2\right )}-\frac {2 x^2 \log (x)}{25 \left (5+2 x^2\right )}+\frac {8}{175} \log (1+x)-\frac {4 \log (1+x)}{25 \left (2-i \sqrt {10}\right )}-\frac {4 \log (1+x)}{25 \left (2+i \sqrt {10}\right )}+\frac {4 \log (1+x)}{25 \left (i \sqrt {10}-2 x\right )}+\frac {4 \log (1+x)}{25 x}-\frac {4 \log (1+x)}{25 \left (i \sqrt {10}+2 x\right )}+\frac {\log (1+x)}{5 \left (5+2 x^2\right )}-\frac {4 (\log (x)+\log (1+x)-\log (x (1+x)))}{25 x}-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}+\frac {4 \log \left (i \sqrt {10}+2 x\right )}{25 \left (2-i \sqrt {10}\right )}-\frac {4}{175} \log \left (5+2 x^2\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (\frac {\sqrt {2} (1+x)}{\sqrt {2}-i \sqrt {5}}\right )-\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (\frac {\sqrt {2} (1+x)}{\sqrt {2}+i \sqrt {5}}\right )-\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \int \frac {\log \left (\frac {i \sqrt {5}-\sqrt {2} x}{\sqrt {2}+i \sqrt {5}}\right )}{1+x} \, dx+\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \int \frac {\log \left (\frac {i \sqrt {5}+\sqrt {2} x}{-\sqrt {2}+i \sqrt {5}}\right )}{1+x} \, dx\\ &=-\frac {4}{35} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )+\frac {4 \log \left (\sqrt {10}+2 i x\right )}{25 \left (2+i \sqrt {10}\right )}+\frac {\log (x)}{25}+\frac {4 \log (x)}{25 x}-\frac {8 x \log (x)}{25 \left (5+2 x^2\right )}-\frac {2 x^2 \log (x)}{25 \left (5+2 x^2\right )}+\frac {8}{175} \log (1+x)-\frac {4 \log (1+x)}{25 \left (2-i \sqrt {10}\right )}-\frac {4 \log (1+x)}{25 \left (2+i \sqrt {10}\right )}+\frac {4 \log (1+x)}{25 \left (i \sqrt {10}-2 x\right )}+\frac {4 \log (1+x)}{25 x}-\frac {4 \log (1+x)}{25 \left (i \sqrt {10}+2 x\right )}+\frac {\log (1+x)}{5 \left (5+2 x^2\right )}-\frac {4 (\log (x)+\log (1+x)-\log (x (1+x)))}{25 x}-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}+\frac {4 \log \left (i \sqrt {10}+2 x\right )}{25 \left (2-i \sqrt {10}\right )}-\frac {4}{175} \log \left (5+2 x^2\right )+\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (\frac {\sqrt {2} (1+x)}{\sqrt {2}-i \sqrt {5}}\right )-\frac {2}{25} i \sqrt {\frac {2}{5}} \text {Li}_2\left (\frac {\sqrt {2} (1+x)}{\sqrt {2}+i \sqrt {5}}\right )+\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {2} x}{-\sqrt {2}+i \sqrt {5}}\right )}{x} \, dx,x,1+x\right )-\frac {1}{25} \left (2 i \sqrt {\frac {2}{5}}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {2} x}{\sqrt {2}+i \sqrt {5}}\right )}{x} \, dx,x,1+x\right )\\ &=-\frac {4}{35} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\sqrt {\frac {2}{5}} x\right )+\frac {4 \log \left (\sqrt {10}+2 i x\right )}{25 \left (2+i \sqrt {10}\right )}+\frac {\log (x)}{25}+\frac {4 \log (x)}{25 x}-\frac {8 x \log (x)}{25 \left (5+2 x^2\right )}-\frac {2 x^2 \log (x)}{25 \left (5+2 x^2\right )}+\frac {8}{175} \log (1+x)-\frac {4 \log (1+x)}{25 \left (2-i \sqrt {10}\right )}-\frac {4 \log (1+x)}{25 \left (2+i \sqrt {10}\right )}+\frac {4 \log (1+x)}{25 \left (i \sqrt {10}-2 x\right )}+\frac {4 \log (1+x)}{25 x}-\frac {4 \log (1+x)}{25 \left (i \sqrt {10}+2 x\right )}+\frac {\log (1+x)}{5 \left (5+2 x^2\right )}-\frac {4 (\log (x)+\log (1+x)-\log (x (1+x)))}{25 x}-\frac {(5-8 x) (\log (x)+\log (1+x)-\log (x (1+x)))}{25 \left (5+2 x^2\right )}+\frac {4 \log \left (i \sqrt {10}+2 x\right )}{25 \left (2-i \sqrt {10}\right )}-\frac {4}{175} \log \left (5+2 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.52, size = 25, normalized size = 1.00 \begin {gather*} \frac {(4+x) \log (x (1+x))}{5 x \left (5+2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20 + 45*x + 18*x^2 + 18*x^3 + 4*x^4 + (-20 - 20*x - 24*x^2 - 28*x^3 - 4*x^4)*Log[x + x^2])/(125*x^2
+ 125*x^3 + 100*x^4 + 100*x^5 + 20*x^6 + 20*x^7),x]

[Out]

((4 + x)*Log[x*(1 + x)])/(5*x*(5 + 2*x^2))

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fricas [A]  time = 0.62, size = 22, normalized size = 0.88 \begin {gather*} \frac {{\left (x + 4\right )} \log \left (x^{2} + x\right )}{5 \, {\left (2 \, x^{3} + 5 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^4-28*x^3-24*x^2-20*x-20)*log(x^2+x)+4*x^4+18*x^3+18*x^2+45*x+20)/(20*x^7+20*x^6+100*x^5+100*x
^4+125*x^3+125*x^2),x, algorithm="fricas")

[Out]

1/5*(x + 4)*log(x^2 + x)/(2*x^3 + 5*x)

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giac [A]  time = 0.62, size = 29, normalized size = 1.16 \begin {gather*} -\frac {1}{25} \, {\left (\frac {8 \, x - 5}{2 \, x^{2} + 5} - \frac {4}{x}\right )} \log \left (x^{2} + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^4-28*x^3-24*x^2-20*x-20)*log(x^2+x)+4*x^4+18*x^3+18*x^2+45*x+20)/(20*x^7+20*x^6+100*x^5+100*x
^4+125*x^3+125*x^2),x, algorithm="giac")

[Out]

-1/25*((8*x - 5)/(2*x^2 + 5) - 4/x)*log(x^2 + x)

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maple [A]  time = 0.16, size = 24, normalized size = 0.96

method	result	size
risch	\(\frac {\ln \left (x^{2}+x \right ) \left (4+x \right )}{5 x \left (2 x^{2}+5\right )}\)	\(24\)
norman	\(\frac {\frac {\ln \left (x^{2}+x \right ) x}{5}+\frac {4 \ln \left (x^{2}+x \right )}{5}}{\left (2 x^{2}+5\right ) x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^4-28*x^3-24*x^2-20*x-20)*ln(x^2+x)+4*x^4+18*x^3+18*x^2+45*x+20)/(20*x^7+20*x^6+100*x^5+100*x^4+125*
x^3+125*x^2),x,method=_RETURNVERBOSE)

[Out]

1/5*ln(x^2+x)/x*(4+x)/(2*x^2+5)

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maxima [B]  time = 1.29, size = 98, normalized size = 3.92 \begin {gather*} \frac {2 \, {\left (28 \, x^{2} + 5 \, {\left (3 \, x^{3} + 11 \, x + 14\right )} \log \left (x + 1\right ) - 35 \, {\left (x^{3} + 2 \, x - 2\right )} \log \relax (x) + 70\right )}}{175 \, {\left (2 \, x^{3} + 5 \, x\right )}} - \frac {4 \, {\left (16 \, x^{2} + 5 \, x + 35\right )}}{175 \, {\left (2 \, x^{3} + 5 \, x\right )}} + \frac {4 \, {\left (2 \, x + 5\right )}}{175 \, {\left (2 \, x^{2} + 5\right )}} - \frac {3}{35} \, \log \left (x + 1\right ) + \frac {1}{5} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^4-28*x^3-24*x^2-20*x-20)*log(x^2+x)+4*x^4+18*x^3+18*x^2+45*x+20)/(20*x^7+20*x^6+100*x^5+100*x
^4+125*x^3+125*x^2),x, algorithm="maxima")

[Out]

2/175*(28*x^2 + 5*(3*x^3 + 11*x + 14)*log(x + 1) - 35*(x^3 + 2*x - 2)*log(x) + 70)/(2*x^3 + 5*x) - 4/175*(16*x
^2 + 5*x + 35)/(2*x^3 + 5*x) + 4/175*(2*x + 5)/(2*x^2 + 5) - 3/35*log(x + 1) + 1/5*log(x)

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mupad [B]  time = 1.18, size = 21, normalized size = 0.84 \begin {gather*} \frac {\ln \left (x^2+x\right )\,\left (\frac {x}{10}+\frac {2}{5}\right )}{x^3+\frac {5\,x}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((45*x - log(x + x^2)*(20*x + 24*x^2 + 28*x^3 + 4*x^4 + 20) + 18*x^2 + 18*x^3 + 4*x^4 + 20)/(125*x^2 + 125*
x^3 + 100*x^4 + 100*x^5 + 20*x^6 + 20*x^7),x)

[Out]

(log(x + x^2)*(x/10 + 2/5))/((5*x)/2 + x^3)

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sympy [A]  time = 0.21, size = 17, normalized size = 0.68 \begin {gather*} \frac {\left (x + 4\right ) \log {\left (x^{2} + x \right )}}{10 x^{3} + 25 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**4-28*x**3-24*x**2-20*x-20)*ln(x**2+x)+4*x**4+18*x**3+18*x**2+45*x+20)/(20*x**7+20*x**6+100*x
**5+100*x**4+125*x**3+125*x**2),x)

[Out]

(x + 4)*log(x**2 + x)/(10*x**3 + 25*x)

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