3.25.39 \(\int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+(-50+10 x-20 x^3+4 x^4) \log (\frac {5-x^3}{x})+(25+5 x-5 x^3-x^4) \log ^2(\frac {5-x^3}{x})}{7500-4500 x+900 x^2-1560 x^3+900 x^4-180 x^5+12 x^6} \, dx\)

Optimal. Leaf size=31 \[ -x+\frac {1}{12} x \left (-2+\frac {\log ^2\left (\frac {5}{x}-x^2\right )}{(-5+x)^2}\right ) \]

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Rubi [C]  time = 11.90, antiderivative size = 3583, normalized size of antiderivative = 115.58, number of steps used = 295, number of rules used = 24, integrand size = 121, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.198, Rules used = {6741, 12, 6725, 1875, 31, 634, 617, 204, 628, 2528, 2525, 2524, 2418, 2394, 2315, 260, 2416, 2393, 2391, 2392, 2390, 2301, 2357, 2337}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(-8750 + 5250*x - 1050*x^2 + 1820*x^3 - 1050*x^4 + 210*x^5 - 14*x^6 + (-50 + 10*x - 20*x^3 + 4*x^4)*Log[(5
- x^3)/x] + (25 + 5*x - 5*x^3 - x^4)*Log[(5 - x^3)/x]^2)/(7500 - 4500*x + 900*x^2 - 1560*x^3 + 900*x^4 - 180*
x^5 + 12*x^6),x]

[Out]

(-7*x)/6 + ((45 + 5^(1/3)*(17 + 5*5^(1/3)))*Log[5^(1/3) - x]^2)/2304 - ((25 + 5^(2/3)*(1 + 5^(2/3)))*Log[5^(1/
3) - x]^2)/1440 - ((25 + 5^(2/3)*(17 + 9*5^(2/3)))*Log[5^(1/3) - x]^2)/11520 + ((25 - 5*(-5)^(1/3) + (-5)^(2/3
))*Log[5]*Log[x])/2160 + ((-5)^(1/3)*(17 - 5*(-5)^(1/3) + 9*(-5)^(2/3))*Log[5]*Log[x])/3456 + ((25 - 45*(-5)^(
1/3) + 17*(-5)^(2/3))*Log[5]*Log[x])/17280 + ((-1)^(2/3)*((-5)^(2/3) - 25*(-1)^(1/3) + 5*5^(1/3))*Log[5]*Log[x
])/2160 + ((-1)^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/3) + 45*5^(1/3))*Log[5]*Log[x])/17280 - ((45 + 5^(1/3)*(17 +
5*5^(1/3)))*Log[5]*Log[x])/3456 + ((-1)^(2/3)*(45*(-1)^(1/3) - 5^(1/3)*(17 + 5*(-1)^(2/3)*5^(1/3)))*Log[5]*Lo
g[x])/3456 + ((25 + 5^(2/3)*(1 + 5^(2/3)))*Log[5]*Log[x])/2160 + ((25 + 5^(2/3)*(17 + 9*5^(2/3)))*Log[5]*Log[x
])/17280 + ((45 + 5^(1/3)*(17 + 5*5^(1/3)))*Log[5^(1/3) - x]*Log[((-5)^(1/3) + x)/((-5)^(1/3) + 5^(1/3))])/115
2 - ((25 + 5^(2/3)*(1 + 5^(2/3)))*Log[5^(1/3) - x]*Log[((-5)^(1/3) + x)/((-5)^(1/3) + 5^(1/3))])/720 - ((25 +
5^(2/3)*(17 + 9*5^(2/3)))*Log[5^(1/3) - x]*Log[((-5)^(1/3) + x)/((-5)^(1/3) + 5^(1/3))])/5760 - ((-1)^(2/3)*((
-5)^(2/3) - 25*(-1)^(1/3) + 5*5^(1/3))*Log[((-1/5)^(1/3)*(5^(1/3) - x))/(1 + (-1)^(1/3))]*Log[5^(1/3) + (-1)^(
1/3)*x])/720 - ((-1)^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/3) + 45*5^(1/3))*Log[((-1/5)^(1/3)*(5^(1/3) - x))/(1 +
(-1)^(1/3))]*Log[5^(1/3) + (-1)^(1/3)*x])/5760 + ((45 - 5^(1/3)*(5*(-5)^(1/3) - 17*(-1)^(2/3)))*Log[((-1/5)^(1
/3)*(5^(1/3) - x))/(1 + (-1)^(1/3))]*Log[5^(1/3) + (-1)^(1/3)*x])/1152 - ((-1)^(2/3)*((-5)^(2/3) - 25*(-1)^(1/
3) + 5*5^(1/3))*Log[-(((-1/5)^(1/3)*((-5)^(1/3) + x))/(1 - (-1)^(2/3)))]*Log[5^(1/3) + (-1)^(1/3)*x])/720 - ((
-1)^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/3) + 45*5^(1/3))*Log[-(((-1/5)^(1/3)*((-5)^(1/3) + x))/(1 - (-1)^(2/3)))
]*Log[5^(1/3) + (-1)^(1/3)*x])/5760 + ((45 - 5^(1/3)*(5*(-5)^(1/3) - 17*(-1)^(2/3)))*Log[-(((-1/5)^(1/3)*((-5)
^(1/3) + x))/(1 - (-1)^(2/3)))]*Log[5^(1/3) + (-1)^(1/3)*x])/1152 - ((-1)^(2/3)*((-5)^(2/3) - 25*(-1)^(1/3) +
5*5^(1/3))*Log[5^(1/3) + (-1)^(1/3)*x]^2)/1440 - ((-1)^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/3) + 45*5^(1/3))*Log[
5^(1/3) + (-1)^(1/3)*x]^2)/11520 + ((45 - 5^(1/3)*(5*(-5)^(1/3) - 17*(-1)^(2/3)))*Log[5^(1/3) + (-1)^(1/3)*x]^
2)/2304 + ((45 + 5^(1/3)*(17 + 5*5^(1/3)))*Log[5^(1/3) - x]*Log[-(((-1)^(2/3)*(5^(1/3) + (-1)^(1/3)*x))/(5^(1/
3)*(1 - (-1)^(2/3))))])/1152 - ((25 + 5^(2/3)*(1 + 5^(2/3)))*Log[5^(1/3) - x]*Log[-(((-1)^(2/3)*(5^(1/3) + (-1
)^(1/3)*x))/(5^(1/3)*(1 - (-1)^(2/3))))])/720 - ((25 + 5^(2/3)*(17 + 9*5^(2/3)))*Log[5^(1/3) - x]*Log[-(((-1)^
(2/3)*(5^(1/3) + (-1)^(1/3)*x))/(5^(1/3)*(1 - (-1)^(2/3))))])/5760 - ((25 - 5*(-5)^(1/3) + (-5)^(2/3))*Log[-((
(-1)^(2/3)*(5^(1/3) - x))/(5^(1/3)*(1 - (-1)^(2/3))))]*Log[5^(1/3) - (-1)^(2/3)*x])/720 - ((-5)^(1/3)*(17 - 5*
(-5)^(1/3) + 9*(-5)^(2/3))*Log[-(((-1)^(2/3)*(5^(1/3) - x))/(5^(1/3)*(1 - (-1)^(2/3))))]*Log[5^(1/3) - (-1)^(2
/3)*x])/1152 - ((25 - 45*(-5)^(1/3) + 17*(-5)^(2/3))*Log[-(((-1)^(2/3)*(5^(1/3) - x))/(5^(1/3)*(1 - (-1)^(2/3)
)))]*Log[5^(1/3) - (-1)^(2/3)*x])/5760 - ((25 - 5*(-5)^(1/3) + (-5)^(2/3))*Log[-(((-1)^(2/3)*((-1)^(2/3)*5^(1/
3) - x))/((-5)^(1/3) + 5^(1/3)))]*Log[5^(1/3) - (-1)^(2/3)*x])/720 - ((-5)^(1/3)*(17 - 5*(-5)^(1/3) + 9*(-5)^(
2/3))*Log[-(((-1)^(2/3)*((-1)^(2/3)*5^(1/3) - x))/((-5)^(1/3) + 5^(1/3)))]*Log[5^(1/3) - (-1)^(2/3)*x])/1152 -
((25 - 45*(-5)^(1/3) + 17*(-5)^(2/3))*Log[-(((-1)^(2/3)*((-1)^(2/3)*5^(1/3) - x))/((-5)^(1/3) + 5^(1/3)))]*Lo
g[5^(1/3) - (-1)^(2/3)*x])/5760 - ((25 - 5*(-5)^(1/3) + (-5)^(2/3))*Log[5^(1/3) - (-1)^(2/3)*x]^2)/1440 - ((-5
)^(1/3)*(17 - 5*(-5)^(1/3) + 9*(-5)^(2/3))*Log[5^(1/3) - (-1)^(2/3)*x]^2)/2304 - ((25 - 45*(-5)^(1/3) + 17*(-5
)^(2/3))*Log[5^(1/3) - (-1)^(2/3)*x]^2)/11520 - ((45 + 5^(1/3)*(17 + 5*5^(1/3)))*Log[5^(1/3) - x]*Log[(5 - x^3
)/x])/1152 + ((25 + 5^(2/3)*(1 + 5^(2/3)))*Log[5^(1/3) - x]*Log[(5 - x^3)/x])/720 + ((25 + 5^(2/3)*(17 + 9*5^(
2/3)))*Log[5^(1/3) - x]*Log[(5 - x^3)/x])/5760 + ((-1)^(2/3)*((-5)^(2/3) - 25*(-1)^(1/3) + 5*5^(1/3))*Log[5^(1
/3) + (-1)^(1/3)*x]*Log[(5 - x^3)/x])/720 + ((-1)^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/3) + 45*5^(1/3))*Log[5^(1/
3) + (-1)^(1/3)*x]*Log[(5 - x^3)/x])/5760 - ((45 - 5^(1/3)*(5*(-5)^(1/3) - 17*(-1)^(2/3)))*Log[5^(1/3) + (-1)^
(1/3)*x]*Log[(5 - x^3)/x])/1152 + ((25 - 5*(-5)^(1/3) + (-5)^(2/3))*Log[5^(1/3) - (-1)^(2/3)*x]*Log[(5 - x^3)/
x])/720 + ((-5)^(1/3)*(17 - 5*(-5)^(1/3) + 9*(-5)^(2/3))*Log[5^(1/3) - (-1)^(2/3)*x]*Log[(5 - x^3)/x])/1152 +
((25 - 45*(-5)^(1/3) + 17*(-5)^(2/3))*Log[5^(1/3) - (-1)^(2/3)*x]*Log[(5 - x^3)/x])/5760 + (5*Log[(5 - x^3)/x]
^2)/(12*(5 - x)^2) - Log[(5 - x^3)/x]^2/(12*(5 - x)) + ((45 + 5^(1/3)*(17 + 5*5^(1/3)))*PolyLog[2, (2*(5^(1/3)
- x))/(5^(1/3)*(3 - I*Sqrt[3]))])/1152 - ((25 + 5^(2/3)*(1 + 5^(2/3)))*PolyLog[2, (2*(5^(1/3) - x))/(5^(1/3)*
(3 - I*Sqrt[3]))])/720 - ((25 + 5^(2/3)*(17 + 9*5^(2/3)))*PolyLog[2, (2*(5^(1/3) - x))/(5^(1/3)*(3 - I*Sqrt[3]
))])/5760 + ((45 + 5^(1/3)*(17 + 5*5^(1/3)))*PolyLog[2, (5^(1/3) - x)/((-5)^(1/3) + 5^(1/3))])/1152 - ((25 + 5
^(2/3)*(1 + 5^(2/3)))*PolyLog[2, (5^(1/3) - x)/((-5)^(1/3) + 5^(1/3))])/720 - ((25 + 5^(2/3)*(17 + 9*5^(2/3)))
*PolyLog[2, (5^(1/3) - x)/((-5)^(1/3) + 5^(1/3))])/5760 - ((-1)^(2/3)*((-5)^(2/3) - 25*(-1)^(1/3) + 5*5^(1/3))
*PolyLog[2, -((-1/5)^(1/3)*x)])/720 - ((-1)^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/3) + 45*5^(1/3))*PolyLog[2, -((-
1/5)^(1/3)*x)])/5760 + ((45 - 5^(1/3)*(5*(-5)^(1/3) - 17*(-1)^(2/3)))*PolyLog[2, -((-1/5)^(1/3)*x)])/1152 + ((
45 + 5^(1/3)*(17 + 5*5^(1/3)))*PolyLog[2, x/5^(1/3)])/1152 - ((25 + 5^(2/3)*(1 + 5^(2/3)))*PolyLog[2, x/5^(1/3
)])/720 - ((25 + 5^(2/3)*(17 + 9*5^(2/3)))*PolyLog[2, x/5^(1/3)])/5760 - ((25 - 5*(-5)^(1/3) + (-5)^(2/3))*Pol
yLog[2, ((-1)^(2/3)*x)/5^(1/3)])/720 - ((-5)^(1/3)*(17 - 5*(-5)^(1/3) + 9*(-5)^(2/3))*PolyLog[2, ((-1)^(2/3)*x
)/5^(1/3)])/1152 - ((25 - 45*(-5)^(1/3) + 17*(-5)^(2/3))*PolyLog[2, ((-1)^(2/3)*x)/5^(1/3)])/5760 - ((25 - 5*(
-5)^(1/3) + (-5)^(2/3))*PolyLog[2, -(((-1)^(2/3)*((-5)^(1/3) + x))/(5^(1/3)*(1 - (-1)^(2/3))))])/720 - ((-5)^(
1/3)*(17 - 5*(-5)^(1/3) + 9*(-5)^(2/3))*PolyLog[2, -(((-1)^(2/3)*((-5)^(1/3) + x))/(5^(1/3)*(1 - (-1)^(2/3))))
])/1152 - ((25 - 45*(-5)^(1/3) + 17*(-5)^(2/3))*PolyLog[2, -(((-1)^(2/3)*((-5)^(1/3) + x))/(5^(1/3)*(1 - (-1)^
(2/3))))])/5760 - ((-1)^(2/3)*((-5)^(2/3) - 25*(-1)^(1/3) + 5*5^(1/3))*PolyLog[2, (5^(1/3) + (-1)^(1/3)*x)/(5^
(1/3)*(1 - (-1)^(2/3)))])/720 - ((-1)^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/3) + 45*5^(1/3))*PolyLog[2, (5^(1/3) +
(-1)^(1/3)*x)/(5^(1/3)*(1 - (-1)^(2/3)))])/5760 + ((45 - 5^(1/3)*(5*(-5)^(1/3) - 17*(-1)^(2/3)))*PolyLog[2, (
5^(1/3) + (-1)^(1/3)*x)/(5^(1/3)*(1 - (-1)^(2/3)))])/1152 - ((-1)^(2/3)*((-5)^(2/3) - 25*(-1)^(1/3) + 5*5^(1/3
))*PolyLog[2, (5^(1/3) + (-1)^(1/3)*x)/((-5)^(1/3) + 5^(1/3))])/720 - ((-1)^(2/3)*(17*(-5)^(2/3) - 25*(-1)^(1/
3) + 45*5^(1/3))*PolyLog[2, (5^(1/3) + (-1)^(1/3)*x)/((-5)^(1/3) + 5^(1/3))])/5760 + ((45 - 5^(1/3)*(5*(-5)^(1
/3) - 17*(-1)^(2/3)))*PolyLog[2, (5^(1/3) + (-1)^(1/3)*x)/((-5)^(1/3) + 5^(1/3))])/1152 - ((25 - 5*(-5)^(1/3)
+ (-5)^(2/3))*PolyLog[2, (5^(1/3) - (-1)^(2/3)*x)/((-5)^(1/3) + 5^(1/3))])/720 - ((-5)^(1/3)*(17 - 5*(-5)^(1/3
) + 9*(-5)^(2/3))*PolyLog[2, (5^(1/3) - (-1)^(2/3)*x)/((-5)^(1/3) + 5^(1/3))])/1152 - ((25 - 45*(-5)^(1/3) + 1
7*(-5)^(2/3))*PolyLog[2, (5^(1/3) - (-1)^(2/3)*x)/((-5)^(1/3) + 5^(1/3))])/5760

Rule 12

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

Rule 31

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1875

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2], q = (-(a/b))^(1/3)}, Dist[(q*(A + B*q + C*q^2))/(3*a), Int[1/(q - x), x], x] + Dist[q/(3*a), Int[(q*(2*A
- B*q - C*q^2) + (A + B*q - 2*C*q^2)*x)/(q^2 + q*x + x^2), x], x] /; NeQ[a*B^3 - b*A^3, 0] && NeQ[A + B*q + C*
q^2, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2] && LtQ[a/b, 0]

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 2337

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

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 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

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 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 6725

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

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{12 (5-x)^3 \left (5-x^3\right )} \, dx\\ &=\frac {1}{12} \int \frac {-8750+5250 x-1050 x^2+1820 x^3-1050 x^4+210 x^5-14 x^6+\left (-50+10 x-20 x^3+4 x^4\right ) \log \left (\frac {5-x^3}{x}\right )+\left (25+5 x-5 x^3-x^4\right ) \log ^2\left (\frac {5-x^3}{x}\right )}{(5-x)^3 \left (5-x^3\right )} \, dx\\ &=\frac {1}{12} \int \left (-\frac {8750}{(-5+x)^3 \left (-5+x^3\right )}+\frac {5250 x}{(-5+x)^3 \left (-5+x^3\right )}-\frac {1050 x^2}{(-5+x)^3 \left (-5+x^3\right )}+\frac {1820 x^3}{(-5+x)^3 \left (-5+x^3\right )}-\frac {1050 x^4}{(-5+x)^3 \left (-5+x^3\right )}+\frac {210 x^5}{(-5+x)^3 \left (-5+x^3\right )}-\frac {14 x^6}{(-5+x)^3 \left (-5+x^3\right )}+\frac {2 \left (5+2 x^3\right ) \log \left (\frac {5-x^3}{x}\right )}{(-5+x)^2 \left (-5+x^3\right )}-\frac {(5+x) \log ^2\left (\frac {5-x^3}{x}\right )}{(-5+x)^3}\right ) \, dx\\ &=-\left (\frac {1}{12} \int \frac {(5+x) \log ^2\left (\frac {5-x^3}{x}\right )}{(-5+x)^3} \, dx\right )+\frac {1}{6} \int \frac {\left (5+2 x^3\right ) \log \left (\frac {5-x^3}{x}\right )}{(-5+x)^2 \left (-5+x^3\right )} \, dx-\frac {7}{6} \int \frac {x^6}{(-5+x)^3 \left (-5+x^3\right )} \, dx+\frac {35}{2} \int \frac {x^5}{(-5+x)^3 \left (-5+x^3\right )} \, dx-\frac {175}{2} \int \frac {x^2}{(-5+x)^3 \left (-5+x^3\right )} \, dx-\frac {175}{2} \int \frac {x^4}{(-5+x)^3 \left (-5+x^3\right )} \, dx+\frac {455}{3} \int \frac {x^3}{(-5+x)^3 \left (-5+x^3\right )} \, dx+\frac {875}{2} \int \frac {x}{(-5+x)^3 \left (-5+x^3\right )} \, dx-\frac {4375}{6} \int \frac {1}{(-5+x)^3 \left (-5+x^3\right )} \, dx\\ &=-\left (\frac {1}{12} \int \left (\frac {10 \log ^2\left (\frac {5-x^3}{x}\right )}{(-5+x)^3}+\frac {\log ^2\left (\frac {5-x^3}{x}\right )}{(-5+x)^2}\right ) \, dx\right )+\frac {1}{6} \int \left (\frac {17 \log \left (\frac {5-x^3}{x}\right )}{8 (-5+x)^2}-\frac {5 \log \left (\frac {5-x^3}{x}\right )}{64 (-5+x)}+\frac {\left (45+17 x+5 x^2\right ) \log \left (\frac {5-x^3}{x}\right )}{64 \left (-5+x^3\right )}\right ) \, dx-\frac {7}{6} \int \left (1+\frac {3125}{24 (-5+x)^3}+\frac {14375}{192 (-5+x)^2}+\frac {23125}{1536 (-5+x)}-\frac {5 \left (89+45 x+17 x^2\right )}{1536 \left (-5+x^3\right )}\right ) \, dx+\frac {35}{2} \int \left (\frac {625}{24 (-5+x)^3}+\frac {625}{64 (-5+x)^2}+\frac {1625}{1536 (-5+x)}+\frac {-225-85 x-89 x^2}{1536 \left (-5+x^3\right )}\right ) \, dx-\frac {175}{2} \int \left (\frac {5}{24 (-5+x)^3}-\frac {3}{64 (-5+x)^2}+\frac {89}{7680 (-5+x)}+\frac {-225-85 x-89 x^2}{7680 \left (-5+x^3\right )}\right ) \, dx-\frac {175}{2} \int \left (\frac {125}{24 (-5+x)^3}+\frac {175}{192 (-5+x)^2}+\frac {15}{512 (-5+x)}+\frac {-85-89 x-45 x^2}{1536 \left (-5+x^3\right )}\right ) \, dx+\frac {455}{3} \int \left (\frac {25}{24 (-5+x)^3}-\frac {5}{192 (-5+x)^2}+\frac {17}{1536 (-5+x)}+\frac {-89-45 x-17 x^2}{1536 \left (-5+x^3\right )}\right ) \, dx+\frac {875}{2} \int \left (\frac {1}{24 (-5+x)^3}-\frac {17}{960 (-5+x)^2}+\frac {3}{512 (-5+x)}+\frac {-85-89 x-45 x^2}{7680 \left (-5+x^3\right )}\right ) \, dx-\frac {4375}{6} \int \left (\frac {1}{120 (-5+x)^3}-\frac {1}{192 (-5+x)^2}+\frac {17}{7680 (-5+x)}+\frac {-89-45 x-17 x^2}{7680 \left (-5+x^3\right )}\right ) \, dx\\ &=-\frac {7 x}{6}+\frac {1}{384} \int \frac {\left (45+17 x+5 x^2\right ) \log \left (\frac {5-x^3}{x}\right )}{-5+x^3} \, dx+\frac {35 \int \frac {89+45 x+17 x^2}{-5+x^3} \, dx}{9216}-\frac {5}{384} \int \frac {\log \left (\frac {5-x^3}{x}\right )}{-5+x} \, dx-\frac {1}{12} \int \frac {\log ^2\left (\frac {5-x^3}{x}\right )}{(-5+x)^2} \, dx-\frac {875 \int \frac {-89-45 x-17 x^2}{-5+x^3} \, dx}{9216}+\frac {455 \int \frac {-89-45 x-17 x^2}{-5+x^3} \, dx}{4608}+\frac {17}{48} \int \frac {\log \left (\frac {5-x^3}{x}\right )}{(-5+x)^2} \, dx-\frac {5}{6} \int \frac {\log ^2\left (\frac {5-x^3}{x}\right )}{(-5+x)^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [C]  time = 64.99, size = 14637, normalized size = 472.16 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-8750 + 5250*x - 1050*x^2 + 1820*x^3 - 1050*x^4 + 210*x^5 - 14*x^6 + (-50 + 10*x - 20*x^3 + 4*x^4)*
Log[(5 - x^3)/x] + (25 + 5*x - 5*x^3 - x^4)*Log[(5 - x^3)/x]^2)/(7500 - 4500*x + 900*x^2 - 1560*x^3 + 900*x^4
- 180*x^5 + 12*x^6),x]

[Out]

Result too large to show

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fricas [A]  time = 1.00, size = 42, normalized size = 1.35 \begin {gather*} -\frac {14 \, x^{3} - x \log \left (-\frac {x^{3} - 5}{x}\right )^{2} - 140 \, x^{2} + 350 \, x}{12 \, {\left (x^{2} - 10 \, x + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-5*x^3+5*x+25)*log((-x^3+5)/x)^2+(4*x^4-20*x^3+10*x-50)*log((-x^3+5)/x)-14*x^6+210*x^5-1050*x^
4+1820*x^3-1050*x^2+5250*x-8750)/(12*x^6-180*x^5+900*x^4-1560*x^3+900*x^2-4500*x+7500),x, algorithm="fricas")

[Out]

-1/12*(14*x^3 - x*log(-(x^3 - 5)/x)^2 - 140*x^2 + 350*x)/(x^2 - 10*x + 25)

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giac [A]  time = 0.38, size = 30, normalized size = 0.97 \begin {gather*} \frac {x \log \left (-\frac {x^{3} - 5}{x}\right )^{2}}{12 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {7}{6} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-5*x^3+5*x+25)*log((-x^3+5)/x)^2+(4*x^4-20*x^3+10*x-50)*log((-x^3+5)/x)-14*x^6+210*x^5-1050*x^
4+1820*x^3-1050*x^2+5250*x-8750)/(12*x^6-180*x^5+900*x^4-1560*x^3+900*x^2-4500*x+7500),x, algorithm="giac")

[Out]

1/12*x*log(-(x^3 - 5)/x)^2/(x^2 - 10*x + 25) - 7/6*x

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maple [A]  time = 0.09, size = 32, normalized size = 1.03

method	result	size
risch	\(\frac {\ln \left (\frac {-x^{3}+5}{x}\right )^{2} x}{12 x^{2}-120 x +300}-\frac {7 x}{6}\)	\(32\)
norman	\(\frac {\frac {175 x}{2}-\frac {7 x^{3}}{6}+\frac {\ln \left (\frac {-x^{3}+5}{x}\right )^{2} x}{12}-\frac {875}{3}}{\left (x -5\right )^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4-5*x^3+5*x+25)*ln((-x^3+5)/x)^2+(4*x^4-20*x^3+10*x-50)*ln((-x^3+5)/x)-14*x^6+210*x^5-1050*x^4+1820*x
^3-1050*x^2+5250*x-8750)/(12*x^6-180*x^5+900*x^4-1560*x^3+900*x^2-4500*x+7500),x,method=_RETURNVERBOSE)

[Out]

1/12*x/(x^2-10*x+25)*ln((-x^3+5)/x)^2-7/6*x

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maxima [B]  time = 5.93, size = 165, normalized size = 5.32 \begin {gather*} -\frac {7}{6} \, x + \frac {x \log \left (-x^{3} + 5\right )^{2} - 2 \, x \log \left (-x^{3} + 5\right ) \log \relax (x) + x \log \relax (x)^{2}}{12 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {4375 \, {\left (23 \, x - 95\right )}}{1152 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {175 \, {\left (17 \, x - 105\right )}}{384 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {175 \, {\left (9 \, x - 65\right )}}{384 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {4375 \, {\left (7 \, x - 15\right )}}{384 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {875 \, {\left (5 \, x - 29\right )}}{1152 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {21875 \, {\left (3 \, x - 11\right )}}{384 \, {\left (x^{2} - 10 \, x + 25\right )}} + \frac {2275 \, {\left (x - 25\right )}}{576 \, {\left (x^{2} - 10 \, x + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4-5*x^3+5*x+25)*log((-x^3+5)/x)^2+(4*x^4-20*x^3+10*x-50)*log((-x^3+5)/x)-14*x^6+210*x^5-1050*x^
4+1820*x^3-1050*x^2+5250*x-8750)/(12*x^6-180*x^5+900*x^4-1560*x^3+900*x^2-4500*x+7500),x, algorithm="maxima")

[Out]

-7/6*x + 1/12*(x*log(-x^3 + 5)^2 - 2*x*log(-x^3 + 5)*log(x) + x*log(x)^2)/(x^2 - 10*x + 25) + 4375/1152*(23*x
- 95)/(x^2 - 10*x + 25) + 175/384*(17*x - 105)/(x^2 - 10*x + 25) - 175/384*(9*x - 65)/(x^2 - 10*x + 25) + 4375
/384*(7*x - 15)/(x^2 - 10*x + 25) - 875/1152*(5*x - 29)/(x^2 - 10*x + 25) - 21875/384*(3*x - 11)/(x^2 - 10*x +
25) + 2275/576*(x - 25)/(x^2 - 10*x + 25)

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mupad [B]  time = 1.63, size = 25, normalized size = 0.81 \begin {gather*} \frac {x\,{\ln \left (-\frac {x^3-5}{x}\right )}^2}{12\,{\left (x-5\right )}^2}-\frac {7\,x}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5250*x + log(-(x^3 - 5)/x)^2*(5*x - 5*x^3 - x^4 + 25) - 1050*x^2 + 1820*x^3 - 1050*x^4 + 210*x^5 - 14*x^6
+ log(-(x^3 - 5)/x)*(10*x - 20*x^3 + 4*x^4 - 50) - 8750)/(900*x^2 - 4500*x - 1560*x^3 + 900*x^4 - 180*x^5 + 1
2*x^6 + 7500),x)

[Out]

(x*log(-(x^3 - 5)/x)^2)/(12*(x - 5)^2) - (7*x)/6

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sympy [A]  time = 0.28, size = 26, normalized size = 0.84 \begin {gather*} - \frac {7 x}{6} + \frac {x \log {\left (\frac {5 - x^{3}}{x} \right )}^{2}}{12 x^{2} - 120 x + 300} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4-5*x**3+5*x+25)*ln((-x**3+5)/x)**2+(4*x**4-20*x**3+10*x-50)*ln((-x**3+5)/x)-14*x**6+210*x**5-
1050*x**4+1820*x**3-1050*x**2+5250*x-8750)/(12*x**6-180*x**5+900*x**4-1560*x**3+900*x**2-4500*x+7500),x)

[Out]

-7*x/6 + x*log((5 - x**3)/x)**2/(12*x**2 - 120*x + 300)

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