3.78.4 \(\int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 (125-225 x^3+135 x^6-27 x^9)+(200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx\)

Optimal. Leaf size=29 \[ 1-\left (e^3+\left (-x+\frac {4}{-5+3 x^3}\right )^2\right ) \log (3 x) \]

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Rubi [C]  time = 2.38, antiderivative size = 772, normalized size of antiderivative = 26.62, number of steps used = 43, number of rules used = 25, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.197, Rules used = {6688, 6742, 1829, 1834, 1875, 31, 634, 617, 204, 628, 2357, 2304, 2338, 266, 44, 199, 200, 2330, 2316, 2315, 2314, 2317, 2391, 27, 12} \begin {gather*} -\frac {48 i \sqrt [6]{3} \text {Li}_2\left (-\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {16 (-1)^{2/3} \text {Li}_2\left (-\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {16 \sqrt [3]{-\frac {1}{3}} \text {Li}_2\left ((-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {32 \text {Li}_2\left (-\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}-\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {16}{15 \left (5-3 x^3\right )}-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}-\frac {16}{75} \log \left (5-3 x^3\right )-x^2 \log (3 x)+\frac {4}{225} \left (12-5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (3^{2/3} x^2+\sqrt [3]{15} x+5^{2/3}\right )-\frac {8 x \log (3 x)}{\sqrt [3]{3} 5^{2/3} \left (3^{2/3} \sqrt [3]{5}-3 x\right )}-\frac {8 \sqrt [3]{-1} x \log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{3} x+\sqrt [3]{-5}\right )}-\frac {8 x \log (3 x)}{15^{2/3} \left (\sqrt [3]{-1} 3^{2/3} x+\sqrt [3]{15}\right )}+\frac {16 \log (45) \log \left (3^{2/3} \sqrt [3]{5}-3 x\right )}{9 \sqrt [3]{3} 5^{2/3}}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)+\frac {16 \log (x)}{25}+\frac {8 \sqrt [3]{-1} 3^{2/3} \log \left (\sqrt [3]{-3} x+\sqrt [3]{5}\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-\frac {48 i \sqrt [6]{3} \log (3 x) \log \left (\sqrt [3]{-\frac {3}{5}} x+1\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {16 (-1)^{2/3} \log (3 x) \log \left (\sqrt [3]{-\frac {3}{5}} x+1\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {16 \sqrt [3]{-\frac {1}{3}} \log (3 x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {16 \log (45) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {8}{225} \left (6+5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )-\frac {8 \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {8 \sqrt [3]{-\frac {1}{3}} \log \left (\sqrt [3]{3} x+\sqrt [3]{-5}\right )}{3\ 5^{2/3}}-\frac {32 \log (3 x) \log \left (1+\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}-\frac {8 \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{5}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6} 5^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(80 + 200*x + 125*x^2 - 48*x^3 - 240*x^4 - 225*x^5 + 72*x^7 + 135*x^8 - 27*x^11 + E^3*(125 - 225*x^3 + 135
*x^6 - 27*x^9) + (200*x + 250*x^2 + 288*x^3 + 120*x^4 - 450*x^5 - 144*x^7 + 270*x^8 - 54*x^11)*Log[3*x])/(-125
*x + 225*x^4 - 135*x^7 + 27*x^10),x]

[Out]

16/(15*(5 - 3*x^3)) - (16*x^3)/(25*(5 - 3*x^3)) - (8*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*5^(1/3))])/(3^(5/6)*5^(
2/3)) + (16*Log[45]*Log[3^(2/3)*5^(1/3) - 3*x])/(9*3^(1/3)*5^(2/3)) + (16*Log[x])/25 - ((16 + 25*E^3)*Log[x])/
25 - (8*x*Log[3*x])/(3^(1/3)*5^(2/3)*(3^(2/3)*5^(1/3) - 3*x)) - x^2*Log[3*x] - (8*(-1)^(1/3)*x*Log[3*x])/(3*5^
(2/3)*((-5)^(1/3) + 3^(1/3)*x)) - (8*x*Log[3*x])/(15^(2/3)*(15^(1/3) + (-1)^(1/3)*3^(2/3)*x)) - (16*Log[3*x])/
(5 - 3*x^3)^2 + (8*(-1)^(1/3)*3^(2/3)*Log[5^(1/3) + (-3)^(1/3)*x])/(5^(2/3)*(1 + (-1)^(1/3))^4) - (16*(-1)^(2/
3)*Log[3*x]*Log[1 + (-3/5)^(1/3)*x])/(3*3^(1/3)*5^(2/3)) - ((48*I)*3^(1/6)*Log[3*x]*Log[1 + (-3/5)^(1/3)*x])/(
5^(2/3)*(1 + (-1)^(1/3))^5) + (16*(-1/3)^(1/3)*Log[3*x]*Log[1 - (-1)^(2/3)*(3/5)^(1/3)*x])/(3*5^(2/3)) - (8*Lo
g[5^(1/3) - 3^(1/3)*x])/(3*3^(1/3)*5^(2/3)) + (8*(6 + 5*3^(2/3)*5^(1/3))*Log[5^(1/3) - 3^(1/3)*x])/225 - (16*L
og[45]*Log[5^(1/3) - 3^(1/3)*x])/(9*3^(1/3)*5^(2/3)) + (8*(-1/3)^(1/3)*Log[(-5)^(1/3) + 3^(1/3)*x])/(3*5^(2/3)
) - (32*Log[3*x]*Log[1 + ((3/5)^(1/3)*(1 - I*Sqrt[3])*x)/2])/(3*3^(1/3)*5^(2/3)*(1 - I*Sqrt[3])) + (4*(12 - 5*
3^(2/3)*5^(1/3))*Log[5^(2/3) + 15^(1/3)*x + 3^(2/3)*x^2])/225 - (16*Log[5 - 3*x^3])/75 - (16*(-1)^(2/3)*PolyLo
g[2, -((-3/5)^(1/3)*x)])/(3*3^(1/3)*5^(2/3)) - ((48*I)*3^(1/6)*PolyLog[2, -((-3/5)^(1/3)*x)])/(5^(2/3)*(1 + (-
1)^(1/3))^5) + (16*(-1/3)^(1/3)*PolyLog[2, (-1)^(2/3)*(3/5)^(1/3)*x])/(3*5^(2/3)) - (32*PolyLog[2, -1/2*((3/5)
^(1/3)*(1 - I*Sqrt[3])*x)])/(3*3^(1/3)*5^(2/3)*(1 - I*Sqrt[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 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 31

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

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], 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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

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 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 2314

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

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 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :
> Simp[(f^m*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p)/(e*r*(q + 1)), x] - Dist[(b*f^m*n*p)/(e*r*(q + 1)), Int[
((d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[
m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -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 6688

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

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 {\left (-5+3 x^3\right ) \left (e^3 \left (5-3 x^3\right )^2+\left (4+5 x-3 x^4\right )^2\right )-2 x \left (100+125 x+144 x^2+60 x^3-225 x^4-72 x^6+135 x^7-27 x^{10}\right ) \log (3 x)}{x \left (5-3 x^3\right )^3} \, dx\\ &=\int \left (\frac {-16 \left (1+\frac {25 e^3}{16}\right )-40 x-25 x^2+30 e^3 x^3+24 x^4+30 x^5-9 e^3 x^6-9 x^8}{x \left (5-3 x^3\right )^2}-\frac {2 \left (-4-5 x+3 x^4\right ) \left (25+36 x^2-30 x^3+9 x^6\right ) \log (3 x)}{\left (-5+3 x^3\right )^3}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-4-5 x+3 x^4\right ) \left (25+36 x^2-30 x^3+9 x^6\right ) \log (3 x)}{\left (-5+3 x^3\right )^3} \, dx\right )+\int \frac {-16 \left (1+\frac {25 e^3}{16}\right )-40 x-25 x^2+30 e^3 x^3+24 x^4+30 x^5-9 e^3 x^6-9 x^8}{x \left (5-3 x^3\right )^2} \, dx\\ &=-\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {1}{405} \int \frac {-81 \left (16+25 e^3\right )-3240 x-2025 x^2+1215 e^3 x^3+1215 x^5}{x \left (5-3 x^3\right )} \, dx-2 \int \left (x \log (3 x)-\frac {144 x^2 \log (3 x)}{\left (-5+3 x^3\right )^3}+\frac {60 \log (3 x)}{\left (-5+3 x^3\right )^2}+\frac {8 \log (3 x)}{-5+3 x^3}\right ) \, dx\\ &=-\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {1}{405} \int \left (-\frac {81 \left (16+25 e^3\right )}{5 x}-405 x+\frac {648 \left (25+6 x^2\right )}{5 \left (-5+3 x^3\right )}\right ) \, dx-2 \int x \log (3 x) \, dx-16 \int \frac {\log (3 x)}{-5+3 x^3} \, dx-120 \int \frac {\log (3 x)}{\left (-5+3 x^3\right )^2} \, dx+288 \int \frac {x^2 \log (3 x)}{\left (-5+3 x^3\right )^3} \, dx\\ &=-\frac {16 x^3}{25 \left (5-3 x^3\right )}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-x^2 \log (3 x)-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}+\frac {8}{25} \int \frac {25+6 x^2}{-5+3 x^3} \, dx+16 \int \frac {1}{x \left (-5+3 x^3\right )^2} \, dx-16 \int \left (-\frac {\log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{5}+\sqrt [3]{-3} x\right )}-\frac {\log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}-\frac {\log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{5}-(-1)^{2/3} \sqrt [3]{3} x\right )}\right ) \, dx-120 \int \left (\frac {2 \log (3 x)}{15 \sqrt [3]{3} 5^{2/3} \left (3^{2/3} \sqrt [3]{5}-3 x\right )}+\frac {3 (-1)^{2/3} \sqrt [3]{\frac {3}{5}} \log (3 x)}{5 \left (1+\sqrt [3]{-1}\right )^4 \left (3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x\right )^2}+\frac {6 (-1)^{5/6} \sqrt [6]{3} \log (3 x)}{5\ 5^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x\right )}+\frac {3 \sqrt [3]{\frac {3}{5}} \log (3 x)}{5 \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (-3^{2/3} \sqrt [3]{5}+3 (-1)^{2/3} x\right )^2}+\frac {4 \log (3 x)}{15 \sqrt [3]{3} 5^{2/3} \left (2\ 3^{2/3} \sqrt [3]{5}+3 \left (1-i \sqrt {3}\right ) x\right )}-\frac {\log (3 x)}{15\ 3^{2/3} \sqrt [3]{5} \left (-\sqrt [3]{3} 5^{2/3}+2\ 3^{2/3} \sqrt [3]{5} x-3 x^2\right )}\right ) \, dx\\ &=-\frac {16 x^3}{25 \left (5-3 x^3\right )}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-x^2 \log (3 x)-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}+\frac {16}{3} \operatorname {Subst}\left (\int \frac {1}{x (-5+3 x)^2} \, dx,x,x^3\right )-\left (8 \sqrt [3]{\frac {3}{5}}\right ) \int \frac {\log (3 x)}{\left (3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x\right )^2} \, dx-\left (8 \sqrt [3]{\frac {3}{5}}\right ) \int \frac {\log (3 x)}{\left (-3^{2/3} \sqrt [3]{5}+3 (-1)^{2/3} x\right )^2} \, dx+\frac {16 \int \frac {\log (3 x)}{\sqrt [3]{5}+\sqrt [3]{-3} x} \, dx}{3\ 5^{2/3}}+\frac {16 \int \frac {\log (3 x)}{\sqrt [3]{5}-\sqrt [3]{3} x} \, dx}{3\ 5^{2/3}}+\frac {16 \int \frac {\log (3 x)}{\sqrt [3]{5}-(-1)^{2/3} \sqrt [3]{3} x} \, dx}{3\ 5^{2/3}}-\frac {8 \int \frac {\sqrt [3]{\frac {5}{3}} \left (50-2 \sqrt [3]{3} 5^{2/3}\right )+\left (25-4 \sqrt [3]{3} 5^{2/3}\right ) x}{\left (\frac {5}{3}\right )^{2/3}+\sqrt [3]{\frac {5}{3}} x+x^2} \, dx}{75 \sqrt [3]{3} 5^{2/3}}-\frac {16 \int \frac {\log (3 x)}{3^{2/3} \sqrt [3]{5}-3 x} \, dx}{\sqrt [3]{3} 5^{2/3}}-\frac {32 \int \frac {\log (3 x)}{2\ 3^{2/3} \sqrt [3]{5}+3 \left (1-i \sqrt {3}\right ) x} \, dx}{\sqrt [3]{3} 5^{2/3}}+\frac {8 \int \frac {\log (3 x)}{-\sqrt [3]{3} 5^{2/3}+2\ 3^{2/3} \sqrt [3]{5} x-3 x^2} \, dx}{3^{2/3} \sqrt [3]{5}}-\frac {\left (144 (-1)^{5/6} \sqrt [6]{3}\right ) \int \frac {\log (3 x)}{3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x} \, dx}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {1}{225} \left (8 \left (6+5\ 3^{2/3} \sqrt [3]{5}\right )\right ) \int \frac {1}{\sqrt [3]{\frac {5}{3}}-x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 35, normalized size = 1.21 \begin {gather*} -e^3 \log (x)-\frac {\left (-4-5 x+3 x^4\right )^2 \log (3 x)}{\left (-5+3 x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(80 + 200*x + 125*x^2 - 48*x^3 - 240*x^4 - 225*x^5 + 72*x^7 + 135*x^8 - 27*x^11 + E^3*(125 - 225*x^3
+ 135*x^6 - 27*x^9) + (200*x + 250*x^2 + 288*x^3 + 120*x^4 - 450*x^5 - 144*x^7 + 270*x^8 - 54*x^11)*Log[3*x])
/(-125*x + 225*x^4 - 135*x^7 + 27*x^10),x]

[Out]

-(E^3*Log[x]) - ((-4 - 5*x + 3*x^4)^2*Log[3*x])/(-5 + 3*x^3)^2

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fricas [B]  time = 0.76, size = 60, normalized size = 2.07 \begin {gather*} -\frac {{\left (9 \, x^{8} - 30 \, x^{5} - 24 \, x^{4} + 25 \, x^{2} + {\left (9 \, x^{6} - 30 \, x^{3} + 25\right )} e^{3} + 40 \, x + 16\right )} \log \left (3 \, x\right )}{9 \, x^{6} - 30 \, x^{3} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x^11+270*x^8-144*x^7-450*x^5+120*x^4+288*x^3+250*x^2+200*x)*log(3*x)+(-27*x^9+135*x^6-225*x^3+
125)*exp(3)-27*x^11+135*x^8+72*x^7-225*x^5-240*x^4-48*x^3+125*x^2+200*x+80)/(27*x^10-135*x^7+225*x^4-125*x),x,
algorithm="fricas")

[Out]

-(9*x^8 - 30*x^5 - 24*x^4 + 25*x^2 + (9*x^6 - 30*x^3 + 25)*e^3 + 40*x + 16)*log(3*x)/(9*x^6 - 30*x^3 + 25)

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giac [B]  time = 0.16, size = 90, normalized size = 3.10 \begin {gather*} -\frac {9 \, x^{8} \log \left (3 \, x\right ) + 9 \, x^{6} e^{3} \log \relax (x) - 30 \, x^{5} \log \left (3 \, x\right ) - 24 \, x^{4} \log \left (3 \, x\right ) - 30 \, x^{3} e^{3} \log \relax (x) + 25 \, x^{2} \log \left (3 \, x\right ) + 40 \, x \log \left (3 \, x\right ) + 25 \, e^{3} \log \relax (x) + 16 \, \log \left (3 \, x\right )}{9 \, x^{6} - 30 \, x^{3} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x^11+270*x^8-144*x^7-450*x^5+120*x^4+288*x^3+250*x^2+200*x)*log(3*x)+(-27*x^9+135*x^6-225*x^3+
125)*exp(3)-27*x^11+135*x^8+72*x^7-225*x^5-240*x^4-48*x^3+125*x^2+200*x+80)/(27*x^10-135*x^7+225*x^4-125*x),x,
algorithm="giac")

[Out]

-(9*x^8*log(3*x) + 9*x^6*e^3*log(x) - 30*x^5*log(3*x) - 24*x^4*log(3*x) - 30*x^3*e^3*log(x) + 25*x^2*log(3*x)
+ 40*x*log(3*x) + 25*e^3*log(x) + 16*log(3*x))/(9*x^6 - 30*x^3 + 25)

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maple [A]  time = 0.17, size = 53, normalized size = 1.83

method	result	size
risch	\(-\frac {\left (9 x^{8}-30 x^{5}-24 x^{4}+25 x^{2}+40 x +16\right ) \ln \left (3 x \right )}{9 x^{6}-30 x^{3}+25}-\ln \relax (x ) {\mathrm e}^{3}\)	\(53\)
derivativedivides	\(-\ln \left (3 x \right ) {\mathrm e}^{3}-\frac {16 \ln \left (3 x \right )}{25}-x^{2} \ln \left (3 x \right )+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}\)	\(66\)
default	\(-\ln \left (3 x \right ) {\mathrm e}^{3}-\frac {16 \ln \left (3 x \right )}{25}-x^{2} \ln \left (3 x \right )+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}\)	\(66\)
norman	\(\frac {-16 \ln \left (3 x \right )-40 x \ln \left (3 x \right )-25 x^{2} \ln \left (3 x \right )+24 \ln \left (3 x \right ) x^{4}+30 \ln \left (3 x \right ) x^{5}-9 \ln \left (3 x \right ) x^{8}}{\left (3 x^{3}-5\right )^{2}}-\ln \relax (x ) {\mathrm e}^{3}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-54*x^11+270*x^8-144*x^7-450*x^5+120*x^4+288*x^3+250*x^2+200*x)*ln(3*x)+(-27*x^9+135*x^6-225*x^3+125)*ex
p(3)-27*x^11+135*x^8+72*x^7-225*x^5-240*x^4-48*x^3+125*x^2+200*x+80)/(27*x^10-135*x^7+225*x^4-125*x),x,method=
_RETURNVERBOSE)

[Out]

-(9*x^8-30*x^5-24*x^4+25*x^2+40*x+16)/(9*x^6-30*x^3+25)*ln(3*x)-ln(x)*exp(3)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x^11+270*x^8-144*x^7-450*x^5+120*x^4+288*x^3+250*x^2+200*x)*log(3*x)+(-27*x^9+135*x^6-225*x^3+
125)*exp(3)-27*x^11+135*x^8+72*x^7-225*x^5-240*x^4-48*x^3+125*x^2+200*x+80)/(27*x^10-135*x^7+225*x^4-125*x),x,
algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 5.65, size = 48, normalized size = 1.66 \begin {gather*} -{\mathrm {e}}^3\,\ln \relax (x)-\frac {\ln \left (3\,x\right )\,\left (x^8-\frac {10\,x^5}{3}-\frac {8\,x^4}{3}+\frac {25\,x^2}{9}+\frac {40\,x}{9}+\frac {16}{9}\right )}{x^6-\frac {10\,x^3}{3}+\frac {25}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(200*x + log(3*x)*(200*x + 250*x^2 + 288*x^3 + 120*x^4 - 450*x^5 - 144*x^7 + 270*x^8 - 54*x^11) - exp(3)*
(225*x^3 - 135*x^6 + 27*x^9 - 125) + 125*x^2 - 48*x^3 - 240*x^4 - 225*x^5 + 72*x^7 + 135*x^8 - 27*x^11 + 80)/(
125*x - 225*x^4 + 135*x^7 - 27*x^10),x)

[Out]

- exp(3)*log(x) - (log(3*x)*((40*x)/9 + (25*x^2)/9 - (8*x^4)/3 - (10*x^5)/3 + x^8 + 16/9))/(x^6 - (10*x^3)/3 +
25/9)

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sympy [B]  time = 0.40, size = 48, normalized size = 1.66 \begin {gather*} - e^{3} \log {\relax (x )} + \frac {\left (- 9 x^{8} + 30 x^{5} + 24 x^{4} - 25 x^{2} - 40 x - 16\right ) \log {\left (3 x \right )}}{9 x^{6} - 30 x^{3} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-54*x**11+270*x**8-144*x**7-450*x**5+120*x**4+288*x**3+250*x**2+200*x)*ln(3*x)+(-27*x**9+135*x**6-
225*x**3+125)*exp(3)-27*x**11+135*x**8+72*x**7-225*x**5-240*x**4-48*x**3+125*x**2+200*x+80)/(27*x**10-135*x**7
+225*x**4-125*x),x)

[Out]

-exp(3)*log(x) + (-9*x**8 + 30*x**5 + 24*x**4 - 25*x**2 - 40*x - 16)*log(3*x)/(9*x**6 - 30*x**3 + 25)

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