3.32.29 \(\int \frac {(-100 x+200 x^2) \log ^3(5-x+x^2)+(125-25 x+25 x^2) \log ^4(5-x+x^2)}{3645-729 x+729 x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac {25}{729} x \log ^4\left (5-x+x^2\right ) \]

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Rubi [A]  time = 115.85, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 131, number of rules used = 23, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.404, Rules used = {6728, 2528, 2523, 773, 634, 618, 204, 628, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2527, 12, 5057, 4920, 4854, 2402, 2315, 31} \begin {gather*} \frac {25}{729} x \log ^4\left (x^2-x+5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-100*x + 200*x^2)*Log[5 - x + x^2]^3 + (125 - 25*x + 25*x^2)*Log[5 - x + x^2]^4)/(3645 - 729*x + 729*x^2
),x]

[Out]

(25*x*Log[5 - x + x^2]^4)/729

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 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 773

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

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 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 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 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*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 2527

Int[Log[(c_.)*(Px_)^(n_.)]/(Qx_), x_Symbol] :> With[{u = IntHide[1/Qx, x]}, Simp[u*Log[c*Px^n], x] - Dist[n, I
nt[SimplifyIntegrand[(u*D[Px, x])/Px, x], x], x]] /; FreeQ[{c, n}, x] && QuadraticQ[{Qx, Px}, x] && EqQ[D[Px/Q
x, x], 0]

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 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 5057

Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_
)^2)^(q_.), x_Symbol] :> Dist[1/d, Subst[Int[((d*e - c*f)/d + (f*x)/d)^m*(C/d^2 + (C*x^2)/d^2)^q*(a + b*ArcTan
[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0]
&& EqQ[2*c*C - B*d, 0]

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 \left (\frac {100 x (-1+2 x) \log ^3\left (5-x+x^2\right )}{729 \left (5-x+x^2\right )}+\frac {25}{729} \log ^4\left (5-x+x^2\right )\right ) \, dx\\ &=\frac {25}{729} \int \log ^4\left (5-x+x^2\right ) \, dx+\frac {100}{729} \int \frac {x (-1+2 x) \log ^3\left (5-x+x^2\right )}{5-x+x^2} \, dx\\ &=\frac {25}{729} x \log ^4\left (5-x+x^2\right )-\frac {100}{729} \int \frac {x (-1+2 x) \log ^3\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {100}{729} \int \left (2 \log ^3\left (5-x+x^2\right )-\frac {(10-x) \log ^3\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx\\ &=\frac {25}{729} x \log ^4\left (5-x+x^2\right )-\frac {100}{729} \int \frac {(10-x) \log ^3\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {100}{729} \int \left (2 \log ^3\left (5-x+x^2\right )-\frac {(10-x) \log ^3\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx+\frac {200}{729} \int \log ^3\left (5-x+x^2\right ) \, dx\\ &=\frac {200}{729} x \log ^3\left (5-x+x^2\right )+\frac {25}{729} x \log ^4\left (5-x+x^2\right )+\frac {100}{729} \int \frac {(10-x) \log ^3\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {100}{729} \int \left (\frac {\left (-1-i \sqrt {19}\right ) \log ^3\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x}+\frac {\left (-1+i \sqrt {19}\right ) \log ^3\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x}\right ) \, dx-\frac {200}{729} \int \log ^3\left (5-x+x^2\right ) \, dx-\frac {200}{243} \int \frac {x (-1+2 x) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx\\ &=\frac {25}{729} x \log ^4\left (5-x+x^2\right )+\frac {100}{729} \int \left (\frac {\left (-1-i \sqrt {19}\right ) \log ^3\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x}+\frac {\left (-1+i \sqrt {19}\right ) \log ^3\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x}\right ) \, dx+\frac {200}{243} \int \frac {x (-1+2 x) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {200}{243} \int \left (2 \log ^2\left (5-x+x^2\right )-\frac {(10-x) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx+\frac {1}{729} \left (100 \left (1-i \sqrt {19}\right )\right ) \int \frac {\log ^3\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x} \, dx+\frac {1}{729} \left (100 \left (1+i \sqrt {19}\right )\right ) \int \frac {\log ^3\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x} \, dx\\ &=\frac {50}{729} \left (1+i \sqrt {19}\right ) \log \left (-1-i \sqrt {19}+2 x\right ) \log ^3\left (5-x+x^2\right )+\frac {50}{729} \left (1-i \sqrt {19}\right ) \log \left (-1+i \sqrt {19}+2 x\right ) \log ^3\left (5-x+x^2\right )+\frac {25}{729} x \log ^4\left (5-x+x^2\right )+\frac {200}{243} \int \frac {(10-x) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {200}{243} \int \left (2 \log ^2\left (5-x+x^2\right )-\frac {(10-x) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx-\frac {400}{243} \int \log ^2\left (5-x+x^2\right ) \, dx-\frac {1}{729} \left (100 \left (1-i \sqrt {19}\right )\right ) \int \frac {\log ^3\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x} \, dx-\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \frac {(-1+2 x) \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{729} \left (100 \left (1+i \sqrt {19}\right )\right ) \int \frac {\log ^3\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x} \, dx-\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \frac {(-1+2 x) \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx\\ &=-\frac {400}{243} x \log ^2\left (5-x+x^2\right )+\frac {25}{729} x \log ^4\left (5-x+x^2\right )-\frac {200}{243} \int \frac {(10-x) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {200}{243} \int \left (\frac {\left (-1-i \sqrt {19}\right ) \log ^2\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x}+\frac {\left (-1+i \sqrt {19}\right ) \log ^2\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x}\right ) \, dx+\frac {400}{243} \int \log ^2\left (5-x+x^2\right ) \, dx+\frac {800}{243} \int \frac {x (-1+2 x) \log \left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \frac {(-1+2 x) \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \left (-\frac {\log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}+\frac {2 x \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx+\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \frac {(-1+2 x) \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \left (-\frac {\log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}+\frac {2 x \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx\\ &=\frac {25}{729} x \log ^4\left (5-x+x^2\right )-\frac {200}{243} \int \left (\frac {\left (-1-i \sqrt {19}\right ) \log ^2\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x}+\frac {\left (-1+i \sqrt {19}\right ) \log ^2\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x}\right ) \, dx-\frac {800}{243} \int \frac {x (-1+2 x) \log \left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {800}{243} \int \left (2 \log \left (5-x+x^2\right )-\frac {(10-x) \log \left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx+\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \frac {\log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {1}{243} \left (50 \left (1-i \sqrt {19}\right )\right ) \int \left (-\frac {\log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}+\frac {2 x \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx-\frac {1}{243} \left (100 \left (1-i \sqrt {19}\right )\right ) \int \frac {x \log \left (-1+i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{243} \left (200 \left (1-i \sqrt {19}\right )\right ) \int \frac {\log ^2\left (5-x+x^2\right )}{-1+i \sqrt {19}+2 x} \, dx+\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \frac {\log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx+\frac {1}{243} \left (50 \left (1+i \sqrt {19}\right )\right ) \int \left (-\frac {\log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}+\frac {2 x \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2}\right ) \, dx-\frac {1}{243} \left (100 \left (1+i \sqrt {19}\right )\right ) \int \frac {x \log \left (-1-i \sqrt {19}+2 x\right ) \log ^2\left (5-x+x^2\right )}{5-x+x^2} \, dx-\frac {1}{243} \left (200 \left (1+i \sqrt {19}\right )\right ) \int \frac {\log ^2\left (5-x+x^2\right )}{-1-i \sqrt {19}+2 x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.49, size = 16, normalized size = 1.00 \begin {gather*} \frac {25}{729} x \log ^4\left (5-x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-100*x + 200*x^2)*Log[5 - x + x^2]^3 + (125 - 25*x + 25*x^2)*Log[5 - x + x^2]^4)/(3645 - 729*x + 7
29*x^2),x]

[Out]

(25*x*Log[5 - x + x^2]^4)/729

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fricas [A]  time = 0.54, size = 14, normalized size = 0.88 \begin {gather*} \frac {25}{729} \, x \log \left (x^{2} - x + 5\right )^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^2-25*x+125)*log(x^2-x+5)^4+(200*x^2-100*x)*log(x^2-x+5)^3)/(729*x^2-729*x+3645),x, algorithm=
"fricas")

[Out]

25/729*x*log(x^2 - x + 5)^4

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giac [A]  time = 0.82, size = 14, normalized size = 0.88 \begin {gather*} \frac {25}{729} \, x \log \left (x^{2} - x + 5\right )^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^2-25*x+125)*log(x^2-x+5)^4+(200*x^2-100*x)*log(x^2-x+5)^3)/(729*x^2-729*x+3645),x, algorithm=
"giac")

[Out]

25/729*x*log(x^2 - x + 5)^4

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maple [A]  time = 0.91, size = 15, normalized size = 0.94

method	result	size
norman	\(\frac {25 x \ln \left (x^{2}-x +5\right )^{4}}{729}\)	\(15\)
risch	\(\frac {25 x \ln \left (x^{2}-x +5\right )^{4}}{729}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((25*x^2-25*x+125)*ln(x^2-x+5)^4+(200*x^2-100*x)*ln(x^2-x+5)^3)/(729*x^2-729*x+3645),x,method=_RETURNVERBO
SE)

[Out]

25/729*x*ln(x^2-x+5)^4

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maxima [A]  time = 1.25, size = 14, normalized size = 0.88 \begin {gather*} \frac {25}{729} \, x \log \left (x^{2} - x + 5\right )^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^2-25*x+125)*log(x^2-x+5)^4+(200*x^2-100*x)*log(x^2-x+5)^3)/(729*x^2-729*x+3645),x, algorithm=
"maxima")

[Out]

25/729*x*log(x^2 - x + 5)^4

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mupad [B]  time = 1.89, size = 14, normalized size = 0.88 \begin {gather*} \frac {25\,x\,{\ln \left (x^2-x+5\right )}^4}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2 - x + 5)^4*(25*x^2 - 25*x + 125) - log(x^2 - x + 5)^3*(100*x - 200*x^2))/(729*x^2 - 729*x + 3645)
,x)

[Out]

(25*x*log(x^2 - x + 5)^4)/729

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sympy [A]  time = 0.14, size = 14, normalized size = 0.88 \begin {gather*} \frac {25 x \log {\left (x^{2} - x + 5 \right )}^{4}}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x**2-25*x+125)*ln(x**2-x+5)**4+(200*x**2-100*x)*ln(x**2-x+5)**3)/(729*x**2-729*x+3645),x)

[Out]

25*x*log(x**2 - x + 5)**4/729

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