\(\int \frac {(216+324 x+72 x^2-80 x^3-32 x^4) \log ^3(\frac {-24-25 x-6 x^2}{4+4 x+x^2})+(3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5) \log ^4(\frac {-24-25 x-6 x^2}{4+4 x+x^2})}{16+14 x+3 x^2} \, dx\) [7628]

Optimal result

Integrand size = 109, antiderivative size = 21 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx=(3+2 x)^4 \log ^4\left (-6-\frac {x}{(2+x)^2}\right ) \]

[Out]

ln(-6-x/(2+x)^2)^4*(3+2*x)^4

Rubi [A] (verified)

Time = 113.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33, number of steps used = 335, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {6860, 2608, 2603, 2604, 2465, 2437, 2338, 2441, 2440, 2438, 6820, 814, 31, 2605, 1642} \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx=(2 x+3)^4 \log ^4\left (-\frac {6 x^2+25 x+24}{(x+2)^2}\right ) \]

[In]

Int[((216 + 324*x + 72*x^2 - 80*x^3 - 32*x^4)*Log[(-24 - 25*x - 6*x^2)/(4 + 4*x + x^2)]^3 + (3456 + 9936*x + 1
1304*x^2 + 6352*x^3 + 1760*x^4 + 192*x^5)*Log[(-24 - 25*x - 6*x^2)/(4 + 4*x + x^2)]^4)/(16 + 14*x + 3*x^2),x]

[Out]

(3 + 2*x)^4*Log[-((24 + 25*x + 6*x^2)/(2 + x)^2)]^4

Rule 31

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

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + 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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1642

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

Rule 2338

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 2437

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 2438

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 2440

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 2441

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 2465

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 2603

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 2604

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 2605

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 2608

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 6820

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

Rule 6860

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{align*} \text {integral}& = \int \left (-\frac {4 (-2+x) (3+2 x)^3 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) (8+3 x)}+8 (3+2 x)^3 \log ^4\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )\right ) \, dx \\ & = -\left (4 \int \frac {(-2+x) (3+2 x)^3 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) (8+3 x)} \, dx\right )+8 \int (3+2 x)^3 \log ^4\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx \\ & = (3+2 x)^4 \log ^4\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-4 \int \frac {(2-x) (3+2 x)^3 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) (8+3 x)} \, dx-4 \int \left (\frac {182}{27} \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {52}{9} x \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )+\frac {8}{3} x^2 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )+\frac {2 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}-\frac {2401 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{27 (8+3 x)}\right ) \, dx \\ & = (3+2 x)^4 \log ^4\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-4 \int \left (-\frac {182}{27} \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )+\frac {52}{9} x \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {8}{3} x^2 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {2 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}+\frac {2401 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{27 (8+3 x)}\right ) \, dx-8 \int \frac {\log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x} \, dx-\frac {32}{3} \int x^2 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx+\frac {208}{9} \int x \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx-\frac {728}{27} \int \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx+\frac {9604}{27} \int \frac {\log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{8+3 x} \, dx \\ & = -\frac {728}{27} x \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )+\frac {104}{9} x^2 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {32}{9} x^3 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-8 \log (2+x) \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )+\frac {9604}{81} \log (8+3 x) \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )+(3+2 x)^4 \log ^4\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )+8 \int \frac {\log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x} \, dx+\frac {32}{3} \int \frac {(2-x) x^3 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx+\frac {32}{3} \int x^2 \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx-\frac {208}{9} \int x \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx-24 \int \frac {(2+x)^2 \left (-\frac {25+12 x}{(2+x)^2}+\frac {2 \left (24+25 x+6 x^2\right )}{(2+x)^3}\right ) \log (2+x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{24+25 x+6 x^2} \, dx+\frac {728}{27} \int \log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx-\frac {104}{3} \int \frac {(2-x) x^2 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx+\frac {728}{9} \int \frac {(2-x) x \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx+\frac {9604}{27} \int \frac {(2+x)^2 \left (-\frac {25+12 x}{(2+x)^2}+\frac {2 \left (24+25 x+6 x^2\right )}{(2+x)^3}\right ) \log (8+3 x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{24+25 x+6 x^2} \, dx-\frac {9604}{27} \int \frac {\log ^3\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{8+3 x} \, dx \\ & = (3+2 x)^4 \log ^4\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {32}{3} \int \frac {(2-x) x^3 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx+\frac {32}{3} \int \left (\frac {49}{36} \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {1}{6} x \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )+\frac {16 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}-\frac {27 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{4 (3+2 x)}-\frac {512 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{9 (8+3 x)}\right ) \, dx-24 \int \frac {(-2+x) \log (2+x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx+24 \int \frac {(2+x)^2 \left (-\frac {25+12 x}{(2+x)^2}+\frac {2 \left (24+25 x+6 x^2\right )}{(2+x)^3}\right ) \log (2+x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{24+25 x+6 x^2} \, dx+\frac {104}{3} \int \frac {(2-x) x^2 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx-\frac {104}{3} \int \left (-\frac {1}{6} \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {8 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}+\frac {9 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2 (3+2 x)}+\frac {64 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{3 (8+3 x)}\right ) \, dx-\frac {728}{9} \int \frac {(2-x) x \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx+\frac {728}{9} \int \left (\frac {4 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}-\frac {3 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{3+2 x}-\frac {8 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{8+3 x}\right ) \, dx+\frac {9604}{27} \int \frac {(-2+x) \log (8+3 x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx-\frac {9604}{27} \int \frac {(2+x)^2 \left (-\frac {25+12 x}{(2+x)^2}+\frac {2 \left (24+25 x+6 x^2\right )}{(2+x)^3}\right ) \log (8+3 x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{24+25 x+6 x^2} \, dx \\ & = (3+2 x)^4 \log ^4\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {16}{9} \int x \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx+\frac {52}{9} \int \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx-\frac {32}{3} \int \left (\frac {49}{36} \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {1}{6} x \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )+\frac {16 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}-\frac {27 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{4 (3+2 x)}-\frac {512 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{9 (8+3 x)}\right ) \, dx+\frac {392}{27} \int \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \, dx+24 \int \frac {(-2+x) \log (2+x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx-24 \int \left (\frac {2 \log (2+x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}-\frac {2 \log (2+x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{3+2 x}-\frac {3 \log (2+x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{8+3 x}\right ) \, dx+\frac {104}{3} \int \left (-\frac {1}{6} \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )-\frac {8 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}+\frac {9 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2 (3+2 x)}+\frac {64 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{3 (8+3 x)}\right ) \, dx-72 \int \frac {\log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{3+2 x} \, dx-\frac {728}{9} \int \left (\frac {4 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}-\frac {3 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{3+2 x}-\frac {8 \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{8+3 x}\right ) \, dx-156 \int \frac {\log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{3+2 x} \, dx+\frac {512}{3} \int \frac {\log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x} \, dx-\frac {728}{3} \int \frac {\log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{3+2 x} \, dx+\frac {832}{3} \int \frac {\log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x} \, dx+\frac {2912}{9} \int \frac {\log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x} \, dx-\frac {9604}{27} \int \frac {(-2+x) \log (8+3 x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{(2+x) \left (24+25 x+6 x^2\right )} \, dx+\frac {9604}{27} \int \left (\frac {2 \log (8+3 x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{2+x}-\frac {2 \log (8+3 x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{3+2 x}-\frac {3 \log (8+3 x) \log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{8+3 x}\right ) \, dx-\frac {16384}{27} \int \frac {\log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{8+3 x} \, dx-\frac {5824}{9} \int \frac {\log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{8+3 x} \, dx-\frac {6656}{9} \int \frac {\log ^2\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right )}{8+3 x} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx=(3+2 x)^4 \log ^4\left (-\frac {24+25 x+6 x^2}{(2+x)^2}\right ) \]

[In]

Integrate[((216 + 324*x + 72*x^2 - 80*x^3 - 32*x^4)*Log[(-24 - 25*x - 6*x^2)/(4 + 4*x + x^2)]^3 + (3456 + 9936
*x + 11304*x^2 + 6352*x^3 + 1760*x^4 + 192*x^5)*Log[(-24 - 25*x - 6*x^2)/(4 + 4*x + x^2)]^4)/(16 + 14*x + 3*x^
2),x]

[Out]

(3 + 2*x)^4*Log[-((24 + 25*x + 6*x^2)/(2 + x)^2)]^4

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57

method	result	size
risch	\(\left (3+2 x \right )^{4} \ln \left (\frac {-6 x^{2}-25 x -24}{x^{2}+4 x +4}\right )^{4}\)	\(33\)
parallelrisch	\(16 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4} x^{4}+96 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4} x^{3}+216 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4} x^{2}+216 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4} x +81 \ln \left (-\frac {6 x^{2}+25 x +24}{x^{2}+4 x +4}\right )^{4}\)	\(147\)

[In]

int(((192*x^5+1760*x^4+6352*x^3+11304*x^2+9936*x+3456)*ln((-6*x^2-25*x-24)/(x^2+4*x+4))^4+(-32*x^4-80*x^3+72*x
^2+324*x+216)*ln((-6*x^2-25*x-24)/(x^2+4*x+4))^3)/(3*x^2+14*x+16),x,method=_RETURNVERBOSE)

[Out]

(3+2*x)^4*ln((-6*x^2-25*x-24)/(x^2+4*x+4))^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).

Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx={\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac {6 \, x^{2} + 25 \, x + 24}{x^{2} + 4 \, x + 4}\right )^{4} \]

[In]

integrate(((192*x^5+1760*x^4+6352*x^3+11304*x^2+9936*x+3456)*log((-6*x^2-25*x-24)/(x^2+4*x+4))^4+(-32*x^4-80*x
^3+72*x^2+324*x+216)*log((-6*x^2-25*x-24)/(x^2+4*x+4))^3)/(3*x^2+14*x+16),x, algorithm="fricas")

[Out]

(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(6*x^2 + 25*x + 24)/(x^2 + 4*x + 4))^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).

Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx=\left (16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81\right ) \log {\left (\frac {- 6 x^{2} - 25 x - 24}{x^{2} + 4 x + 4} \right )}^{4} \]

[In]

integrate(((192*x**5+1760*x**4+6352*x**3+11304*x**2+9936*x+3456)*ln((-6*x**2-25*x-24)/(x**2+4*x+4))**4+(-32*x*
*4-80*x**3+72*x**2+324*x+216)*ln((-6*x**2-25*x-24)/(x**2+4*x+4))**3)/(3*x**2+14*x+16),x)

[Out]

(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81)*log((-6*x**2 - 25*x - 24)/(x**2 + 4*x + 4))**4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (21) = 42\).

Time = 0.27 (sec) , antiderivative size = 491, normalized size of antiderivative = 23.38 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx={\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (2 \, x + 3\right )^{4} + 16 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{4} - 32 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{3} \log \left (-3 \, x - 8\right ) + 24 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{2} \log \left (-3 \, x - 8\right )^{2} - 8 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right ) \log \left (-3 \, x - 8\right )^{3} + {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-3 \, x - 8\right )^{4} - 4 \, {\left (2 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right ) - {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-3 \, x - 8\right )\right )} \log \left (2 \, x + 3\right )^{3} + 6 \, {\left (4 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{2} - 4 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right ) \log \left (-3 \, x - 8\right ) + {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-3 \, x - 8\right )^{2}\right )} \log \left (2 \, x + 3\right )^{2} - 4 \, {\left (8 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{3} - 12 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right )^{2} \log \left (-3 \, x - 8\right ) + 6 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (x + 2\right ) \log \left (-3 \, x - 8\right )^{2} - {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-3 \, x - 8\right )^{3}\right )} \log \left (2 \, x + 3\right ) \]

[In]

integrate(((192*x^5+1760*x^4+6352*x^3+11304*x^2+9936*x+3456)*log((-6*x^2-25*x-24)/(x^2+4*x+4))^4+(-32*x^4-80*x
^3+72*x^2+324*x+216)*log((-6*x^2-25*x-24)/(x^2+4*x+4))^3)/(3*x^2+14*x+16),x, algorithm="maxima")

[Out]

(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(2*x + 3)^4 + 16*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x +
2)^4 - 32*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)^3*log(-3*x - 8) + 24*(16*x^4 + 96*x^3 + 216*x^2
+ 216*x + 81)*log(x + 2)^2*log(-3*x - 8)^2 - 8*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)*log(-3*x -
8)^3 + (16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-3*x - 8)^4 - 4*(2*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81
)*log(x + 2) - (16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-3*x - 8))*log(2*x + 3)^3 + 6*(4*(16*x^4 + 96*x^3
+ 216*x^2 + 216*x + 81)*log(x + 2)^2 - 4*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)*log(-3*x - 8) + (
16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-3*x - 8)^2)*log(2*x + 3)^2 - 4*(8*(16*x^4 + 96*x^3 + 216*x^2 + 21
6*x + 81)*log(x + 2)^3 - 12*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)^2*log(-3*x - 8) + 6*(16*x^4 +
96*x^3 + 216*x^2 + 216*x + 81)*log(x + 2)*log(-3*x - 8)^2 - (16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-3*x
- 8)^3)*log(2*x + 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).

Time = 4.55 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx={\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac {6 \, x^{2} + 25 \, x + 24}{x^{2} + 4 \, x + 4}\right )^{4} \]

[In]

integrate(((192*x^5+1760*x^4+6352*x^3+11304*x^2+9936*x+3456)*log((-6*x^2-25*x-24)/(x^2+4*x+4))^4+(-32*x^4-80*x
^3+72*x^2+324*x+216)*log((-6*x^2-25*x-24)/(x^2+4*x+4))^3)/(3*x^2+14*x+16),x, algorithm="giac")

[Out]

(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(6*x^2 + 25*x + 24)/(x^2 + 4*x + 4))^4

Mupad [B] (verification not implemented)

Time = 14.57 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {\left (216+324 x+72 x^2-80 x^3-32 x^4\right ) \log ^3\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )+\left (3456+9936 x+11304 x^2+6352 x^3+1760 x^4+192 x^5\right ) \log ^4\left (\frac {-24-25 x-6 x^2}{4+4 x+x^2}\right )}{16+14 x+3 x^2} \, dx={\ln \left (-\frac {6\,x^2+25\,x+24}{x^2+4\,x+4}\right )}^4\,{\left (2\,x+3\right )}^4 \]

[In]

int((log(-(25*x + 6*x^2 + 24)/(4*x + x^2 + 4))^3*(324*x + 72*x^2 - 80*x^3 - 32*x^4 + 216) + log(-(25*x + 6*x^2
+ 24)/(4*x + x^2 + 4))^4*(9936*x + 11304*x^2 + 6352*x^3 + 1760*x^4 + 192*x^5 + 3456))/(14*x + 3*x^2 + 16),x)

[Out]

log(-(25*x + 6*x^2 + 24)/(4*x + x^2 + 4))^4*(2*x + 3)^4