3.4.98 \(\int (a+b \log (c (d+e x)^n))^3 (f+g \log (h (i+j x)^m)) \, dx\) [398]

Optimal. Leaf size=1147 \[ 6 a b^2 f n^2 x-18 a b^2 g m n^2 x-6 b^3 f n^3 x+24 b^3 g m n^3 x+\frac {6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {18 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^2 g i m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}+\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{e}-\frac {3 b g i m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{e}+\frac {g i m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}-\frac {6 b^3 g n^3 (i+j x) \log \left (h (i+j x)^m\right )}{j}+\frac {6 b^3 d g n^3 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{e}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )-\frac {3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (i+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {6 b^3 g i m n^3 \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{j}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{e}-\frac {6 b^2 g i m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{j}-\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{e}+\frac {3 b g i m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{e i-d j}\right )}{j}+\frac {6 b^3 d g m n^3 \text {Li}_2\left (\frac {e (i+j x)}{e i-d j}\right )}{e}-\frac {6 b^3 d g m n^3 \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{e}+\frac {6 b^3 g i m n^3 \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{j}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{e}-\frac {6 b^2 g i m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{j}-\frac {6 b^3 d g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{e i-d j}\right )}{e}+\frac {6 b^3 g i m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{e i-d j}\right )}{j} \]

[Out]

-g*m*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^3/e+d*g*(a+b*ln(c*(e*x+d)^n))^3*ln(h*(j*x+i)^m)/e+d*f*(a+b*ln(c*(e*x+d)^n))
^3/e+6*a*b^2*f*n^2*x+24*b^3*g*m*n^3*x+6*b^2*g*i*m*n^2*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/j+3*b*d*g
*m*n*(a+b*ln(c*(e*x+d)^n))^2*ln(e*(j*x+i)/(-d*j+e*i))/e-3*b*g*i*m*n*(a+b*ln(c*(e*x+d)^n))^2*ln(e*(j*x+i)/(-d*j
+e*i))/j+6*b^2*d*g*m*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(2,-j*(e*x+d)/(-d*j+e*i))/e-6*b^2*g*i*m*n^2*(a+b*ln(c*(e
*x+d)^n))*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j-3*b*d*g*m*n*(a+b*ln(c*(e*x+d)^n))^2*polylog(2,-j*(e*x+d)/(-d*j+e*
i))/e+3*b*g*i*m*n*(a+b*ln(c*(e*x+d)^n))^2*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j+6*b^2*d*g*m*n^2*(a+b*ln(c*(e*x+d)
^n))*polylog(3,-j*(e*x+d)/(-d*j+e*i))/e-6*b^2*g*i*m*n^2*(a+b*ln(c*(e*x+d)^n))*polylog(3,-j*(e*x+d)/(-d*j+e*i))
/j+x*(a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m))-6*b^3*f*n^3*x+6*b^3*g*i*m*n^3*polylog(2,-j*(e*x+d)/(-d*j+e*
i))/j+6*b^3*d*g*m*n^3*polylog(2,e*(j*x+i)/(-d*j+e*i))/e-6*b^3*d*g*m*n^3*polylog(3,-j*(e*x+d)/(-d*j+e*i))/e+6*b
^3*g*i*m*n^3*polylog(3,-j*(e*x+d)/(-d*j+e*i))/j-6*b^3*d*g*m*n^3*polylog(4,-j*(e*x+d)/(-d*j+e*i))/e+6*b^3*g*i*m
*n^3*polylog(4,-j*(e*x+d)/(-d*j+e*i))/j-18*a*b^2*g*m*n^2*x-d*g*m*(a+b*ln(c*(e*x+d)^n))^3*ln(e*(j*x+i)/(-d*j+e*
i))/e+g*i*m*(a+b*ln(c*(e*x+d)^n))^3*ln(e*(j*x+i)/(-d*j+e*i))/j+6*b^3*f*n^2*(e*x+d)*ln(c*(e*x+d)^n)/e-3*b*f*n*(
e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e-6*b^3*g*n^3*(j*x+i)*ln(h*(j*x+i)^m)/j+6*b^2*g*n^2*x*(a+b*ln(c*(e*x+d)^n))*ln(
h*(j*x+i)^m)-3*b*g*n*x*(a+b*ln(c*(e*x+d)^n))^2*ln(h*(j*x+i)^m)-18*b^3*g*m*n^2*(e*x+d)*ln(c*(e*x+d)^n)/e+6*b*g*
m*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e+6*b^3*d*g*n^3*ln(-j*(e*x+d)/(-d*j+e*i))*ln(h*(j*x+i)^m)/e-3*b*d*g*n*(a+b
*ln(c*(e*x+d)^n))^2*ln(h*(j*x+i)^m)/e

________________________________________________________________________________________

Rubi [A]
time = 2.17, antiderivative size = 1147, normalized size of antiderivative = 1.00, number of steps used = 64, number of rules used = 22, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.710, Rules used = {2479, 2463, 2436, 2333, 2332, 2443, 2481, 2421, 2430, 6724, 6874, 2458, 2388, 2339, 30, 2338, 45, 2441, 2440, 2438, 2422, 2354} \begin {gather*} -6 f n^3 x b^3+24 g m n^3 x b^3+\frac {6 f n^2 (d+e x) \log \left (c (d+e x)^n\right ) b^3}{e}-\frac {18 g m n^2 (d+e x) \log \left (c (d+e x)^n\right ) b^3}{e}-\frac {6 g n^3 (i+j x) \log \left (h (i+j x)^m\right ) b^3}{j}+\frac {6 d g n^3 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right ) b^3}{e}+\frac {6 g i m n^3 \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b^3}{j}+\frac {6 d g m n^3 \text {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right ) b^3}{e}-\frac {6 d g m n^3 \text {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^3}{e}+\frac {6 g i m n^3 \text {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^3}{j}-\frac {6 d g m n^3 \text {PolyLog}\left (4,-\frac {j (d+e x)}{e i-d j}\right ) b^3}{e}+\frac {6 g i m n^3 \text {PolyLog}\left (4,-\frac {j (d+e x)}{e i-d j}\right ) b^3}{j}+6 a f n^2 x b^2-18 a g m n^2 x b^2+\frac {6 g i m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right ) b^2}{j}+6 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) b^2+\frac {6 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{e}-\frac {6 g i m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{j}+\frac {6 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{e}-\frac {6 g i m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{j}-\frac {3 f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 b}{e}+\frac {6 g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 b}{e}+\frac {3 d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right ) b}{e}-\frac {3 g i m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right ) b}{j}-\frac {3 d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right ) b}{e}-3 g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right ) b-\frac {3 d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b}{e}+\frac {3 g i m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b}{j}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{e}+\frac {g i m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (i+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]),x]

[Out]

6*a*b^2*f*n^2*x - 18*a*b^2*g*m*n^2*x - 6*b^3*f*n^3*x + 24*b^3*g*m*n^3*x + (6*b^3*f*n^2*(d + e*x)*Log[c*(d + e*
x)^n])/e - (18*b^3*g*m*n^2*(d + e*x)*Log[c*(d + e*x)^n])/e - (3*b*f*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/
e + (6*b*g*m*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/e + (d*f*(a + b*Log[c*(d + e*x)^n])^3)/e - (g*m*(d + e*
x)*(a + b*Log[c*(d + e*x)^n])^3)/e + (6*b^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)
])/j + (3*b*d*g*m*n*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/e - (3*b*g*i*m*n*(a + b*Log[c
*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/j - (d*g*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*
i - d*j)])/e + (g*i*m*(a + b*Log[c*(d + e*x)^n])^3*Log[(e*(i + j*x))/(e*i - d*j)])/j - (6*b^3*g*n^3*(i + j*x)*
Log[h*(i + j*x)^m])/j + (6*b^3*d*g*n^3*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/e + 6*b^2*g*n^2*x
*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m] - (3*b*d*g*n*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/e
- 3*b*g*n*x*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m] + (d*g*(a + b*Log[c*(d + e*x)^n])^3*Log[h*(i + j*
x)^m])/e + x*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]) + (6*b^3*g*i*m*n^3*PolyLog[2, -((j*(d + e
*x))/(e*i - d*j))])/j + (6*b^2*d*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/
e - (6*b^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j - (3*b*d*g*m*n*(a
+ b*Log[c*(d + e*x)^n])^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e + (3*b*g*i*m*n*(a + b*Log[c*(d + e*x)^n]
)^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j + (6*b^3*d*g*m*n^3*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/e -
(6*b^3*d*g*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e + (6*b^3*g*i*m*n^3*PolyLog[3, -((j*(d + e*x))/(e*
i - d*j))])/j + (6*b^2*d*g*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e - (6*b
^2*g*i*m*n^2*(a + b*Log[c*(d + e*x)^n])*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j - (6*b^3*d*g*m*n^3*PolyLog
[4, -((j*(d + e*x))/(e*i - d*j))])/e + (6*b^3*g*i*m*n^3*PolyLog[4, -((j*(d + e*x))/(e*i - d*j))])/j

Rule 30

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2354

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 2388

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[(d
+ e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x), x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2422

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

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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 2443

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

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2463

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 2479

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.)), x_Symbol] :> Simp[x*(a + b*Log[c*(d + e*x)^n])^p*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[g*j*m, Int[x*
((a + b*Log[c*(d + e*x)^n])^p/(i + j*x)), x], x] - Dist[b*e*n*p, Int[x*(a + b*Log[c*(d + e*x)^n])^(p - 1)*((f
+ g*Log[h*(i + j*x)^m])/(d + e*x)), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0]

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6874

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

Rubi steps

\begin {align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right ) \, dx &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-(g j m) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{398+j x} \, dx-(3 b e n) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (398+j x)^m\right )\right )}{d+e x} \, dx\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-(g j m) \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{j}-\frac {398 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{j (398+j x)}\right ) \, dx-(3 b e n) \int \left (\frac {f x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x}+\frac {g x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{d+e x}\right ) \, dx\\ &=x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-(g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx+(398 g m) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{398+j x} \, dx-(3 b e f n) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx-(3 b e g n) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{d+e x} \, dx\\ &=\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac {(g m) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e}-(3 b f n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e x\right )-(3 b e g n) \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-\frac {d \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e (d+e x)}\right ) \, dx-\frac {(1194 b e g m n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{d+e x} \, dx}{j}\\ &=-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac {(3 b f n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}+\frac {(3 b d f n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e x\right )}{e}-(3 b g n) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right ) \, dx+(3 b d g n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{d+e x} \, dx+\frac {(3 b g m n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}-\frac {(1194 b g m n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (\frac {e \left (\frac {398 e-d j}{e}+\frac {j x}{e}\right )}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {(3 d f) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac {(3 b d g n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (h \left (\frac {398 e-d j}{e}+\frac {j x}{e}\right )^m\right )}{x} \, dx,x,d+e x\right )}{e}+(3 b g j m n) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{398+j x} \, dx+\frac {\left (6 b^2 f n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\left (6 b^2 e g n^2\right ) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )}{d+e x} \, dx-\frac {\left (6 b^2 g m n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}-\frac {\left (2388 b^2 g m n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=6 a b^2 f n^2 x-6 a b^2 g m n^2 x-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {(d g j m) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\frac {398 e-d j}{e}+\frac {j x}{e}} \, dx,x,d+e x\right )}{e^2}+(3 b g j m n) \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}-\frac {398 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j (398+j x)}\right ) \, dx+\frac {\left (6 b^3 f n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\left (6 b^2 e g n^2\right ) \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )}{e}-\frac {d \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )}{e (d+e x)}\right ) \, dx-\frac {\left (6 b^3 g m n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}+\frac {\left (2388 b^3 g m n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=6 a b^2 f n^2 x-6 a b^2 g m n^2 x-6 b^3 f n^3 x+6 b^3 g m n^3 x+\frac {6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {2388 b^3 g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+(3 b g m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx-(1194 b g m n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{398+j x} \, dx+\frac {(3 b d g m n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}+\left (6 b^2 g n^2\right ) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right ) \, dx-\left (6 b^2 d g n^2\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )}{d+e x} \, dx\\ &=6 a b^2 f n^2 x-6 a b^2 g m n^2 x-6 b^3 f n^3 x+6 b^3 g m n^3 x+\frac {6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {3 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {2388 b^3 g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {(3 b g m n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e}-\frac {\left (6 b^2 d g n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (h \left (\frac {398 e-d j}{e}+\frac {j x}{e}\right )^m\right )}{x} \, dx,x,d+e x\right )}{e}+\frac {\left (6 b^2 d g m n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}+\frac {\left (2388 b^2 e g m n^2\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{d+e x} \, dx}{j}-\left (6 b^2 g j m n^2\right ) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{398+j x} \, dx-\left (6 b^3 e g n^3\right ) \int \frac {x \log \left (h (398+j x)^m\right )}{d+e x} \, dx\\ &=6 a b^2 f n^2 x-6 a b^2 g m n^2 x-6 b^3 f n^3 x+6 b^3 g m n^3 x+\frac {6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac {3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {2388 b^3 g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {(3 b d g j m n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\frac {398 e-d j}{e}+\frac {j x}{e}} \, dx,x,d+e x\right )}{e^2}-\frac {\left (6 b^2 g m n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e}+\frac {\left (2388 b^2 g m n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {398 e-d j}{e}+\frac {j x}{e}\right )}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}-\left (6 b^2 g j m n^2\right ) \int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{j}-\frac {398 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j (398+j x)}\right ) \, dx-\left (6 b^3 e g n^3\right ) \int \left (\frac {\log \left (h (398+j x)^m\right )}{e}-\frac {d \log \left (h (398+j x)^m\right )}{e (d+e x)}\right ) \, dx-\frac {\left (6 b^3 d g m n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=6 a b^2 f n^2 x-12 a b^2 g m n^2 x-6 b^3 f n^3 x+6 b^3 g m n^3 x+\frac {6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {6 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac {3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {6 b^3 d g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {2388 b^3 g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\left (6 b^2 g m n^2\right ) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx+\left (2388 b^2 g m n^2\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{398+j x} \, dx-\frac {\left (6 b^3 g m n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}-\frac {\left (6 b^2 d g m n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}-\left (6 b^3 g n^3\right ) \int \log \left (h (398+j x)^m\right ) \, dx+\left (6 b^3 d g n^3\right ) \int \frac {\log \left (h (398+j x)^m\right )}{d+e x} \, dx+\frac {\left (2388 b^3 g m n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=6 a b^2 f n^2 x-18 a b^2 g m n^2 x-6 b^3 f n^3 x+12 b^3 g m n^3 x+\frac {6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {12 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}-\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+\frac {6 b^3 d g n^3 \log \left (-\frac {j (d+e x)}{398 e-d j}\right ) \log \left (h (398+j x)^m\right )}{e}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac {3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {2388 b^3 g m n^3 \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {6 b^3 d g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {2388 b^3 g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\left (6 b^3 g m n^2\right ) \int \log \left (c (d+e x)^n\right ) \, dx-\frac {\left (6 b^3 g n^3\right ) \text {Subst}\left (\int \log \left (h x^m\right ) \, dx,x,398+j x\right )}{j}-\frac {\left (6 b^3 d g m n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{e}-\frac {\left (2388 b^3 e g m n^3\right ) \int \frac {\log \left (\frac {e (398+j x)}{398 e-d j}\right )}{d+e x} \, dx}{j}-\frac {\left (6 b^3 d g j m n^3\right ) \int \frac {\log \left (\frac {j (d+e x)}{-398 e+d j}\right )}{398+j x} \, dx}{e}\\ &=6 a b^2 f n^2 x-18 a b^2 g m n^2 x-6 b^3 f n^3 x+18 b^3 g m n^3 x+\frac {6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {12 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}-\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}-\frac {6 b^3 g n^3 (398+j x) \log \left (h (398+j x)^m\right )}{j}+\frac {6 b^3 d g n^3 \log \left (-\frac {j (d+e x)}{398 e-d j}\right ) \log \left (h (398+j x)^m\right )}{e}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac {3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {6 b^3 d g m n^3 \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {2388 b^3 g m n^3 \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {6 b^3 d g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {2388 b^3 g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {\left (6 b^3 g m n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e}-\frac {\left (6 b^3 d g m n^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-398 e+d j}\right )}{x} \, dx,x,398+j x\right )}{e}-\frac {\left (2388 b^3 g m n^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {j x}{398 e-d j}\right )}{x} \, dx,x,d+e x\right )}{j}\\ &=6 a b^2 f n^2 x-18 a b^2 g m n^2 x-6 b^3 f n^3 x+24 b^3 g m n^3 x+\frac {6 b^3 f n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {18 b^3 g m n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {6 b g m n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}-\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}+\frac {398 g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (398+j x)}{398 e-d j}\right )}{j}-\frac {6 b^3 g n^3 (398+j x) \log \left (h (398+j x)^m\right )}{j}+\frac {6 b^3 d g n^3 \log \left (-\frac {j (d+e x)}{398 e-d j}\right ) \log \left (h (398+j x)^m\right )}{e}+6 b^2 g n^2 x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (398+j x)^m\right )-\frac {3 b d g n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )}{e}-3 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (398+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (h (398+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (398+j x)^m\right )\right )+\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (1+\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {2388 b^3 g m n^3 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {3 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {1194 b g m n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \text {Li}_2\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {6 b^3 d g m n^3 \text {Li}_2\left (\frac {e (398+j x)}{398 e-d j}\right )}{e}-\frac {6 b^3 d g m n^3 \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {2388 b^3 g m n^3 \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}+\frac {6 b^2 d g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}-\frac {2388 b^2 g m n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_3\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}-\frac {6 b^3 d g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{e}+\frac {2388 b^3 g m n^3 \text {Li}_4\left (-\frac {j (d+e x)}{398 e-d j}\right )}{j}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(3163\) vs. \(2(1147)=2294\).
time = 0.63, size = 3163, normalized size = 2.76 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]),x]

[Out]

(-3*a^2*b*d*f*j*n + 3*a^2*b*d*g*j*m*n - 6*a*b^2*d*g*j*m*n^2 + 6*b^3*d*g*j*m*n^3 + a^3*e*f*j*x - a^3*e*g*j*m*x
- 3*a^2*b*e*f*j*n*x + 6*a^2*b*e*g*j*m*n*x + 6*a*b^2*e*f*j*n^2*x - 18*a*b^2*e*g*j*m*n^2*x - 6*b^3*e*f*j*n^3*x +
24*b^3*e*g*j*m*n^3*x + 3*a^2*b*d*f*j*n*Log[d + e*x] - 3*a^2*b*d*g*j*m*n*Log[d + e*x] + 6*a*b^2*d*g*j*m*n^2*Lo
g[d + e*x] + 6*b^3*d*f*j*n^3*Log[d + e*x] - 12*b^3*d*g*j*m*n^3*Log[d + e*x] - 3*a*b^2*d*f*j*n^2*Log[d + e*x]^2
+ 3*a*b^2*d*g*j*m*n^2*Log[d + e*x]^2 - 3*b^3*d*g*j*m*n^3*Log[d + e*x]^2 + b^3*d*f*j*n^3*Log[d + e*x]^3 - b^3*
d*g*j*m*n^3*Log[d + e*x]^3 - 6*a*b^2*d*f*j*n*Log[c*(d + e*x)^n] + 6*a*b^2*d*g*j*m*n*Log[c*(d + e*x)^n] - 6*b^3
*d*g*j*m*n^2*Log[c*(d + e*x)^n] + 3*a^2*b*e*f*j*x*Log[c*(d + e*x)^n] - 3*a^2*b*e*g*j*m*x*Log[c*(d + e*x)^n] -
6*a*b^2*e*f*j*n*x*Log[c*(d + e*x)^n] + 12*a*b^2*e*g*j*m*n*x*Log[c*(d + e*x)^n] + 6*b^3*e*f*j*n^2*x*Log[c*(d +
e*x)^n] - 18*b^3*e*g*j*m*n^2*x*Log[c*(d + e*x)^n] + 6*a*b^2*d*f*j*n*Log[d + e*x]*Log[c*(d + e*x)^n] - 6*a*b^2*
d*g*j*m*n*Log[d + e*x]*Log[c*(d + e*x)^n] + 6*b^3*d*g*j*m*n^2*Log[d + e*x]*Log[c*(d + e*x)^n] - 3*b^3*d*f*j*n^
2*Log[d + e*x]^2*Log[c*(d + e*x)^n] + 3*b^3*d*g*j*m*n^2*Log[d + e*x]^2*Log[c*(d + e*x)^n] - 3*b^3*d*f*j*n*Log[
c*(d + e*x)^n]^2 + 3*b^3*d*g*j*m*n*Log[c*(d + e*x)^n]^2 + 3*a*b^2*e*f*j*x*Log[c*(d + e*x)^n]^2 - 3*a*b^2*e*g*j
*m*x*Log[c*(d + e*x)^n]^2 - 3*b^3*e*f*j*n*x*Log[c*(d + e*x)^n]^2 + 6*b^3*e*g*j*m*n*x*Log[c*(d + e*x)^n]^2 + 3*
b^3*d*f*j*n*Log[d + e*x]*Log[c*(d + e*x)^n]^2 - 3*b^3*d*g*j*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]^2 + b^3*e*f*j*
x*Log[c*(d + e*x)^n]^3 - b^3*e*g*j*m*x*Log[c*(d + e*x)^n]^3 + a^3*e*g*i*m*Log[i + j*x] - 3*a^2*b*e*g*i*m*n*Log
[i + j*x] + 3*a^2*b*d*g*j*m*n*Log[i + j*x] + 6*a*b^2*e*g*i*m*n^2*Log[i + j*x] - 6*b^3*e*g*i*m*n^3*Log[i + j*x]
- 3*a^2*b*e*g*i*m*n*Log[d + e*x]*Log[i + j*x] + 6*a*b^2*e*g*i*m*n^2*Log[d + e*x]*Log[i + j*x] - 6*a*b^2*d*g*j
*m*n^2*Log[d + e*x]*Log[i + j*x] - 6*b^3*e*g*i*m*n^3*Log[d + e*x]*Log[i + j*x] + 3*a*b^2*e*g*i*m*n^2*Log[d + e
*x]^2*Log[i + j*x] - 3*b^3*e*g*i*m*n^3*Log[d + e*x]^2*Log[i + j*x] + 3*b^3*d*g*j*m*n^3*Log[d + e*x]^2*Log[i +
j*x] - b^3*e*g*i*m*n^3*Log[d + e*x]^3*Log[i + j*x] + 3*a^2*b*e*g*i*m*Log[c*(d + e*x)^n]*Log[i + j*x] - 6*a*b^2
*e*g*i*m*n*Log[c*(d + e*x)^n]*Log[i + j*x] + 6*a*b^2*d*g*j*m*n*Log[c*(d + e*x)^n]*Log[i + j*x] + 6*b^3*e*g*i*m
*n^2*Log[c*(d + e*x)^n]*Log[i + j*x] - 6*a*b^2*e*g*i*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[i + j*x] + 6*b^3*
e*g*i*m*n^2*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[i + j*x] - 6*b^3*d*g*j*m*n^2*Log[d + e*x]*Log[c*(d + e*x)^n]*L
og[i + j*x] + 3*b^3*e*g*i*m*n^2*Log[d + e*x]^2*Log[c*(d + e*x)^n]*Log[i + j*x] + 3*a*b^2*e*g*i*m*Log[c*(d + e*
x)^n]^2*Log[i + j*x] - 3*b^3*e*g*i*m*n*Log[c*(d + e*x)^n]^2*Log[i + j*x] + 3*b^3*d*g*j*m*n*Log[c*(d + e*x)^n]^
2*Log[i + j*x] - 3*b^3*e*g*i*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]^2*Log[i + j*x] + b^3*e*g*i*m*Log[c*(d + e*x)^
n]^3*Log[i + j*x] + 3*a^2*b*e*g*i*m*n*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] - 3*a^2*b*d*g*j*m*n*Log[d +
e*x]*Log[(e*(i + j*x))/(e*i - d*j)] - 6*a*b^2*e*g*i*m*n^2*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] + 6*a*b^
2*d*g*j*m*n^2*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] + 6*b^3*e*g*i*m*n^3*Log[d + e*x]*Log[(e*(i + j*x))/(
e*i - d*j)] - 6*b^3*d*g*j*m*n^3*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] - 3*a*b^2*e*g*i*m*n^2*Log[d + e*x]
^2*Log[(e*(i + j*x))/(e*i - d*j)] + 3*a*b^2*d*g*j*m*n^2*Log[d + e*x]^2*Log[(e*(i + j*x))/(e*i - d*j)] + 3*b^3*
e*g*i*m*n^3*Log[d + e*x]^2*Log[(e*(i + j*x))/(e*i - d*j)] - 3*b^3*d*g*j*m*n^3*Log[d + e*x]^2*Log[(e*(i + j*x))
/(e*i - d*j)] + b^3*e*g*i*m*n^3*Log[d + e*x]^3*Log[(e*(i + j*x))/(e*i - d*j)] - b^3*d*g*j*m*n^3*Log[d + e*x]^3
*Log[(e*(i + j*x))/(e*i - d*j)] + 6*a*b^2*e*g*i*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d
*j)] - 6*a*b^2*d*g*j*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] - 6*b^3*e*g*i*m*n^2*Lo
g[d + e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] + 6*b^3*d*g*j*m*n^2*Log[d + e*x]*Log[c*(d + e*x)^
n]*Log[(e*(i + j*x))/(e*i - d*j)] - 3*b^3*e*g*i*m*n^2*Log[d + e*x]^2*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i
- d*j)] + 3*b^3*d*g*j*m*n^2*Log[d + e*x]^2*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] + 3*b^3*e*g*i*m*
n*Log[d + e*x]*Log[c*(d + e*x)^n]^2*Log[(e*(i + j*x))/(e*i - d*j)] - 3*b^3*d*g*j*m*n*Log[d + e*x]*Log[c*(d + e
*x)^n]^2*Log[(e*(i + j*x))/(e*i - d*j)] - 3*a^2*b*d*g*j*n*Log[h*(i + j*x)^m] + a^3*e*g*j*x*Log[h*(i + j*x)^m]
- 3*a^2*b*e*g*j*n*x*Log[h*(i + j*x)^m] + 6*a*b^2*e*g*j*n^2*x*Log[h*(i + j*x)^m] - 6*b^3*e*g*j*n^3*x*Log[h*(i +
j*x)^m] + 3*a^2*b*d*g*j*n*Log[d + e*x]*Log[h*(i + j*x)^m] + 6*b^3*d*g*j*n^3*Log[d + e*x]*Log[h*(i + j*x)^m] -
3*a*b^2*d*g*j*n^2*Log[d + e*x]^2*Log[h*(i + j*x)^m] + b^3*d*g*j*n^3*Log[d + e*x]^3*Log[h*(i + j*x)^m] - 6*a*b
^2*d*g*j*n*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] + 3*a^2*b*e*g*j*x*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] - 6*a
*b^2*e*g*j*n*x*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] + 6*b^3*e*g*j*n^2*x*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m]
+ 6*a*b^2*d*g*j*n*Log[d + e*x]*Log[c*(d + e*x)...

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{3} \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m)),x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m)),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m)),x, algorithm="maxima")

[Out]

b^3*f*x*log((x*e + d)^n*c)^3 - a^3*g*j*m*(x/j - I*log(j*x + I)/j^2) + 3*(d*e^(-2)*log(x*e + d) - x*e^(-1))*a^2
*b*f*n*e + 3*a*b^2*f*x*log((x*e + d)^n*c)^2 + 3*a^2*b*f*x*log((x*e + d)^n*c) + a^3*g*x*log((j*x + I)^m*h) - 3*
((d*log(x*e + d)^2 - 2*x*e + 2*d*log(x*e + d))*n^2*e^(-1) - 2*(d*e^(-2)*log(x*e + d) - x*e^(-1))*n*e*log((x*e
+ d)^n*c))*a*b^2*f + (3*(d*e^(-2)*log(x*e + d) - x*e^(-1))*n*e*log((x*e + d)^n*c)^2 + ((d*log(x*e + d)^3 + 3*d
*log(x*e + d)^2 - 6*x*e + 6*d*log(x*e + d))*n^2*e^(-2) - 3*(d*log(x*e + d)^2 - 2*x*e + 2*d*log(x*e + d))*n*e^(
-2)*log((x*e + d)^n*c))*n*e)*b^3*f + a^3*f*x - (((j*m - j*log(h))*b^3*g*x*e - I*b^3*g*m*e*log(j*x + I))*log((x
*e + d)^n)^3 - (b^3*d*g*j*n^3*log(x*e + d)^3 + b^3*g*j*x*e*log((x*e + d)^n)^3 - (3*(g*j*n - g*j*log(c))*a^2*b
- 3*(2*g*j*n^2 - 2*g*j*n*log(c) + g*j*log(c)^2)*a*b^2 + (6*g*j*n^3 - 6*g*j*n^2*log(c) + 3*g*j*n*log(c)^2 - g*j
*log(c)^3)*b^3)*x*e - 3*(a*b^2*d*g*j*n^2 - (d*g*j*n^3 - d*g*j*n^2*log(c))*b^3)*log(x*e + d)^2 + 3*(b^3*d*g*j*n
*log(x*e + d) + (a*b^2*g*j - (g*j*n - g*j*log(c))*b^3)*x*e)*log((x*e + d)^n)^2 + 3*(a^2*b*d*g*j*n - 2*(d*g*j*n
^2 - d*g*j*n*log(c))*a*b^2 + (2*d*g*j*n^3 - 2*d*g*j*n^2*log(c) + d*g*j*n*log(c)^2)*b^3)*log(x*e + d) - 3*(b^3*
d*g*j*n^2*log(x*e + d)^2 - (a^2*b*g*j - 2*(g*j*n - g*j*log(c))*a*b^2 + (2*g*j*n^2 - 2*g*j*n*log(c) + g*j*log(c
)^2)*b^3)*x*e - 2*(a*b^2*d*g*j*n - (d*g*j*n^2 - d*g*j*n*log(c))*b^3)*log(x*e + d))*log((x*e + d)^n))*log((j*x
+ I)^m))*e^(-1)/j - integrate(-((3*(g*j^2*m*n - (j^2*m - j^2*log(h))*g*log(c))*a^2*b - 3*(2*g*j^2*m*n^2 - 2*g*
j^2*m*n*log(c) + (j^2*m - j^2*log(h))*g*log(c)^2)*a*b^2 + (6*g*j^2*m*n^3 - 6*g*j^2*m*n^2*log(c) + 3*g*j^2*m*n*
log(c)^2 - (j^2*m - j^2*log(h))*g*log(c)^3)*b^3)*x^2*e^2 - (b^3*d*g*j^2*m*n^3*x*e + b^3*d^2*g*j^2*m*n^3)*log(x
*e + d)^3 + 3*(a*b^2*d^2*g*j^2*m*n^2 - (d^2*g*j^2*m*n^3 - d^2*g*j^2*m*n^2*log(c))*b^3 + (a*b^2*d*g*j^2*m*n^2 -
(d*g*j^2*m*n^3 - d*g*j^2*m*n^2*log(c))*b^3)*x*e)*log(x*e + d)^2 - 3*(((j^2*m - j^2*log(h))*a*b^2*g + ((j^2*m
- j^2*log(h))*g*log(c) - (2*j^2*m*n - j^2*n*log(h))*g)*b^3)*x^2*e^2 - ((I*a*b^2*g*j*log(h) + (I*g*j*log(c)*log
(h) + (I*j*m*n - I*j*n*log(h))*g)*b^3)*e^2 - ((j^2*m - j^2*log(h))*a*b^2*d*g - (d*g*j^2*m*n - (j^2*m - j^2*log
(h))*d*g*log(c))*b^3)*e)*x - (I*b^3*d*g*j*log(c)*log(h) + I*a*b^2*d*g*j*log(h))*e - (-I*b^3*g*j*m*n*x*e^2 + b^
3*g*m*n*e^2)*log(j*x + I) + (b^3*d*g*j^2*m*n*x*e + b^3*d^2*g*j^2*m*n)*log(x*e + d))*log((x*e + d)^n)^2 - ((-I*
b^3*g*j*log(c)^3*log(h) - 3*I*a*b^2*g*j*log(c)^2*log(h) - 3*I*a^2*b*g*j*log(c)*log(h))*e^2 - (3*(d*g*j^2*m*n -
(j^2*m - j^2*log(h))*d*g*log(c))*a^2*b - 3*(2*d*g*j^2*m*n^2 - 2*d*g*j^2*m*n*log(c) + (j^2*m - j^2*log(h))*d*g
*log(c)^2)*a*b^2 + (6*d*g*j^2*m*n^3 - 6*d*g*j^2*m*n^2*log(c) + 3*d*g*j^2*m*n*log(c)^2 - (j^2*m - j^2*log(h))*d
*g*log(c)^3)*b^3)*e)*x - (-I*b^3*d*g*j*log(c)^3*log(h) - 3*I*a*b^2*d*g*j*log(c)^2*log(h) - 3*I*a^2*b*d*g*j*log
(c)*log(h))*e - 3*(a^2*b*d^2*g*j^2*m*n - 2*(d^2*g*j^2*m*n^2 - d^2*g*j^2*m*n*log(c))*a*b^2 + (2*d^2*g*j^2*m*n^3
- 2*d^2*g*j^2*m*n^2*log(c) + d^2*g*j^2*m*n*log(c)^2)*b^3 + (a^2*b*d*g*j^2*m*n - 2*(d*g*j^2*m*n^2 - d*g*j^2*m*
n*log(c))*a*b^2 + (2*d*g*j^2*m*n^3 - 2*d*g*j^2*m*n^2*log(c) + d*g*j^2*m*n*log(c)^2)*b^3)*x*e)*log(x*e + d) - 3
*(((j^2*m - j^2*log(h))*a^2*b*g - 2*(g*j^2*m*n - (j^2*m - j^2*log(h))*g*log(c))*a*b^2 + (2*g*j^2*m*n^2 - 2*g*j
^2*m*n*log(c) + (j^2*m - j^2*log(h))*g*log(c)^2)*b^3)*x^2*e^2 - (b^3*d*g*j^2*m*n^2*x*e + b^3*d^2*g*j^2*m*n^2)*
log(x*e + d)^2 - ((I*b^3*g*j*log(c)^2*log(h) + 2*I*a*b^2*g*j*log(c)*log(h) + I*a^2*b*g*j*log(h))*e^2 - ((j^2*m
- j^2*log(h))*a^2*b*d*g - 2*(d*g*j^2*m*n - (j^2*m - j^2*log(h))*d*g*log(c))*a*b^2 + (2*d*g*j^2*m*n^2 - 2*d*g*
j^2*m*n*log(c) + (j^2*m - j^2*log(h))*d*g*log(c)^2)*b^3)*e)*x - (I*b^3*d*g*j*log(c)^2*log(h) + 2*I*a*b^2*d*g*j
*log(c)*log(h) + I*a^2*b*d*g*j*log(h))*e + 2*(a*b^2*d^2*g*j^2*m*n - (d^2*g*j^2*m*n^2 - d^2*g*j^2*m*n*log(c))*b
^3 + (a*b^2*d*g*j^2*m*n - (d*g*j^2*m*n^2 - d*g*j^2*m*n*log(c))*b^3)*x*e)*log(x*e + d))*log((x*e + d)^n))/(j^2*
x^2*e^2 + I*d*j*e + (d*j^2*e + I*j*e^2)*x), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m)),x, algorithm="fricas")

[Out]

((b^3*g*j*n^3*x*log(h) - (b^3*g*j*m - b^3*f*j)*n^3*x + (b^3*g*j*m*n^3*x + I*b^3*g*m*n^3)*log(j*x + I))*log(x*e
+ d)^3 + j*integral((a^3*f*j*x*e + a^3*d*f*j + (b^3*f*j*x*e + b^3*d*f*j)*log(c)^3 + 3*(a*b^2*d*f*j*n^2 + (a*b
^2*f*j*n^2 + (b^3*g*j*m - b^3*f*j)*n^3)*x*e + (a*b^2*d*g*j*m*n^2 - (I*b^3*g*m*n^3 + (b^3*g*j*m*n^3 - a*b^2*g*j
*m*n^2)*x)*e + (b^3*g*j*m*n^2*x*e + b^3*d*g*j*m*n^2)*log(c))*log(j*x + I) + (b^3*f*j*n^2*x*e + b^3*d*f*j*n^2)*
log(c) + (a*b^2*d*g*j*n^2 - (b^3*g*j*n^3 - a*b^2*g*j*n^2)*x*e + (b^3*g*j*n^2*x*e + b^3*d*g*j*n^2)*log(c))*log(
h))*log(x*e + d)^2 + 3*(a*b^2*f*j*x*e + a*b^2*d*f*j)*log(c)^2 + (a^3*g*j*m*x*e + a^3*d*g*j*m + (b^3*g*j*m*x*e
+ b^3*d*g*j*m)*log(c)^3 + 3*(a*b^2*g*j*m*x*e + a*b^2*d*g*j*m)*log(c)^2 + 3*(a^2*b*g*j*m*x*e + a^2*b*d*g*j*m)*l
og(c))*log(j*x + I) + 3*(a^2*b*f*j*n*x*e + a^2*b*d*f*j*n + (b^3*f*j*n*x*e + b^3*d*f*j*n)*log(c)^2 + (a^2*b*g*j
*m*n*x*e + a^2*b*d*g*j*m*n + (b^3*g*j*m*n*x*e + b^3*d*g*j*m*n)*log(c)^2 + 2*(a*b^2*g*j*m*n*x*e + a*b^2*d*g*j*m
*n)*log(c))*log(j*x + I) + 2*(a*b^2*f*j*n*x*e + a*b^2*d*f*j*n)*log(c) + (a^2*b*g*j*n*x*e + a^2*b*d*g*j*n + (b^
3*g*j*n*x*e + b^3*d*g*j*n)*log(c)^2 + 2*(a*b^2*g*j*n*x*e + a*b^2*d*g*j*n)*log(c))*log(h))*log(x*e + d) + 3*(a^
2*b*f*j*x*e + a^2*b*d*f*j)*log(c) + (a^3*g*j*x*e + a^3*d*g*j + (b^3*g*j*x*e + b^3*d*g*j)*log(c)^3 + 3*(a*b^2*g
*j*x*e + a*b^2*d*g*j)*log(c)^2 + 3*(a^2*b*g*j*x*e + a^2*b*d*g*j)*log(c))*log(h))/(j*x*e + d*j), x))/j

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))**3*(f+g*ln(h*(j*x+i)**m)),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m)),x, algorithm="giac")

[Out]

integrate((b*log((x*e + d)^n*c) + a)^3*(g*log((j*x + I)^m*h) + f), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x)^n))^3*(f + g*log(h*(i + j*x)^m)),x)

[Out]

int((a + b*log(c*(d + e*x)^n))^3*(f + g*log(h*(i + j*x)^m)), x)

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