\(\int x (a+b \log (c (d+e x)^n))^2 (f+g \log (h (i+j x)^m)) \, dx\) [393]

Optimal result

Integrand size = 32, antiderivative size = 1210 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=-\frac {2 a b d g m n x}{e}-\frac {3 a b g i m n x}{2 j}-\frac {2 b^2 d f n^2 x}{e}+\frac {15 b^2 d g m n^2 x}{4 e}+\frac {7 b^2 g i m n^2 x}{4 j}-\frac {1}{4} b^2 g m n^2 x^2+\frac {b^2 f n^2 (d+e x)^2}{4 e^2}-\frac {b^2 g m n^2 (d+e x)^2}{8 e^2}-\frac {b^2 d^2 g m n^2 \log (d+e x)}{4 e^2}+\frac {b^2 d^2 f n^2 \log ^2(d+e x)}{2 e^2}-\frac {2 b^2 d g m n (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac {3 b^2 g i m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}+\frac {1}{4} b g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {2 b d f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}-\frac {b d^2 f n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac {b^2 g i^2 m n^2 \log (i+j x)}{4 j^2}+\frac {b g i^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 j^2}+\frac {b d g i m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{e j}+\frac {d^2 g m \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 )}{2 e^2}-\frac {g i^2 m \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 )}{2 j^2}+\frac {1}{4} b^2 g n^2 x^2 \log \left (h (i+j x)^m\right )-\frac {3 b^2 d g n^2 (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}+\frac {3 b^2 d^2 g n^2 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{2 e^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )+\frac {b^2 g i^2 m n^2 \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 j^2}+\frac {b^2 d g i m n^2 \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{e j}+\frac {b d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{e^2}-\frac {b g i^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{j^2}+\frac {3 b^2 d^2 g m n^2 \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{2 e^2}-\frac {b^2 d^2 g m n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{e^2}+\frac {b^2 g i^2 m n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{j^2} \]

[Out]

-1/4*b^2*d^2*g*m*n^2*ln(e*x+d)/e^2+2*b*d*f*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))/e^2+1/4*b*g*m*n*(e*x+d)^2*(a+b*ln(c
*(e*x+d)^n))/e^2+1/2*g*i*m*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e/j-1/4*b^2*g*i^2*m*n^2*ln(j*x+i)/j^2+3/2*b^2*d^2*g
*n^2*ln(-j*(e*x+d)/(-d*j+e*i))*ln(h*(j*x+i)^m)/e^2+15/4*b^2*d*g*m*n^2*x/e+7/4*b^2*g*i*m*n^2*x/j+1/2*b^2*g*i^2*
m*n^2*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^2+3/2*b^2*d^2*g*m*n^2*polylog(2,e*(j*x+i)/(-d*j+e*i))/e^2-3/2*b^2*g*i
*m*n*(e*x+d)*ln(c*(e*x+d)^n)/e/j-1/2*d^2*g*(a+b*ln(c*(e*x+d)^n))^2*ln(h*(j*x+i)^m)/e^2-1/4*g*m*(e*x+d)^2*(a+b*
ln(c*(e*x+d)^n))^2/e^2+1/4*b^2*g*n^2*x^2*ln(h*(j*x+i)^m)-1/4*b^2*g*m*n^2*x^2+1/4*b^2*f*n^2*(e*x+d)^2/e^2+b*d*g
*n*x*(a+b*ln(c*(e*x+d)^n))*ln(h*(j*x+i)^m)/e+b*d^2*g*m*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,-j*(e*x+d)/(-d*j+e*i)
)/e^2-b*g*i^2*m*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^2-2*b^2*d*g*m*n*(e*x+d)*ln(c*(e*x+d
)^n)/e^2+1/2*b*g*i^2*m*n*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/j^2-3/2*b^2*d*g*n^2*(j*x+i)*ln(h*(j*x+
i)^m)/e/j-2*a*b*d*g*m*n*x/e-3/2*a*b*g*i*m*n*x/j-b^2*d^2*g*m*n^2*polylog(3,-j*(e*x+d)/(-d*j+e*i))/e^2+b^2*g*i^2
*m*n^2*polylog(3,-j*(e*x+d)/(-d*j+e*i))/j^2-b*d^2*f*n*ln(e*x+d)*(a+b*ln(c*(e*x+d)^n))/e^2+b^2*d*g*i*m*n^2*poly
log(2,-j*(e*x+d)/(-d*j+e*i))/e/j+b*d*g*i*m*n*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/e/j-2*b^2*d*f*n^2*
x/e-1/8*b^2*g*m*n^2*(e*x+d)^2/e^2+1/2*b^2*d^2*f*n^2*ln(e*x+d)^2/e^2+1/4*b*g*m*n*x^2*(a+b*ln(c*(e*x+d)^n))-1/2*
b*f*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))/e^2+1/2*d*g*m*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e^2+1/2*d^2*g*m*(a+b*ln(c*
(e*x+d)^n))^2*ln(e*(j*x+i)/(-d*j+e*i))/e^2-1/2*g*i^2*m*(a+b*ln(c*(e*x+d)^n))^2*ln(e*(j*x+i)/(-d*j+e*i))/j^2-1/
2*b*g*n*x^2*(a+b*ln(c*(e*x+d)^n))*ln(h*(j*x+i)^m)+1/2*x^2*(a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m))

Rubi [A] (verified)

Time = 1.96 (sec) , antiderivative size = 1210, normalized size of antiderivative = 1.00, number of steps used = 73, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.844, Rules used = {2489, 2463, 2436, 2333, 2332, 2448, 2437, 2342, 2341, 2443, 2481, 2421, 6724, 6874, 2458, 45, 2372, 12, 14, 2338, 2479, 2441, 2440, 2438, 2442, 2422, 2354} \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=-\frac {1}{4} g m n^2 x^2 b^2+\frac {f n^2 (d+e x)^2 b^2}{4 e^2}-\frac {g m n^2 (d+e x)^2 b^2}{8 e^2}+\frac {d^2 f n^2 \log ^2(d+e x) b^2}{2 e^2}-\frac {2 d f n^2 x b^2}{e}+\frac {15 d g m n^2 x b^2}{4 e}+\frac {7 g i m n^2 x b^2}{4 j}-\frac {d^2 g m n^2 \log (d+e x) b^2}{4 e^2}-\frac {2 d g m n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{e^2}-\frac {3 g i m n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{2 e j}-\frac {g i^2 m n^2 \log (i+j x) b^2}{4 j^2}+\frac {1}{4} g n^2 x^2 \log \left (h (i+j x)^m\right ) b^2-\frac {3 d g n^2 (i+j x) \log \left (h (i+j x)^m\right ) b^2}{2 e j}+\frac {3 d^2 g n^2 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right ) b^2}{2 e^2}+\frac {d g i m n^2 \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{e j}+\frac {g i^2 m n^2 \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{2 j^2}+\frac {3 d^2 g m n^2 \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right ) b^2}{2 e^2}-\frac {d^2 g m n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{e^2}+\frac {g i^2 m n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{j^2}-\frac {2 a d g m n x b}{e}-\frac {3 a g i m n x b}{2 j}+\frac {1}{4} g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b-\frac {f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{2 e^2}+\frac {g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{4 e^2}+\frac {2 d f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{e^2}-\frac {d^2 f n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{e^2}+\frac {d g i m n \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}{e j}+\frac {g i^2 m n \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^2}-\frac {1}{2} g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) b+\frac {d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) b}{e}+\frac {d^2 g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b}{e^2}-\frac {g i^2 m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) b}{j^2}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}+\frac {d^2 g m \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 )}{2 e^2}-\frac {g i^2 m \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 )}{2 j^2}-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \]

[In]

Int[x*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]),x]

[Out]

(-2*a*b*d*g*m*n*x)/e - (3*a*b*g*i*m*n*x)/(2*j) - (2*b^2*d*f*n^2*x)/e + (15*b^2*d*g*m*n^2*x)/(4*e) + (7*b^2*g*i
*m*n^2*x)/(4*j) - (b^2*g*m*n^2*x^2)/4 + (b^2*f*n^2*(d + e*x)^2)/(4*e^2) - (b^2*g*m*n^2*(d + e*x)^2)/(8*e^2) -
(b^2*d^2*g*m*n^2*Log[d + e*x])/(4*e^2) + (b^2*d^2*f*n^2*Log[d + e*x]^2)/(2*e^2) - (2*b^2*d*g*m*n*(d + e*x)*Log
[c*(d + e*x)^n])/e^2 - (3*b^2*g*i*m*n*(d + e*x)*Log[c*(d + e*x)^n])/(2*e*j) + (b*g*m*n*x^2*(a + b*Log[c*(d + e
*x)^n]))/4 + (2*b*d*f*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/e^2 - (b*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^
n]))/(2*e^2) + (b*g*m*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(4*e^2) - (b*d^2*f*n*Log[d + e*x]*(a + b*Log[c
*(d + e*x)^n]))/e^2 + (d*g*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2) + (g*i*m*(d + e*x)*(a + b*Log[c*(
d + e*x)^n])^2)/(2*e*j) - (g*m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(4*e^2) - (b^2*g*i^2*m*n^2*Log[i + j*
x])/(4*j^2) + (b*g*i^2*m*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(2*j^2) + (b*d*g*i*m*n*(
a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(e*j) + (d^2*g*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e
*(i + j*x))/(e*i - d*j)])/(2*e^2) - (g*i^2*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(2*j
^2) + (b^2*g*n^2*x^2*Log[h*(i + j*x)^m])/4 - (3*b^2*d*g*n^2*(i + j*x)*Log[h*(i + j*x)^m])/(2*e*j) + (3*b^2*d^2
*g*n^2*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x)^m])/(2*e^2) + (b*d*g*n*x*(a + b*Log[c*(d + e*x)^n])*L
og[h*(i + j*x)^m])/e - (b*g*n*x^2*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/2 - (d^2*g*(a + b*Log[c*(d +
e*x)^n])^2*Log[h*(i + j*x)^m])/(2*e^2) + (x^2*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]))/2 + (b^
2*g*i^2*m*n^2*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*j^2) + (b^2*d*g*i*m*n^2*PolyLog[2, -((j*(d + e*x))/
(e*i - d*j))])/(e*j) + (b*d^2*g*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e^2 -
(b*g*i^2*m*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/j^2 + (3*b^2*d^2*g*m*n^2*Po
lyLog[2, (e*(i + j*x))/(e*i - d*j)])/(2*e^2) - (b^2*d^2*g*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/e^2
+ (b^2*g*i^2*m*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))])/j^2

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

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 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

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 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 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 2442

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

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 2448

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

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 2489

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*(x_)^(r_.), x_Symbol] :> Simp[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^p*((f + g*Log[h*(i + j*x)^m])/(r + 1
)), x] + (-Dist[g*j*(m/(r + 1)), Int[x^(r + 1)*((a + b*Log[c*(d + e*x)^n])^p/(i + j*x)), x], x] - Dist[b*e*n*(
p/(r + 1)), Int[x^(r + 1)*(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] && IntegerQ[r] && (EqQ[p, 1] || GtQ[r, 0]) && N
eQ[r, -1]

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*} \text {integral}& = \frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {1}{2} (g j m) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{i+j x} \, dx-(b e n) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d+e x} \, dx \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {1}{2} (g j m) \int \left (-\frac {i \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}+\frac {i^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^2 (i+j x)}\right ) \, dx-(b e n) \int \left (\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}+\frac {g x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{d+e x}\right ) \, dx \\ & = \frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {1}{2} (g m) \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx+\frac {(g i m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{2 j}-\frac {\left (g i^2 m\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{i+j x} \, dx}{2 j}-(b e f n) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-(b e g n) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{d+e x} \, dx \\ & = -\frac {g i^2 m \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 )}{2 j^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {1}{2} (g m) \int \left (-\frac {d \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}\right ) \, dx+\frac {(g i m) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e j}-(b f n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )-(b e g n) \int \left (-\frac {d \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{e^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{e}+\frac {d^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{e^2 (d+e x)}\right ) \, dx+\frac {\left (b e g i^2 m n\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{d+e x} \, dx}{j^2} \\ & = \frac {2 b d f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b d^2 f n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g i^2 m \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 )}{2 j^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {(g m) \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{2 e}+\frac {(d g m) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{2 e}-(b g n) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) \, dx+\frac {(b d g n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) \, dx}{e}-\frac {\left (b d^2 g n\right ) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{d+e x} \, dx}{e}+\frac {\left (b g i^2 m n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e i-d j}{e}+\frac {j x}{e}\right )}{e i-d j}\right )}{x} \, dx,x,d+e x\right )}{j^2}-\frac {(b g i m n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e j}+\left (b^2 f n^2\right ) \text {Subst}\left (\int \frac {x (-4 d+x)+2 d^2 \log (x)}{2 e^2 x} \, dx,x,d+e x\right ) \\ & = -\frac {a b g i m n x}{j}+\frac {2 b d f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b d^2 f n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g i^2 m \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 )}{2 j^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^2 m n \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^2}-\frac {(g m) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}+\frac {(d g m) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{2 e^2}-\frac {\left (b d^2 g n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (h \left (\frac {e i-d j}{e}+\frac {j x}{e}\right )^m\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac {\left (b^2 g i m n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e j}+\frac {1}{2} (b g j m n) \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{i+j x} \, dx-\frac {(b d g j m n) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{i+j x} \, dx}{e}+\frac {\left (b^2 f n^2\right ) \text {Subst}\left (\int \frac {x (-4 d+x)+2 d^2 \log (x)}{x} \, dx,x,d+e x\right )}{2 e^2}-\left (b^2 d g n^2\right ) \int \frac {x \log \left (h (i+j x)^m\right )}{d+e x} \, dx+\frac {1}{2} \left (b^2 e g n^2\right ) \int \frac {x^2 \log \left (h (i+j x)^m\right )}{d+e x} \, dx+\frac {\left (b^2 g i^2 m n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {j x}{e i-d j}\right )}{x} \, dx,x,d+e x\right )}{j^2} \\ & = -\frac {a b g i m n x}{j}+\frac {b^2 g i m n^2 x}{j}-\frac {b^2 g i m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac {2 b d f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}-\frac {b d^2 f n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}-\frac {g i^2 m \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 )}{2 j^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^2 m n \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^2}+\frac {b^2 g i^2 m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{j^2}+\frac {\left (d^2 g j m\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\frac {e i-d j}{e}+\frac {j x}{e}} \, dx,x,d+e x\right )}{2 e^3}+\frac {(b g m n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{2 e^2}-\frac {(b d g m n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2}+\frac {1}{2} (b g j m n) \int \left (-\frac {i \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{j}+\frac {i^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2 (i+j x)}\right ) \, dx-\frac {(b d g j m n) \int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{j}-\frac {i \left (a+b \log \left (c (d+e x)^n\right )\right )}{j (i+j x)}\right ) \, dx}{e}+\frac {\left (b^2 f n^2\right ) \text {Subst}\left (\int \left (-4 d+x+\frac {2 d^2 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{2 e^2}-\left (b^2 d g n^2\right ) \int \left (\frac {\log \left (h (i+j x)^m\right )}{e}-\frac {d \log \left (h (i+j x)^m\right )}{e (d+e x)}\right ) \, dx+\frac {1}{2} \left (b^2 e g n^2\right ) \int \left (-\frac {d \log \left (h (i+j x)^m\right )}{e^2}+\frac {x \log \left (h (i+j x)^m\right )}{e}+\frac {d^2 \log \left (h (i+j x)^m\right )}{e^2 (d+e x)}\right ) \, dx \\ & = -\frac {a b d g m n x}{e}-\frac {a b g i m n x}{j}-\frac {2 b^2 d f n^2 x}{e}+\frac {b^2 g i m n^2 x}{j}+\frac {b^2 f n^2 (d+e x)^2}{4 e^2}-\frac {b^2 g m n^2 (d+e x)^2}{8 e^2}-\frac {b^2 g i m n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac {2 b d f n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}-\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2}+\frac {b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2}-\frac {b d^2 f n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2}+\frac {g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e j}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2}+\frac {d^2 g m \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 )}{2 e^2}-\frac {g i^2 m \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 )}{2 j^2}+\frac {b d g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{e}-\frac {1}{2} b g n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )-\frac {d^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{2 e^2}+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {b g i^2 m n \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^2}+\frac {b^2 g i^2 m n^2 \text {Li}_3\left (-\frac {j (d+e x)}{e i-d j}\right )}{j^2}+\frac {1}{2} (b g m n) \int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx-\frac {\left (b^2 d g m n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2}-\frac {\left (b d^2 g m n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {j x}{e i-d j}\right )}{x} \, dx,x,d+e x\right )}{e^2}-\frac {(b d g m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{e}+\frac {(b d g i m n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{i+j x} \, dx}{e}-\frac {(b g i m n) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{2 j}+\frac {\left (b g i^2 m n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{i+j x} \, dx}{2 j}+\frac {\left (b^2 d^2 f n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x\right )}{e^2}+\frac {1}{2} \left (b^2 g n^2\right ) \int x \log \left (h (i+j x)^m\right ) \, dx-\frac {\left (b^2 d g n^2\right ) \int \log \left (h (i+j x)^m\right ) \, dx}{2 e}-\frac {\left (b^2 d g n^2\right ) \int \log \left (h (i+j x)^m\right ) \, dx}{e}+\frac {\left (b^2 d^2 g n^2\right ) \int \frac {\log \left (h (i+j x)^m\right )}{d+e x} \, dx}{2 e}+\frac {\left (b^2 d^2 g n^2\right ) \int \frac {\log \left (h (i+j x)^m\right )}{d+e x} \, dx}{e} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 2067, normalized size of antiderivative = 1.71 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Result too large to show} \]

[In]

Integrate[x*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]),x]

[Out]

(-8*a*b*d*e*g*i*j*m*n + 4*b^2*d*e*g*i*j*m*n^2 + 8*b^2*d^2*g*j^2*m*n^2 + 4*a^2*e^2*g*i*j*m*x + 8*a*b*d*e*f*j^2*
n*x - 12*a*b*e^2*g*i*j*m*n*x - 12*a*b*d*e*g*j^2*m*n*x - 12*b^2*d*e*f*j^2*n^2*x + 14*b^2*e^2*g*i*j*m*n^2*x + 28
*b^2*d*e*g*j^2*m*n^2*x + 4*a^2*e^2*f*j^2*x^2 - 2*a^2*e^2*g*j^2*m*x^2 - 4*a*b*e^2*f*j^2*n*x^2 + 4*a*b*e^2*g*j^2
*m*n*x^2 + 2*b^2*e^2*f*j^2*n^2*x^2 - 3*b^2*e^2*g*j^2*m*n^2*x^2 - 8*a*b*d^2*f*j^2*n*Log[d + e*x] + 8*a*b*d*e*g*
i*j*m*n*Log[d + e*x] + 4*a*b*d^2*g*j^2*m*n*Log[d + e*x] + 12*b^2*d^2*f*j^2*n^2*Log[d + e*x] - 4*b^2*d*e*g*i*j*
m*n^2*Log[d + e*x] - 16*b^2*d^2*g*j^2*m*n^2*Log[d + e*x] + 4*b^2*d^2*f*j^2*n^2*Log[d + e*x]^2 - 4*b^2*d*e*g*i*
j*m*n^2*Log[d + e*x]^2 - 2*b^2*d^2*g*j^2*m*n^2*Log[d + e*x]^2 - 8*b^2*d*e*g*i*j*m*n*Log[c*(d + e*x)^n] + 8*a*b
*e^2*g*i*j*m*x*Log[c*(d + e*x)^n] + 8*b^2*d*e*f*j^2*n*x*Log[c*(d + e*x)^n] - 12*b^2*e^2*g*i*j*m*n*x*Log[c*(d +
e*x)^n] - 12*b^2*d*e*g*j^2*m*n*x*Log[c*(d + e*x)^n] + 8*a*b*e^2*f*j^2*x^2*Log[c*(d + e*x)^n] - 4*a*b*e^2*g*j^
2*m*x^2*Log[c*(d + e*x)^n] - 4*b^2*e^2*f*j^2*n*x^2*Log[c*(d + e*x)^n] + 4*b^2*e^2*g*j^2*m*n*x^2*Log[c*(d + e*x
)^n] - 8*b^2*d^2*f*j^2*n*Log[d + e*x]*Log[c*(d + e*x)^n] + 8*b^2*d*e*g*i*j*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]
+ 4*b^2*d^2*g*j^2*m*n*Log[d + e*x]*Log[c*(d + e*x)^n] + 4*b^2*e^2*g*i*j*m*x*Log[c*(d + e*x)^n]^2 + 4*b^2*e^2*
f*j^2*x^2*Log[c*(d + e*x)^n]^2 - 2*b^2*e^2*g*j^2*m*x^2*Log[c*(d + e*x)^n]^2 - 4*a^2*e^2*g*i^2*m*Log[i + j*x] +
4*a*b*e^2*g*i^2*m*n*Log[i + j*x] + 8*a*b*d*e*g*i*j*m*n*Log[i + j*x] - 2*b^2*e^2*g*i^2*m*n^2*Log[i + j*x] - 12
*b^2*d*e*g*i*j*m*n^2*Log[i + j*x] + 8*a*b*e^2*g*i^2*m*n*Log[d + e*x]*Log[i + j*x] - 4*b^2*e^2*g*i^2*m*n^2*Log[
d + e*x]*Log[i + j*x] - 8*b^2*d*e*g*i*j*m*n^2*Log[d + e*x]*Log[i + j*x] - 4*b^2*e^2*g*i^2*m*n^2*Log[d + e*x]^2
*Log[i + j*x] - 8*a*b*e^2*g*i^2*m*Log[c*(d + e*x)^n]*Log[i + j*x] + 4*b^2*e^2*g*i^2*m*n*Log[c*(d + e*x)^n]*Log
[i + j*x] + 8*b^2*d*e*g*i*j*m*n*Log[c*(d + e*x)^n]*Log[i + j*x] + 8*b^2*e^2*g*i^2*m*n*Log[d + e*x]*Log[c*(d +
e*x)^n]*Log[i + j*x] - 4*b^2*e^2*g*i^2*m*Log[c*(d + e*x)^n]^2*Log[i + j*x] - 8*a*b*e^2*g*i^2*m*n*Log[d + e*x]*
Log[(e*(i + j*x))/(e*i - d*j)] + 8*a*b*d^2*g*j^2*m*n*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] + 4*b^2*e^2*g
*i^2*m*n^2*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] + 8*b^2*d*e*g*i*j*m*n^2*Log[d + e*x]*Log[(e*(i + j*x))/
(e*i - d*j)] - 12*b^2*d^2*g*j^2*m*n^2*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] + 4*b^2*e^2*g*i^2*m*n^2*Log[
d + e*x]^2*Log[(e*(i + j*x))/(e*i - d*j)] - 4*b^2*d^2*g*j^2*m*n^2*Log[d + e*x]^2*Log[(e*(i + j*x))/(e*i - d*j)
] - 8*b^2*e^2*g*i^2*m*n*Log[d + e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] + 8*b^2*d^2*g*j^2*m*n*L
og[d + e*x]*Log[c*(d + e*x)^n]*Log[(e*(i + j*x))/(e*i - d*j)] + 8*a*b*d*e*g*j^2*n*x*Log[h*(i + j*x)^m] - 12*b^
2*d*e*g*j^2*n^2*x*Log[h*(i + j*x)^m] + 4*a^2*e^2*g*j^2*x^2*Log[h*(i + j*x)^m] - 4*a*b*e^2*g*j^2*n*x^2*Log[h*(i
+ j*x)^m] + 2*b^2*e^2*g*j^2*n^2*x^2*Log[h*(i + j*x)^m] - 8*a*b*d^2*g*j^2*n*Log[d + e*x]*Log[h*(i + j*x)^m] +
12*b^2*d^2*g*j^2*n^2*Log[d + e*x]*Log[h*(i + j*x)^m] + 4*b^2*d^2*g*j^2*n^2*Log[d + e*x]^2*Log[h*(i + j*x)^m] +
8*b^2*d*e*g*j^2*n*x*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] + 8*a*b*e^2*g*j^2*x^2*Log[c*(d + e*x)^n]*Log[h*(i +
j*x)^m] - 4*b^2*e^2*g*j^2*n*x^2*Log[c*(d + e*x)^n]*Log[h*(i + j*x)^m] - 8*b^2*d^2*g*j^2*n*Log[d + e*x]*Log[c*
(d + e*x)^n]*Log[h*(i + j*x)^m] + 4*b^2*e^2*g*j^2*x^2*Log[c*(d + e*x)^n]^2*Log[h*(i + j*x)^m] - 4*b*g*(e*i - d
*j)*m*n*(2*a*(e*i + d*j) - b*(e*i + 3*d*j)*n + 2*b*(e*i + d*j)*Log[c*(d + e*x)^n])*PolyLog[2, (j*(d + e*x))/(-
(e*i) + d*j)] + 8*b^2*g*(e^2*i^2 - d^2*j^2)*m*n^2*PolyLog[3, (j*(d + e*x))/(-(e*i) + d*j)])/(8*e^2*j^2)

Maple [F]

\[\int x {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2} \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )d x\]

[In]

int(x*(a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m)),x)

[Out]

int(x*(a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m)),x)

Fricas [F]

\[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x \,d x } \]

[In]

integrate(x*(a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m)),x, algorithm="fricas")

[Out]

integral(b^2*f*x*log((e*x + d)^n*c)^2 + 2*a*b*f*x*log((e*x + d)^n*c) + a^2*f*x + (b^2*g*x*log((e*x + d)^n*c)^2
+ 2*a*b*g*x*log((e*x + d)^n*c) + a^2*g*x)*log((j*x + i)^m*h), x)

Sympy [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Timed out} \]

[In]

integrate(x*(a+b*ln(c*(e*x+d)**n))**2*(f+g*ln(h*(j*x+i)**m)),x)

[Out]

Timed out

Maxima [F]

\[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x \,d x } \]

[In]

integrate(x*(a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m)),x, algorithm="maxima")

[Out]

1/2*b^2*f*x^2*log((e*x + d)^n*c)^2 - 1/2*a*b*e*f*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) - 1/4*a^2*g*
j*m*(2*i^2*log(j*x + i)/j^3 + (j*x^2 - 2*i*x)/j^2) + a*b*f*x^2*log((e*x + d)^n*c) + 1/2*a^2*g*x^2*log((j*x + i
)^m*h) + 1/2*a^2*f*x^2 - 1/4*(2*e*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2)*log((e*x + d)^n*c) - (e^2*x
^2 + 2*d^2*log(e*x + d)^2 - 6*d*e*x + 6*d^2*log(e*x + d))*n^2/e^2)*b^2*f + 1/4*((2*b^2*e^2*g*i*j*m*x - 2*b^2*e
^2*g*i^2*m*log(j*x + i) - (j^2*m - 2*j^2*log(h))*b^2*e^2*g*x^2)*log((e*x + d)^n)^2 + (2*b^2*d^2*g*j^2*n^2*log(
e*x + d)^2 + 2*b^2*e^2*g*j^2*x^2*log((e*x + d)^n)^2 - (2*(e^2*g*j^2*n - 2*e^2*g*j^2*log(c))*a*b - (e^2*g*j^2*n
^2 - 2*e^2*g*j^2*n*log(c) + 2*e^2*g*j^2*log(c)^2)*b^2)*x^2 + 2*(2*a*b*d*e*g*j^2*n - (3*d*e*g*j^2*n^2 - 2*d*e*g
*j^2*n*log(c))*b^2)*x - 2*(2*a*b*d^2*g*j^2*n - (3*d^2*g*j^2*n^2 - 2*d^2*g*j^2*n*log(c))*b^2)*log(e*x + d) + 2*
(2*b^2*d*e*g*j^2*n*x - 2*b^2*d^2*g*j^2*n*log(e*x + d) + (2*a*b*e^2*g*j^2 - (e^2*g*j^2*n - 2*e^2*g*j^2*log(c))*
b^2)*x^2)*log((e*x + d)^n))*log((j*x + i)^m))/(e^2*j^2) + integrate(1/4*((2*(e^3*g*j^3*m*n - 2*(j^3*m - 2*j^3*
log(h))*e^3*g*log(c))*a*b - (e^3*g*j^3*m*n^2 - 2*e^3*g*j^3*m*n*log(c) + 2*(j^3*m - 2*j^3*log(h))*e^3*g*log(c)^
2)*b^2)*x^3 - (2*(d*e^2*g*j^3*m*n - 2*(2*e^3*g*i*j^2*log(h) - (j^3*m - 2*j^3*log(h))*d*e^2*g)*log(c))*a*b - (5
*d*e^2*g*j^3*m*n^2 - 2*d*e^2*g*j^3*m*n*log(c) + 2*(2*e^3*g*i*j^2*log(h) - (j^3*m - 2*j^3*log(h))*d*e^2*g)*log(
c)^2)*b^2)*x^2 - 2*(b^2*d^2*e*g*j^3*m*n^2*x + b^2*d^3*g*j^3*m*n^2)*log(e*x + d)^2 - 2*(2*(d^2*e*g*j^3*m*n - 2*
d*e^2*g*i*j^2*log(c)*log(h))*a*b - (3*d^2*e*g*j^3*m*n^2 - 2*d^2*e*g*j^3*m*n*log(c) + 2*d*e^2*g*i*j^2*log(c)^2*
log(h))*b^2)*x + 2*(2*a*b*d^3*g*j^3*m*n - (3*d^3*g*j^3*m*n^2 - 2*d^3*g*j^3*m*n*log(c))*b^2 + (2*a*b*d^2*e*g*j^
3*m*n - (3*d^2*e*g*j^3*m*n^2 - 2*d^2*e*g*j^3*m*n*log(c))*b^2)*x)*log(e*x + d) - 2*(2*((j^3*m - 2*j^3*log(h))*a
*b*e^3*g + ((j^3*m - 2*j^3*log(h))*e^3*g*log(c) - (j^3*m*n - j^3*n*log(h))*e^3*g)*b^2)*x^3 - (2*(2*e^3*g*i*j^2
*log(h) - (j^3*m - 2*j^3*log(h))*d*e^2*g)*a*b - (d*e^2*g*j^3*m*n + (i*j^2*m*n + 2*i*j^2*n*log(h))*e^3*g - 2*(2
*e^3*g*i*j^2*log(h) - (j^3*m - 2*j^3*log(h))*d*e^2*g)*log(c))*b^2)*x^2 - 2*(2*a*b*d*e^2*g*i*j^2*log(h) - (e^3*
g*i^2*j*m*n + d^2*e*g*j^3*m*n - 2*d*e^2*g*i*j^2*log(c)*log(h))*b^2)*x - 2*(b^2*d^2*e*g*j^3*m*n*x + b^2*d^3*g*j
^3*m*n)*log(e*x + d) - 2*(b^2*e^3*g*i^2*j*m*n*x + b^2*e^3*g*i^3*m*n)*log(j*x + i))*log((e*x + d)^n))/(e^3*j^3*
x^2 + d*e^2*i*j^2 + (e^3*i*j^2 + d*e^2*j^3)*x), x)

Giac [F]

\[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x \,d x } \]

[In]

integrate(x*(a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m)),x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^2*(g*log((j*x + i)^m*h) + f)*x, x)

Mupad [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \]

[In]

int(x*(a + b*log(c*(d + e*x)^n))^2*(f + g*log(h*(i + j*x)^m)),x)

[Out]

int(x*(a + b*log(c*(d + e*x)^n))^2*(f + g*log(h*(i + j*x)^m)), x)