\(\int (s+t x) (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\) [419]

Optimal result

Integrand size = 34, antiderivative size = 527 \[ \int (s+t x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=-\frac {b f n (e s-d t) x}{2 e}+\frac {3 b g m n (e s-d t) x}{4 e}-\frac {a g m (j s-i t) x}{2 j}+\frac {3 b g m n (j s-i t) x}{4 j}+\frac {b g m n (s+t x)^2}{4 t}+\frac {b g m n (e s-d t)^2 \log (d+e x)}{4 e^2 t}-\frac {b g m (j s-i t) (d+e x) \log \left (c (d+e x)^n\right )}{2 e j}-\frac {g m (s+t x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 t}+\frac {b g m n (j s-i t)^2 \log (i+j x)}{4 j^2 t}-\frac {g m (j s-i t)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 j^2 t}-\frac {b g n (e s-d t) (i+j x) \log \left (h (i+j x)^m\right )}{2 e j}-\frac {b n (s+t x)^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{4 t}-\frac {b n (e s-d t)^2 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 e^2 t}+\frac {(s+t x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 t}-\frac {b g m n (j s-i t)^2 \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 j^2 t}-\frac {b g m n (e s-d t)^2 \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{2 e^2 t} \] Output:

-1/2*b*f*n*(-d*t+e*s)*x/e+3/4*b*g*m*n*(-d*t+e*s)*x/e-1/2*a*g*m*(-i*t+j*s)* 
x/j+3/4*b*g*m*n*(-i*t+j*s)*x/j+1/4*b*g*m*n*(t*x+s)^2/t+1/4*b*g*m*n*(-d*t+e 
*s)^2*ln(e*x+d)/e^2/t-1/2*b*g*m*(-i*t+j*s)*(e*x+d)*ln(c*(e*x+d)^n)/e/j-1/4 
*g*m*(t*x+s)^2*(a+b*ln(c*(e*x+d)^n))/t+1/4*b*g*m*n*(-i*t+j*s)^2*ln(j*x+i)/ 
j^2/t-1/2*g*m*(-i*t+j*s)^2*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/ 
j^2/t-1/2*b*g*n*(-d*t+e*s)*(j*x+i)*ln(h*(j*x+i)^m)/e/j-1/4*b*n*(t*x+s)^2*( 
f+g*ln(h*(j*x+i)^m))/t-1/2*b*n*(-d*t+e*s)^2*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g 
*ln(h*(j*x+i)^m))/e^2/t+1/2*(t*x+s)^2*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x 
+i)^m))/t-1/2*b*g*m*n*(-i*t+j*s)^2*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^2/t- 
1/2*b*g*m*n*(-d*t+e*s)^2*polylog(2,e*(j*x+i)/(-d*j+e*i))/e^2/t
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 482, normalized size of antiderivative = 0.91 \[ \int (s+t x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\frac {b n \log (d+e x) \left (2 e^2 g i m (-2 j s+i t) \log (i+j x)-2 g (e i-d j) m (-2 e j s+e i t+d j t) \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (d j (-2 f+g m) t+2 e (2 f j s-2 g j m s+g i m t)+2 g j (2 e s-d t) \log \left (h (i+j x)^m\right )\right )\right )+e \left (g m (-2 a e i (-2 j s+i t)+b n (e i (-4 j s+i t)+2 d j (2 j s+i t))) \log (i+j x)-j \left (a e x (-2 f j (2 s+t x)+g m (4 j s-2 i t+j t x))+b n (d f j (4 s-2 t x)+e f j x (4 s+t x)-e g m x (8 j s-3 i t+j t x)+d g m (-4 j s+2 i t+3 j t x))+g j \left (-2 a e x (2 s+t x)+b n \left (4 d s+4 e s x-2 d t x+e t x^2\right )\right ) \log \left (h (i+j x)^m\right )\right )+b e \log \left (c (d+e x)^n\right ) \left (-2 g i m (-2 j s+i t) \log (i+j x)+j x \left (2 f j (2 s+t x)-g m (4 j s-2 i t+j t x)+2 g j (2 s+t x) \log \left (h (i+j x)^m\right )\right )\right )\right )-2 b g (e i-d j) m n (-2 e j s+e i t+d j t) \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{4 e^2 j^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.28 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.39, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2890, 2889, 27, 25, 2863, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (s+t x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx\)

\(\Big \downarrow \) 2890

\(\displaystyle \frac {\int (s+t x) \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right ) \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )d(s+t x)}{t}\)

\(\Big \downarrow \) 2889

\(\displaystyle \frac {-\frac {g j m \int \frac {t (s+t x)^2 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right )}{\left (i-\frac {j s}{t}\right ) t+j (s+t x)}d(s+t x)}{2 t}-\frac {b e n \int \frac {t (s+t x)^2 \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{\left (d-\frac {e s}{t}\right ) t+e (s+t x)}d(s+t x)}{2 t}+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right ) \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{t}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{2} g j m \int -\frac {(s+t x)^2 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right )}{j s-i t-j (s+t x)}d(s+t x)-\frac {1}{2} b e n \int -\frac {(s+t x)^2 \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{e s-d t-e (s+t x)}d(s+t x)+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right ) \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{t}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{2} g j m \int \frac {(s+t x)^2 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right )}{j s-i t-j (s+t x)}d(s+t x)+\frac {1}{2} b e n \int \frac {(s+t x)^2 \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{e s-d t-e (s+t x)}d(s+t x)+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right ) \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{t}\)

\(\Big \downarrow \) 2863

\(\displaystyle \frac {\frac {1}{2} g j m \int \left (\frac {\left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right ) (j s-i t)^2}{j^2 (j s-i t-j (s+t x))}+\frac {(i t-j s) \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right )}{j^2}-\frac {(s+t x) \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right )}{j}\right )d(s+t x)+\frac {1}{2} b e n \int \left (\frac {\left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right ) (e s-d t)^2}{e^2 (e s-d t-e (s+t x))}+\frac {(d t-e s) \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{e^2}-\frac {(s+t x) \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{e}\right )d(s+t x)+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right ) \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{t}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{2} g j m \left (-\frac {(j s-i t)^2 \log \left (-\frac {e (-i t-j (s+t x)+j s)}{t (e i-d j)}\right ) \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right )}{j^3}-\frac {(s+t x)^2 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right )}{2 j}-\frac {a (s+t x) (j s-i t)}{j^2}+\frac {b (j s-i t) (-d t-e (s+t x)+e s) \log \left (c (d+e x)^n\right )}{e j^2}+\frac {b n (e s-d t)^2 \log (-d t-e (s+t x)+e s)}{2 e^2 j}-\frac {b n (j s-i t)^2 \operatorname {PolyLog}\left (2,\frac {j (e s-d t-e (s+t x))}{(e i-d j) t}\right )}{j^3}+\frac {b n (s+t x) (e s-d t)}{2 e j}+\frac {b n (s+t x) (j s-i t)}{j^2}+\frac {b n (s+t x)^2}{4 j}\right )+\frac {1}{2} (s+t x)^2 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right ) \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )+\frac {1}{2} b e n \left (-\frac {(e s-d t)^2 \log \left (\frac {j (-d t-e (s+t x)+e s)}{t (e i-d j)}\right ) \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{e^3}-\frac {g m (e s-d t)^2 \operatorname {PolyLog}\left (2,\frac {e (i t+j x t)}{(e i-d j) t}\right )}{e^3}-\frac {f (s+t x) (e s-d t)}{e^2}+\frac {g (e s-d t) (-i t-j (s+t x)+j s) \log \left (h (i+j x)^m\right )}{e^2 j}+\frac {g m (s+t x) (e s-d t)}{e^2}-\frac {(s+t x)^2 \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{2 e}+\frac {g m (j s-i t)^2 \log (-i t-j (s+t x)+j s)}{2 e j^2}+\frac {g m (s+t x) (j s-i t)}{2 e j}+\frac {g m (s+t x)^2}{4 e}\right )}{t}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2863

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 2889

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 
] + (-Simp[g*j*(m/(r + 1))   Int[x^(r + 1)*((a + b*Log[c*(d + e*x)^n])^p/(i 
 + j*x)), x], x] - Simp[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] && (E 
qQ[p, 1] || GtQ[r, 0]) && NeQ[r, -1]
 

rule 2890

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 2215, normalized size of antiderivative = 4.20

\[\text {Expression too large to display}\]

Input:

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

Output:

-5/8/e^2*b*d^2*g*m*n*t+(1/2*b*g*x*(t*x+2*s)*ln((j*x+i)^m)-1/4*b*(-I*Pi*g*j 
^2*t*x^2*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+I*Pi*g*j^2*t*x^2*csgn(I*h)*csgn(I 
*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+2*I*Pi*g*j^2*s*x*csgn(I*h*(j*x+i)^m)^3+I*P 
i*g*j^2*t*x^2*csgn(I*h*(j*x+i)^m)^3-2*I*Pi*g*j^2*s*x*csgn(I*(j*x+i)^m)*csg 
n(I*h*(j*x+i)^m)^2-I*Pi*g*j^2*t*x^2*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^ 
2-2*I*Pi*g*j^2*s*x*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+2*I*Pi*g*j^2*s*x*csgn(I 
*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)-2*ln(h)*g*j^2*t*x^2+g*j^2*m*t*x^ 
2+2*ln(j*x+i)*g*i^2*m*t-4*ln(j*x+i)*g*i*j*m*s-4*ln(h)*g*j^2*s*x-2*f*j^2*t* 
x^2-2*g*i*j*m*t*x+4*g*j^2*m*s*x-4*f*j^2*s*x)/j^2)*ln((e*x+d)^n)-1/2*I/e*n* 
b*d*ln(e*x+d)*Pi*g*s*csgn(I*h*(j*x+i)^m)^3-1/4*I/e*n*b*x*Pi*d*g*t*csgn(I*h 
*(j*x+i)^m)^3+(1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/4*I*b* 
Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-1/4*I*b*Pi*csgn(I*c*(e* 
x+d)^n)^3+1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+1/2*b*ln(c)+1/2*a)*(( 
-I*g*Pi*csgn(I*h)*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)+I*g*Pi*csgn(I*h)*c 
sgn(I*h*(j*x+i)^m)^2+I*g*Pi*csgn(I*(j*x+i)^m)*csgn(I*h*(j*x+i)^m)^2-I*g*Pi 
*csgn(I*h*(j*x+i)^m)^3+2*g*ln(h)+2*f)*(1/2*t*x^2+s*x)+2*g*(1/2*ln((j*x+i)^ 
m)*x^2*t+ln((j*x+i)^m)*x*s-1/2*m*j*(1/j^2*(1/2*j*t*x^2-i*t*x+2*j*s*x)+i*(i 
*t-2*j*s)/j^3*ln(j*x+i))))+1/e*b*d*g*m*n*s-1/4*I/e^2*n*b*d^2*ln(e*x+d)*Pi* 
g*t*csgn(I*h)*csgn(I*h*(j*x+i)^m)^2+1/2*I/e*n*b*d*ln(e*x+d)*Pi*g*s*csgn(I* 
h)*csgn(I*h*(j*x+i)^m)^2-1/4*I/e^2*n*b*d^2*ln(e*x+d)*Pi*g*t*csgn(I*(j*x...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.61 \[ \int (s+t x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

-b*e*f*n*s*(x/e - d*log(e*x + d)/e^2) - a*g*j*m*s*(x/j - i*log(j*x + i)/j^ 
2) - 1/4*b*e*f*n*t*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) - 1/4*a* 
g*j*m*t*(2*i^2*log(j*x + i)/j^3 + (j*x^2 - 2*i*x)/j^2) + 1/2*b*f*t*x^2*log 
((e*x + d)^n*c) + 1/2*a*g*t*x^2*log((j*x + i)^m*h) + 1/2*a*f*t*x^2 + b*f*s 
*x*log((e*x + d)^n*c) + a*g*s*x*log((j*x + i)^m*h) + a*f*s*x + 1/4*(2*d*g* 
i*j*m*n*t + 2*(2*i*j*m*s - i^2*m*t)*e*g*log(c) - (4*i*j*m*n*s - i^2*m*n*t) 
*e*g)*b*log(j*x + i)/(e*j^2) - 1/2*(2*d*e*g*j^2*m*n*s - d^2*g*j^2*m*n*t - 
(2*i*j*m*n*s - i^2*m*n*t)*e^2*g)*(log(e*x + d)*log((e*j*x + d*j)/(e*i - d* 
j) + 1) + dilog(-(e*j*x + d*j)/(e*i - d*j)))*b/(e^2*j^2) - 1/4*(2*(2*i*j*m 
*n*s - i^2*m*n*t)*b*e^2*g*log(e*x + d)*log(j*x + i) + ((j^2*m*t - 2*j^2*t* 
log(h))*e^2*g*log(c) - (j^2*m*n*t - j^2*n*t*log(h))*e^2*g)*b*x^2 + (2*(2*j 
^2*m*s - i*j*m*t - 2*j^2*s*log(h))*e^2*g*log(c) + (3*j^2*m*n*t - 2*j^2*n*t 
*log(h))*d*e*g - (8*j^2*m*n*s - 3*i*j*m*n*t - 4*j^2*n*s*log(h))*e^2*g)*b*x 
 - ((j^2*m*n*t - 2*j^2*n*t*log(h))*d^2*g - 2*(2*j^2*m*n*s - i*j*m*n*t - 2* 
j^2*n*s*log(h))*d*e*g)*b*log(e*x + d) + ((j^2*m*t - 2*j^2*t*log(h))*b*e^2* 
g*x^2 + 2*(2*j^2*m*s - i*j*m*t - 2*j^2*s*log(h))*b*e^2*g*x - 2*(2*i*j*m*s 
- i^2*m*t)*b*e^2*g*log(j*x + i))*log((e*x + d)^n) + ((e^2*g*j^2*n*t - 2*e^ 
2*g*j^2*t*log(c))*b*x^2 + 2*(2*e^2*g*j^2*n*s - d*e*g*j^2*n*t - 2*e^2*g*j^2 
*s*log(c))*b*x - 2*(2*d*e*g*j^2*n*s - d^2*g*j^2*n*t)*b*log(e*x + d) - 2*(b 
*e^2*g*j^2*t*x^2 + 2*b*e^2*g*j^2*s*x)*log((e*x + d)^n))*log((j*x + i)^m...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (s+t x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {too large to display} \] Input:

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

Output:

(4*atan(j*x)*b*d**5*e*g*i*j**5*m*n*t - 8*atan(j*x)*b*d**4*e**2*g*i*j**5*m* 
n*s + 14*atan(j*x)*b*d**4*e**2*g*j**4*m*n*t - 16*atan(j*x)*b*d**3*e**3*g*i 
*j**3*m*n*t - 32*atan(j*x)*b*d**3*e**3*g*j**4*m*n*s + 48*atan(j*x)*b*d**2* 
e**4*g*i*j**3*m*n*s - 4*atan(j*x)*b*d**2*e**4*g*j**2*m*n*t - 4*atan(j*x)*b 
*d*e**5*g*i*j*m*n*t + 32*atan(j*x)*b*d*e**5*g*j**2*m*n*s - 8*atan(j*x)*b*e 
**6*g*i*j*m*n*s - 2*atan(j*x)*b*e**6*g*m*n*t + 4*int(log((i + j*x)**m*h)/( 
d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2*i + 2*d*e**2*j*x - e** 
3*i*x),x)*b*d**8*e*g*j**8*n*t - 24*int(log((i + j*x)**m*h)/(d**3*i*j**2 + 
d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2*i + 2*d*e**2*j*x - e**3*i*x),x)*b*d* 
*7*e**2*g*i*j**7*n*t - 8*int(log((i + j*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j 
**2*x + 2*d**2*e*j - d*e**2*i + 2*d*e**2*j*x - e**3*i*x),x)*b*d**7*e**2*g* 
j**8*n*s + 56*int(log((i + j*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d 
**2*e*j - d*e**2*i + 2*d*e**2*j*x - e**3*i*x),x)*b*d**6*e**3*g*i*j**7*n*s 
- 56*int(log((i + j*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - 
 d*e**2*i + 2*d*e**2*j*x - e**3*i*x),x)*b*d**6*e**3*g*j**6*n*t + 56*int(lo 
g((i + j*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2*i + 
 2*d*e**2*j*x - e**3*i*x),x)*b*d**5*e**4*g*i*j**5*n*t + 168*int(log((i + j 
*x)**m*h)/(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2*i + 2*d*e** 
2*j*x - e**3*i*x),x)*b*d**5*e**4*g*j**6*n*s - 280*int(log((i + j*x)**m*h)/ 
(d**3*i*j**2 + d**2*e*i*j**2*x + 2*d**2*e*j - d*e**2*i + 2*d*e**2*j*x -...