\(\int (s+t x)^2 (a+b \log (c (d+e x)^n)) (f+g \log (h (i+j x)^m)) \, dx\) [420]

Optimal result

Integrand size = 36, antiderivative size = 792 \[ \int (s+t x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=-\frac {b f n (e s-d t)^2 x}{3 e^2}+\frac {4 b g m n (e s-d t)^2 x}{9 e^2}+\frac {b g m n (e s-d t) (j s-i t) x}{3 e j}-\frac {a g m (j s-i t)^2 x}{3 j^2}+\frac {4 b g m n (j s-i t)^2 x}{9 j^2}+\frac {5 b g m n (e s-d t) (s+t x)^2}{36 e t}+\frac {5 b g m n (j s-i t) (s+t x)^2}{36 j t}+\frac {2 b g m n (s+t x)^3}{27 t}+\frac {b g m n (e s-d t)^3 \log (d+e x)}{9 e^3 t}+\frac {b g m n (e s-d t)^2 (j s-i t) \log (d+e x)}{6 e^2 j t}-\frac {b g m (j s-i t)^2 (d+e x) \log \left (c (d+e x)^n\right )}{3 e j^2}-\frac {g m (j s-i t) (s+t x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{6 j t}-\frac {g m (s+t x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{9 t}+\frac {b g m n (e s-d t) (j s-i t)^2 \log (i+j x)}{6 e j^2 t}+\frac {b g m n (j s-i t)^3 \log (i+j x)}{9 j^3 t}-\frac {g m (j s-i t)^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{3 j^3 t}-\frac {b g n (e s-d t)^2 (i+j x) \log \left (h (i+j x)^m\right )}{3 e^2 j}-\frac {b n (e s-d t) (s+t x)^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{6 e t}-\frac {b n (s+t x)^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{9 t}-\frac {b n (e s-d t)^3 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{3 e^3 t}+\frac {(s+t x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{3 t}-\frac {b g m n (j s-i t)^3 \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{3 j^3 t}-\frac {b g m n (e s-d t)^3 \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{3 e^3 t} \] Output:

-1/3*b*f*n*(-d*t+e*s)^2*x/e^2+4/9*b*g*m*n*(-d*t+e*s)^2*x/e^2+1/3*b*g*m*n*( 
-d*t+e*s)*(-i*t+j*s)*x/e/j-1/3*a*g*m*(-i*t+j*s)^2*x/j^2+4/9*b*g*m*n*(-i*t+ 
j*s)^2*x/j^2+5/36*b*g*m*n*(-d*t+e*s)*(t*x+s)^2/e/t+5/36*b*g*m*n*(-i*t+j*s) 
*(t*x+s)^2/j/t+2/27*b*g*m*n*(t*x+s)^3/t+1/9*b*g*m*n*(-d*t+e*s)^3*ln(e*x+d) 
/e^3/t+1/6*b*g*m*n*(-d*t+e*s)^2*(-i*t+j*s)*ln(e*x+d)/e^2/j/t-1/3*b*g*m*(-i 
*t+j*s)^2*(e*x+d)*ln(c*(e*x+d)^n)/e/j^2-1/6*g*m*(-i*t+j*s)*(t*x+s)^2*(a+b* 
ln(c*(e*x+d)^n))/j/t-1/9*g*m*(t*x+s)^3*(a+b*ln(c*(e*x+d)^n))/t+1/6*b*g*m*n 
*(-d*t+e*s)*(-i*t+j*s)^2*ln(j*x+i)/e/j^2/t+1/9*b*g*m*n*(-i*t+j*s)^3*ln(j*x 
+i)/j^3/t-1/3*g*m*(-i*t+j*s)^3*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e* 
i))/j^3/t-1/3*b*g*n*(-d*t+e*s)^2*(j*x+i)*ln(h*(j*x+i)^m)/e^2/j-1/6*b*n*(-d 
*t+e*s)*(t*x+s)^2*(f+g*ln(h*(j*x+i)^m))/e/t-1/9*b*n*(t*x+s)^3*(f+g*ln(h*(j 
*x+i)^m))/t-1/3*b*n*(-d*t+e*s)^3*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+ 
i)^m))/e^3/t+1/3*(t*x+s)^3*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m))/t-1 
/3*b*g*m*n*(-i*t+j*s)^3*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^3/t-1/3*b*g*m*n 
*(-d*t+e*s)^3*polylog(2,e*(j*x+i)/(-d*j+e*i))/e^3/t
 

Mathematica [A] (verified)

Time = 1.96 (sec) , antiderivative size = 986, normalized size of antiderivative = 1.24 \[ \int (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 ) \, dx=\frac {6 b n \log (d+e x) \left (-6 e^3 g i m \left (3 j^2 s^2-3 i j s t+i^2 t^2\right ) \log (i+j x)+6 g (e i-d j) m \left (d^2 j^2 t^2+d e j t (-3 j s+i t)+e^2 \left (3 j^2 s^2-3 i j s t+i^2 t^2\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )+d j \left (2 d^2 j^2 (3 f-g m) t^2-3 d e j t (6 f j s+g m (-3 j s+i t))+6 e^2 \left (3 f j^2 s^2-g m \left (3 j^2 s^2-3 i j s t+i^2 t^2\right )\right )+6 g j^2 \left (3 e^2 s^2-3 d e s t+d^2 t^2\right ) \log \left (h (i+j x)^m\right )\right )\right )+e \left (6 g m \left (6 a e^2 i \left (3 j^2 s^2-3 i j s t+i^2 t^2\right )+b n \left (-6 d^2 i j^2 t^2+e^2 i \left (-18 j^2 s^2+9 i j s t-2 i^2 t^2\right )+3 d e j \left (6 j^2 s^2+6 i j s t-i^2 t^2\right )\right )\right ) \log (i+j x)+j \left (6 a e^2 x \left (6 f j^2 \left (3 s^2+3 s t x+t^2 x^2\right )-g m \left (6 i^2 t^2-3 i j t (6 s+t x)+j^2 \left (18 s^2+9 s t x+2 t^2 x^2\right )\right )\right )+b n \left (12 d^2 j^2 (-3 f+4 g m) t^2 x-3 d e \left (6 f j^2 \left (6 s^2-6 s t x-t^2 x^2\right )+g m \left (-12 i^2 t^2-12 i j t (-3 s+t x)+j^2 \left (-36 s^2+54 s t x+5 t^2 x^2\right )\right )\right )+e^2 x \left (-6 f j^2 \left (18 s^2+9 s t x+2 t^2 x^2\right )+g m \left (48 i^2 t^2-3 i j t (54 s+5 t x)+j^2 \left (216 s^2+54 s t x+8 t^2 x^2\right )\right )\right )\right )-6 g j^2 \left (-6 a e^2 x \left (3 s^2+3 s t x+t^2 x^2\right )+b n \left (6 d^2 t^2 x+3 d e \left (6 s^2-6 s t x-t^2 x^2\right )+e^2 x \left (18 s^2+9 s t x+2 t^2 x^2\right )\right )\right ) \log \left (h (i+j x)^m\right )\right )+6 b e^2 \log \left (c (d+e x)^n\right ) \left (6 g i m \left (3 j^2 s^2-3 i j s t+i^2 t^2\right ) \log (i+j x)+j x \left (6 f j^2 \left (3 s^2+3 s t x+t^2 x^2\right )-g m \left (6 i^2 t^2-3 i j t (6 s+t x)+j^2 \left (18 s^2+9 s t x+2 t^2 x^2\right )\right )+6 g j^2 \left (3 s^2+3 s t x+t^2 x^2\right ) \log \left (h (i+j x)^m\right )\right )\right )\right )+36 b g (e i-d j) m n \left (d^2 j^2 t^2+d e j t (-3 j s+i t)+e^2 \left (3 j^2 s^2-3 i j s t+i^2 t^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{108 e^3 j^3} \] Input:

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

Output:

(6*b*n*Log[d + e*x]*(-6*e^3*g*i*m*(3*j^2*s^2 - 3*i*j*s*t + i^2*t^2)*Log[i 
+ j*x] + 6*g*(e*i - d*j)*m*(d^2*j^2*t^2 + d*e*j*t*(-3*j*s + i*t) + e^2*(3* 
j^2*s^2 - 3*i*j*s*t + i^2*t^2))*Log[(e*(i + j*x))/(e*i - d*j)] + d*j*(2*d^ 
2*j^2*(3*f - g*m)*t^2 - 3*d*e*j*t*(6*f*j*s + g*m*(-3*j*s + i*t)) + 6*e^2*( 
3*f*j^2*s^2 - g*m*(3*j^2*s^2 - 3*i*j*s*t + i^2*t^2)) + 6*g*j^2*(3*e^2*s^2 
- 3*d*e*s*t + d^2*t^2)*Log[h*(i + j*x)^m])) + e*(6*g*m*(6*a*e^2*i*(3*j^2*s 
^2 - 3*i*j*s*t + i^2*t^2) + b*n*(-6*d^2*i*j^2*t^2 + e^2*i*(-18*j^2*s^2 + 9 
*i*j*s*t - 2*i^2*t^2) + 3*d*e*j*(6*j^2*s^2 + 6*i*j*s*t - i^2*t^2)))*Log[i 
+ j*x] + j*(6*a*e^2*x*(6*f*j^2*(3*s^2 + 3*s*t*x + t^2*x^2) - g*m*(6*i^2*t^ 
2 - 3*i*j*t*(6*s + t*x) + j^2*(18*s^2 + 9*s*t*x + 2*t^2*x^2))) + b*n*(12*d 
^2*j^2*(-3*f + 4*g*m)*t^2*x - 3*d*e*(6*f*j^2*(6*s^2 - 6*s*t*x - t^2*x^2) + 
 g*m*(-12*i^2*t^2 - 12*i*j*t*(-3*s + t*x) + j^2*(-36*s^2 + 54*s*t*x + 5*t^ 
2*x^2))) + e^2*x*(-6*f*j^2*(18*s^2 + 9*s*t*x + 2*t^2*x^2) + g*m*(48*i^2*t^ 
2 - 3*i*j*t*(54*s + 5*t*x) + j^2*(216*s^2 + 54*s*t*x + 8*t^2*x^2)))) - 6*g 
*j^2*(-6*a*e^2*x*(3*s^2 + 3*s*t*x + t^2*x^2) + b*n*(6*d^2*t^2*x + 3*d*e*(6 
*s^2 - 6*s*t*x - t^2*x^2) + e^2*x*(18*s^2 + 9*s*t*x + 2*t^2*x^2)))*Log[h*( 
i + j*x)^m]) + 6*b*e^2*Log[c*(d + e*x)^n]*(6*g*i*m*(3*j^2*s^2 - 3*i*j*s*t 
+ i^2*t^2)*Log[i + j*x] + j*x*(6*f*j^2*(3*s^2 + 3*s*t*x + t^2*x^2) - g*m*( 
6*i^2*t^2 - 3*i*j*t*(6*s + t*x) + j^2*(18*s^2 + 9*s*t*x + 2*t^2*x^2)) + 6* 
g*j^2*(3*s^2 + 3*s*t*x + t^2*x^2)*Log[h*(i + j*x)^m]))) + 36*b*g*(e*i -...
 

Rubi [A] (verified)

Time = 3.03 (sec) , antiderivative size = 1113, normalized size of antiderivative = 1.41, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.

\(\displaystyle \int (s+t x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx\)

\(\Big \downarrow \) 2890

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

\(\Big \downarrow \) 2889

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{3} \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 ) (s+t x)^3+\frac {1}{3} b e n \left (-\frac {\log \left (\frac {j (e s-d t-e (s+t x))}{(e i-d j) t}\right ) \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)^3}{e^4}-\frac {g m \operatorname {PolyLog}\left (2,\frac {e (i t+j x t)}{(e i-d j) t}\right ) (e s-d t)^3}{e^4}-\frac {f (s+t x) (e s-d t)^2}{e^3}+\frac {g m (s+t x) (e s-d t)^2}{e^3}+\frac {g (j s-i t-j (s+t x)) \log \left (h (i+j x)^m\right ) (e s-d t)^2}{e^3 j}+\frac {g m (s+t x)^2 (e s-d t)}{4 e^2}+\frac {g m (j s-i t) (s+t x) (e s-d t)}{2 e^2 j}+\frac {g m (j s-i t)^2 \log (j s-i t-j (s+t x)) (e s-d t)}{2 e^2 j^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 ) (e s-d t)}{2 e^2}+\frac {g m (s+t x)^3}{9 e}+\frac {g m (j s-i t) (s+t x)^2}{6 e j}+\frac {g m (j s-i t)^2 (s+t x)}{3 e j^2}+\frac {g m (j s-i t)^3 \log (j s-i t-j (s+t x))}{3 e j^3}-\frac {(s+t x)^3 \left (f+g \log \left (h \left (i+\frac {j (s+t x)}{t}-\frac {j s}{t}\right )^m\right )\right )}{3 e}\right )+\frac {1}{3} g j m \left (\frac {b n \log (e s-d t-e (s+t x)) (e s-d t)^3}{3 e^3 j}+\frac {b n (s+t x) (e s-d t)^2}{3 e^2 j}+\frac {b n (j s-i t) \log (e s-d t-e (s+t x)) (e s-d t)^2}{2 e^2 j^2}+\frac {b n (s+t x)^2 (e s-d t)}{6 e j}+\frac {b n (j s-i t) (s+t x) (e s-d t)}{2 e j^2}+\frac {b n (s+t x)^3}{9 j}+\frac {b n (j s-i t) (s+t x)^2}{4 j^2}+\frac {b n (j s-i t)^2 (s+t x)}{j^3}-\frac {a (j s-i t)^2 (s+t x)}{j^3}+\frac {b (j s-i t)^2 (e s-d t-e (s+t x)) \log \left (c (d+e x)^n\right )}{e j^3}-\frac {(s+t x)^3 \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right )}{3 j}-\frac {(j s-i 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 )}{2 j^2}-\frac {(j s-i t)^3 \log \left (-\frac {e (j s-i t-j (s+t x))}{(e i-d j) t}\right ) \left (a+b \log \left (c \left (d+\frac {e (s+t x)}{t}-\frac {e s}{t}\right )^n\right )\right )}{j^4}-\frac {b n (j s-i t)^3 \operatorname {PolyLog}\left (2,\frac {j (e s-d t-e (s+t x))}{(e i-d j) t}\right )}{j^4}\right )}{t}\)

Input:

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

Output:

(((s + t*x)^3*(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]))/3 + (b*e*n*(-((f*(e*s - d*t)^2*(s + 
 t*x))/e^3) + (g*m*(e*s - d*t)^2*(s + t*x))/e^3 + (g*m*(e*s - d*t)*(j*s - 
i*t)*(s + t*x))/(2*e^2*j) + (g*m*(j*s - i*t)^2*(s + t*x))/(3*e*j^2) + (g*m 
*(e*s - d*t)*(s + t*x)^2)/(4*e^2) + (g*m*(j*s - i*t)*(s + t*x)^2)/(6*e*j) 
+ (g*m*(s + t*x)^3)/(9*e) + (g*(e*s - d*t)^2*(j*s - i*t - j*(s + t*x))*Log 
[h*(i + j*x)^m])/(e^3*j) + (g*m*(e*s - d*t)*(j*s - i*t)^2*Log[j*s - i*t - 
j*(s + t*x)])/(2*e^2*j^2) + (g*m*(j*s - i*t)^3*Log[j*s - i*t - j*(s + t*x) 
])/(3*e*j^3) - ((e*s - d*t)*(s + t*x)^2*(f + g*Log[h*(i - (j*s)/t + (j*(s 
+ t*x))/t)^m]))/(2*e^2) - ((s + t*x)^3*(f + g*Log[h*(i - (j*s)/t + (j*(s + 
 t*x))/t)^m]))/(3*e) - ((e*s - d*t)^3*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^4 - (g* 
m*(e*s - d*t)^3*PolyLog[2, (e*(i*t + j*t*x))/((e*i - d*j)*t)])/e^4))/3 + ( 
g*j*m*((b*n*(e*s - d*t)^2*(s + t*x))/(3*e^2*j) + (b*n*(e*s - d*t)*(j*s - i 
*t)*(s + t*x))/(2*e*j^2) - (a*(j*s - i*t)^2*(s + t*x))/j^3 + (b*n*(j*s - i 
*t)^2*(s + t*x))/j^3 + (b*n*(e*s - d*t)*(s + t*x)^2)/(6*e*j) + (b*n*(j*s - 
 i*t)*(s + t*x)^2)/(4*j^2) + (b*n*(s + t*x)^3)/(9*j) + (b*(j*s - i*t)^2*(e 
*s - d*t - e*(s + t*x))*Log[c*(d + e*x)^n])/(e*j^3) + (b*n*(e*s - d*t)^3*L 
og[e*s - d*t - e*(s + t*x)])/(3*e^3*j) + (b*n*(e*s - d*t)^2*(j*s - i*t)*Lo 
g[e*s - d*t - e*(s + t*x)])/(2*e^2*j^2) - ((j*s - i*t)*(s + t*x)^2*(a +...
 

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 = 4224, normalized size of antiderivative = 5.33

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

Input:

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

Output:

1/3/e^3*n*b*d^3*ln(e*x+d)*f*t^2+1/e*n*b*d*ln(e*x+d)*f*s^2-1/2*n*b*ln(h)*x^ 
2*g*s*t-1/3*n*b*g/t*ln((j*x+i)^m)*ln(e*x+d)*s^3-1/2*n*b*g*t*ln((j*x+i)^m)* 
x^2*s+1/3*n*b*g/t*m*dilog(((e*x+d)*j-d*j+e*i)/(-d*j+e*i))*s^3-3/2/j*t*x*b* 
g*i*m*n*s-3/2*t/e*x*b*d*g*m*n*s+t/e*x*b*d*f*n*s+2*x*b*g*m*n*s^2-1/2*t*x^2* 
b*f*n*s+2/27*x^3*t^2*b*g*m*n-5/36/j*t^2*x^2*b*g*i*m*n+4/9/j^2*t^2*x*b*g*i^ 
2*m*n-1/3*t^2/e^2*x*b*d^2*f*n+1/6*t^2/e*x^2*b*d*f*n+1/2*t*x^2*b*g*m*n*s+1/ 
3/t*b*g*m*n*dilog(((j*x+i)*e+d*j-e*i)/(d*j-e*i))*s^3+4/9*t^2/e^2*x*b*d^2*g 
*m*n-5/36*t^2/e*x^2*b*d*g*m*n-1/3/j^3*t^2*b*g*m*n*dilog(((j*x+i)*e+d*j-e*i 
)/(d*j-e*i))*i^3-1/j*b*g*m*n*dilog(((j*x+i)*e+d*j-e*i)/(d*j-e*i))*i*s^2+1/ 
3/t*b*g*m*n*ln(j*x+i)*ln(((j*x+i)*e+d*j-e*i)/(d*j-e*i))*s^3-1/9*n*b*ln(h)* 
x^3*g*t^2-1/9*n*b*g*t^2*ln((j*x+i)^m)*x^3-n*b*g*ln((j*x+i)^m)*x*s^2-n*b*x* 
ln(h)*g*s^2+2/9/j*t^2/e^2*b*d^2*g*i*m*n+(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)*(1/3*(-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)*csgn(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)*(t*x+s 
)^3/t+2*g*(1/3*ln((j*x+i)^m)*t^2*x^3+ln((j*x+i)^m)*t*x^2*s+ln((j*x+i)^m)*x 
*s^2+1/3*ln((j*x+i)^m)/t*s^3-1/3/t*m*j*(t/j^3*(1/3*t^2*x^3*j^2-1/2*x^2*i*j 
*t^2+3/2*x^2*j^2*s*t+x*i^2*t^2-3*x*i*j*s*t+3*x*j^2*s^2)+(-i^3*t^3+3*i^2...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1574 vs. \(2 (744) = 1488\).

Time = 0.47 (sec) , antiderivative size = 1574, normalized size of antiderivative = 1.99 \[ \int (s+t x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*b*f*t^2*x^3*log((e*x + d)^n*c) + 1/3*a*g*t^2*x^3*log((j*x + i)^m*h) + 
1/3*a*f*t^2*x^3 - b*e*f*n*s^2*(x/e - d*log(e*x + d)/e^2) - a*g*j*m*s^2*(x/ 
j - i*log(j*x + i)/j^2) + 1/18*b*e*f*n*t^2*(6*d^3*log(e*x + d)/e^4 - (2*e^ 
2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3) - 1/2*b*e*f*n*s*t*(2*d^2*log(e*x + d)/e^ 
3 + (e*x^2 - 2*d*x)/e^2) + 1/18*a*g*j*m*t^2*(6*i^3*log(j*x + i)/j^4 - (2*j 
^2*x^3 - 3*i*j*x^2 + 6*i^2*x)/j^3) - 1/2*a*g*j*m*s*t*(2*i^2*log(j*x + i)/j 
^3 + (j*x^2 - 2*i*x)/j^2) + b*f*s*t*x^2*log((e*x + d)^n*c) + a*g*s*t*x^2*l 
og((j*x + i)^m*h) + a*f*s*t*x^2 + b*f*s^2*x*log((e*x + d)^n*c) + a*g*s^2*x 
*log((j*x + i)^m*h) + a*f*s^2*x - 1/18*(6*d^2*g*i*j^2*m*n*t^2 - 6*(3*i*j^2 
*m*s^2 - 3*i^2*j*m*s*t + i^3*m*t^2)*e^2*g*log(c) - 3*(6*i*j^2*m*n*s*t - i^ 
2*j*m*n*t^2)*d*e*g + (18*i*j^2*m*n*s^2 - 9*i^2*j*m*n*s*t + 2*i^3*m*n*t^2)* 
e^2*g)*b*log(j*x + i)/(e^2*j^3) - 1/3*(3*d*e^2*g*j^3*m*n*s^2 - 3*d^2*e*g*j 
^3*m*n*s*t + d^3*g*j^3*m*n*t^2 - (3*i*j^2*m*n*s^2 - 3*i^2*j*m*n*s*t + i^3* 
m*n*t^2)*e^3*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^3*j^3) - 1/108*(36*(3*i*j^2*m*n*s^2 - 3*i 
^2*j*m*n*s*t + i^3*m*n*t^2)*b*e^3*g*log(e*x + d)*log(j*x + i) + 4*(3*(j^3* 
m*t^2 - 3*j^3*t^2*log(h))*e^3*g*log(c) - (2*j^3*m*n*t^2 - 3*j^3*n*t^2*log( 
h))*e^3*g)*b*x^3 + 3*(6*(3*j^3*m*s*t - i*j^2*m*t^2 - 6*j^3*s*t*log(h))*e^3 
*g*log(c) + (5*j^3*m*n*t^2 - 6*j^3*n*t^2*log(h))*d*e^2*g - (18*j^3*m*n*s*t 
 - 5*i*j^2*m*n*t^2 - 18*j^3*n*s*t*log(h))*e^3*g)*b*x^2 + 6*(6*(3*j^3*m*...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (s+t x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {too large to display} \] Input:

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

Output:

( - 36*atan(j*x)*b*d**6*e*g*i*j**6*m*n*t**2 + 108*atan(j*x)*b*d**5*e**2*g* 
i*j**6*m*n*s*t - 126*atan(j*x)*b*d**5*e**2*g*j**5*m*n*t**2 - 108*atan(j*x) 
*b*d**4*e**3*g*i*j**6*m*n*s**2 + 156*atan(j*x)*b*d**4*e**3*g*i*j**4*m*n*t* 
*2 + 378*atan(j*x)*b*d**4*e**3*g*j**5*m*n*s*t - 432*atan(j*x)*b*d**3*e**4* 
g*i*j**4*m*n*s*t - 432*atan(j*x)*b*d**3*e**4*g*j**5*m*n*s**2 + 84*atan(j*x 
)*b*d**3*e**4*g*j**3*m*n*t**2 + 648*atan(j*x)*b*d**2*e**5*g*i*j**4*m*n*s** 
2 - 36*atan(j*x)*b*d**2*e**5*g*i*j**2*m*n*t**2 - 108*atan(j*x)*b*d**2*e**5 
*g*j**3*m*n*s*t - 108*atan(j*x)*b*d*e**6*g*i*j**2*m*n*s*t + 432*atan(j*x)* 
b*d*e**6*g*j**3*m*n*s**2 - 30*atan(j*x)*b*d*e**6*g*j*m*n*t**2 - 108*atan(j 
*x)*b*e**7*g*i*j**2*m*n*s**2 + 12*atan(j*x)*b*e**7*g*i*m*n*t**2 - 54*atan( 
j*x)*b*e**7*g*j*m*n*s*t - 36*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**9*e*g 
*j**9*n*t**2 + 216*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**2*g*i*j**8 
*n*t**2 + 108*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**2*g*j**9*n*s*t 
- 648*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**3*g*i*j**8*n*s*t - 108* 
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**3*g*j**9*n*s**2 + 540*int(...