Integrand size = 31, antiderivative size = 649 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=-2 a b f n x+4 a b g m n x+2 b^2 f n^2 x-6 b^2 g m n^2 x-\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {4 b^2 g m n (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}-\frac {2 b g i m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}-\frac {d g m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{e}+\frac {g i m \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}+\frac {2 b^2 g n^2 (i+j x) \log \left (h (i+j x)^m\right )}{j}-\frac {2 b^2 d g n^2 \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{e}-2 b g n x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )+\frac {d g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right )}{e}+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )-\frac {2 b^2 g i m n^2 \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{j}-\frac {2 b d g m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{e}+\frac {2 b g i m n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{j}-\frac {2 b^2 d g m n^2 \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{e}+\frac {2 b^2 d g m n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{e}-\frac {2 b^2 g i m n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{j} \]
-2*a*b*f*n*x+4*a*b*g*m*n*x+2*b^2*f*n^2*x-6*b^2*g*m*n^2*x-2*b^2*f*n*(e*x+d) *ln(c*(e*x+d)^n)/e+4*b^2*g*m*n*(e*x+d)*ln(c*(e*x+d)^n)/e+d*f*(a+b*ln(c*(e* x+d)^n))^2/e-g*m*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e-2*b*g*i*m*n*(a+b*ln(c*( e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/j-d*g*m*(a+b*ln(c*(e*x+d)^n))^2*ln(e*( j*x+i)/(-d*j+e*i))/e+g*i*m*(a+b*ln(c*(e*x+d)^n))^2*ln(e*(j*x+i)/(-d*j+e*i) )/j+2*b^2*g*n^2*(j*x+i)*ln(h*(j*x+i)^m)/j-2*b^2*d*g*n^2*ln(-j*(e*x+d)/(-d* j+e*i))*ln(h*(j*x+i)^m)/e-2*b*g*n*x*(a+b*ln(c*(e*x+d)^n))*ln(h*(j*x+i)^m)+ d*g*(a+b*ln(c*(e*x+d)^n))^2*ln(h*(j*x+i)^m)/e+x*(a+b*ln(c*(e*x+d)^n))^2*(f +g*ln(h*(j*x+i)^m))-2*b^2*g*i*m*n^2*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j-2*b *d*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* m*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j-2*b^2*d*g*m*n ^2*polylog(2,e*(j*x+i)/(-d*j+e*i))/e+2*b^2*d*g*m*n^2*polylog(3,-j*(e*x+d)/ (-d*j+e*i))/e-2*b^2*g*i*m*n^2*polylog(3,-j*(e*x+d)/(-d*j+e*i))/j
Leaf count is larger than twice the leaf count of optimal. \(1355\) vs. \(2(649)=1298\).
Time = 0.32 (sec) , antiderivative size = 1355, normalized size of antiderivative = 2.09 \[ \int \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 f j n+2 a b d g j m n-2 b^2 d g j m n^2+a^2 e f j x-a^2 e g j m x-2 a b e f j n x+4 a b e g j m n x+2 b^2 e f j n^2 x-6 b^2 e g j m n^2 x+2 a b d f j n \log (d+e x)-2 a b d g j m n \log (d+e x)+2 b^2 d g j m n^2 \log (d+e x)-b^2 d f j n^2 \log ^2(d+e x)+b^2 d g j m n^2 \log ^2(d+e x)-2 b^2 d f j n \log \left (c (d+e x)^n\right )+2 b^2 d g j m n \log \left (c (d+e x)^n\right )+2 a b e f j x \log \left (c (d+e x)^n\right )-2 a b e g j m x \log \left (c (d+e x)^n\right )-2 b^2 e f j n x \log \left (c (d+e x)^n\right )+4 b^2 e g j m n x \log \left (c (d+e x)^n\right )+2 b^2 d f j n \log (d+e x) \log \left (c (d+e x)^n\right )-2 b^2 d g j m n \log (d+e x) \log \left (c (d+e x)^n\right )+b^2 e f j x \log ^2\left (c (d+e x)^n\right )-b^2 e g j m x \log ^2\left (c (d+e x)^n\right )+a^2 e g i m \log (i+j x)-2 a b e g i m n \log (i+j x)+2 a b d g j m n \log (i+j x)+2 b^2 e g i m n^2 \log (i+j x)-2 a b e g i m n \log (d+e x) \log (i+j x)+2 b^2 e g i m n^2 \log (d+e x) \log (i+j x)-2 b^2 d g j m n^2 \log (d+e x) \log (i+j x)+b^2 e g i m n^2 \log ^2(d+e x) \log (i+j x)+2 a b e g i m \log \left (c (d+e x)^n\right ) \log (i+j x)-2 b^2 e g i m n \log \left (c (d+e x)^n\right ) \log (i+j x)+2 b^2 d g j m n \log \left (c (d+e x)^n\right ) \log (i+j x)-2 b^2 e g i m n \log (d+e x) \log \left (c (d+e x)^n\right ) \log (i+j x)+b^2 e g i m \log ^2\left (c (d+e x)^n\right ) \log (i+j x)+2 a b e g i m n \log (d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )-2 a b d g j m n \log (d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )-2 b^2 e g i m n^2 \log (d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )+2 b^2 d g j m n^2 \log (d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )-b^2 e g i m n^2 \log ^2(d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )+b^2 d g j m n^2 \log ^2(d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )+2 b^2 e g i m n \log (d+e x) \log \left (c (d+e x)^n\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )-2 b^2 d g j m n \log (d+e x) \log \left (c (d+e x)^n\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )-2 a b d g j n \log \left (h (i+j x)^m\right )+a^2 e g j x \log \left (h (i+j x)^m\right )-2 a b e g j n x \log \left (h (i+j x)^m\right )+2 b^2 e g j n^2 x \log \left (h (i+j x)^m\right )+2 a b d g j n \log (d+e x) \log \left (h (i+j x)^m\right )-b^2 d g j n^2 \log ^2(d+e x) \log \left (h (i+j x)^m\right )-2 b^2 d g j n \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+2 a b e g j x \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )-2 b^2 e g j n x \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+2 b^2 d g j n \log (d+e x) \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+b^2 e g j x \log ^2\left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+2 b g (e i-d j) m n \left (a-b n+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )+2 b^2 g (-e i+d j) m n^2 \operatorname {PolyLog}\left (3,\frac {j (d+e x)}{-e i+d j}\right )}{e j} \]
(-2*a*b*d*f*j*n + 2*a*b*d*g*j*m*n - 2*b^2*d*g*j*m*n^2 + a^2*e*f*j*x - a^2* e*g*j*m*x - 2*a*b*e*f*j*n*x + 4*a*b*e*g*j*m*n*x + 2*b^2*e*f*j*n^2*x - 6*b^ 2*e*g*j*m*n^2*x + 2*a*b*d*f*j*n*Log[d + e*x] - 2*a*b*d*g*j*m*n*Log[d + e*x ] + 2*b^2*d*g*j*m*n^2*Log[d + e*x] - b^2*d*f*j*n^2*Log[d + e*x]^2 + b^2*d* g*j*m*n^2*Log[d + e*x]^2 - 2*b^2*d*f*j*n*Log[c*(d + e*x)^n] + 2*b^2*d*g*j* m*n*Log[c*(d + e*x)^n] + 2*a*b*e*f*j*x*Log[c*(d + e*x)^n] - 2*a*b*e*g*j*m* x*Log[c*(d + e*x)^n] - 2*b^2*e*f*j*n*x*Log[c*(d + e*x)^n] + 4*b^2*e*g*j*m* n*x*Log[c*(d + e*x)^n] + 2*b^2*d*f*j*n*Log[d + e*x]*Log[c*(d + e*x)^n] - 2 *b^2*d*g*j*m*n*Log[d + e*x]*Log[c*(d + e*x)^n] + b^2*e*f*j*x*Log[c*(d + e* x)^n]^2 - b^2*e*g*j*m*x*Log[c*(d + e*x)^n]^2 + a^2*e*g*i*m*Log[i + j*x] - 2*a*b*e*g*i*m*n*Log[i + j*x] + 2*a*b*d*g*j*m*n*Log[i + j*x] + 2*b^2*e*g*i* m*n^2*Log[i + j*x] - 2*a*b*e*g*i*m*n*Log[d + e*x]*Log[i + j*x] + 2*b^2*e*g *i*m*n^2*Log[d + e*x]*Log[i + j*x] - 2*b^2*d*g*j*m*n^2*Log[d + e*x]*Log[i + j*x] + b^2*e*g*i*m*n^2*Log[d + e*x]^2*Log[i + j*x] + 2*a*b*e*g*i*m*Log[c *(d + e*x)^n]*Log[i + j*x] - 2*b^2*e*g*i*m*n*Log[c*(d + e*x)^n]*Log[i + j* x] + 2*b^2*d*g*j*m*n*Log[c*(d + e*x)^n]*Log[i + j*x] - 2*b^2*e*g*i*m*n*Log [d + e*x]*Log[c*(d + e*x)^n]*Log[i + j*x] + b^2*e*g*i*m*Log[c*(d + e*x)^n] ^2*Log[i + j*x] + 2*a*b*e*g*i*m*n*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d* j)] - 2*a*b*d*g*j*m*n*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] - 2*b^2* e*g*i*m*n^2*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] + 2*b^2*d*g*j*m...
Time = 2.03 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2879, 2863, 2009, 7293, 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.
| \(\displaystyle \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2879 |
| \(\displaystyle -2 b e n \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d+e x}dx-g j m \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{i+j x}dx+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle -2 b e n \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d+e x}dx-g j m \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j}-\frac {i \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j (i+j x)}\right )dx+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b e n \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d+e x}dx-g j m \left (-\frac {2 b i n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}-\frac {i \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^2}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e j}-\frac {2 a b n x}{j}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac {2 b^2 i n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{j^2}+\frac {2 b^2 n^2 x}{j}\right )+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 b e n \int \left (\frac {g x \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}+\frac {f x \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}\right )dx-g j m \left (-\frac {2 b i n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}-\frac {i \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^2}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e j}-\frac {2 a b n x}{j}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac {2 b^2 i n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{j^2}+\frac {2 b^2 n^2 x}{j}\right )+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -g j m \left (-\frac {2 b i n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^2}-\frac {i \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^2}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e j}-\frac {2 a b n x}{j}-\frac {2 b^2 n (d+e x) \log \left (c (d+e x)^n\right )}{e j}+\frac {2 b^2 i n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{j^2}+\frac {2 b^2 n^2 x}{j}\right )-2 b e n \left (-\frac {d f \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b e^2 n}-\frac {d g \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b e^2 n}+\frac {d g m \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {d g m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b e^2 n}+\frac {g x \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac {g i m \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e j}+\frac {a f x}{e}-\frac {a g m x}{e}+\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e^2}-\frac {b g m (d+e x) \log \left (c (d+e x)^n\right )}{e^2}+\frac {b d g n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right )}{e^2}+\frac {b d g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{e^2}-\frac {b d g m n \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{e^2}+\frac {b g i m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{e j}-\frac {b f n x}{e}-\frac {b g n (i+j x) \log \left (h (i+j x)^m\right )}{e j}+\frac {2 b g m n x}{e}\right )+x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )\) |
x*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]) - 2*b*e*n*((a*f* x)/e - (a*g*m*x)/e - (b*f*n*x)/e + (2*b*g*m*n*x)/e + (b*f*(d + e*x)*Log[c* (d + e*x)^n])/e^2 - (b*g*m*(d + e*x)*Log[c*(d + e*x)^n])/e^2 - (d*f*(a + b *Log[c*(d + e*x)^n])^2)/(2*b*e^2*n) + (g*i*m*(a + b*Log[c*(d + e*x)^n])*Lo g[(e*(i + j*x))/(e*i - d*j)])/(e*j) + (d*g*m*(a + b*Log[c*(d + e*x)^n])^2* Log[(e*(i + j*x))/(e*i - d*j)])/(2*b*e^2*n) - (b*g*n*(i + j*x)*Log[h*(i + j*x)^m])/(e*j) + (b*d*g*n*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h*(i + j*x )^m])/e^2 + (g*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/e - (d*g*( a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/(2*b*e^2*n) + (b*g*i*m*n*P olyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(e*j) + (d*g*m*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e^2 + (b*d*g*m*n*PolyLo g[2, (e*(i + j*x))/(e*i - d*j)])/e^2 - (b*d*g*m*n*PolyLog[3, -((j*(d + e*x ))/(e*i - d*j))])/e^2) - g*j*m*((-2*a*b*n*x)/j + (2*b^2*n^2*x)/j - (2*b^2* n*(d + e*x)*Log[c*(d + e*x)^n])/(e*j) + ((d + e*x)*(a + b*Log[c*(d + e*x)^ n])^2)/(e*j) - (i*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d* j)])/j^2 - (2*b*i*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/ (e*i - d*j))])/j^2 + (2*b^2*i*n^2*PolyLog[3, -((j*(d + e*x))/(e*i - d*j))] )/j^2)
3.4.94.3.1 Defintions of rubi rules used
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]
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] + (-Simp[g*j*m Int[x*((a + b*Log[c*(d + e*x)^n])^p/(i + j*x)), x], x] - Simp[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]
\[\int {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2} \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )d x\]
\[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} \,d x } \]
integral(b^2*f*log((e*x + d)^n*c)^2 + 2*a*b*f*log((e*x + d)^n*c) + a^2*f + (b^2*g*log((e*x + d)^n*c)^2 + 2*a*b*g*log((e*x + d)^n*c) + a^2*g)*log((j* x + i)^m*h), x)
Timed out. \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Timed out} \]
\[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} \,d x } \]
-2*a*b*e*f*n*(x/e - d*log(e*x + d)/e^2) - a^2*g*j*m*(x/j - i*log(j*x + i)/ j^2) + b^2*f*x*log((e*x + d)^n*c)^2 + 2*a*b*f*x*log((e*x + d)^n*c) + a^2*g *x*log((j*x + i)^m*h) - (2*e*n*(x/e - d*log(e*x + d)/e^2)*log((e*x + d)^n* c) + (d*log(e*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*n^2/e)*b^2*f + a^2*f*x + ((b^2*e*g*i*m*log(j*x + i) - (j*m - j*log(h))*b^2*e*g*x)*log((e*x + d)^n )^2 - (b^2*d*g*j*n^2*log(e*x + d)^2 - b^2*e*g*j*x*log((e*x + d)^n)^2 + (2* (e*g*j*n - e*g*j*log(c))*a*b - (2*e*g*j*n^2 - 2*e*g*j*n*log(c) + e*g*j*log (c)^2)*b^2)*x - 2*(a*b*d*g*j*n - (d*g*j*n^2 - d*g*j*n*log(c))*b^2)*log(e*x + d) - 2*(b^2*d*g*j*n*log(e*x + d) + (a*b*e*g*j - (e*g*j*n - e*g*j*log(c) )*b^2)*x)*log((e*x + d)^n))*log((j*x + i)^m))/(e*j) - integrate(-(b^2*d*e* g*i*j*log(c)^2*log(h) + 2*a*b*d*e*g*i*j*log(c)*log(h) + (2*(e^2*g*j^2*m*n - (j^2*m - j^2*log(h))*e^2*g*log(c))*a*b - (2*e^2*g*j^2*m*n^2 - 2*e^2*g*j^ 2*m*n*log(c) + (j^2*m - j^2*log(h))*e^2*g*log(c)^2)*b^2)*x^2 + (b^2*d*e*g* j^2*m*n^2*x + b^2*d^2*g*j^2*m*n^2)*log(e*x + d)^2 + (2*(d*e*g*j^2*m*n + (e ^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d*e*g)*log(c))*a*b - (2*d*e*g*j^2*m *n^2 - 2*d*e*g*j^2*m*n*log(c) - (e^2*g*i*j*log(h) - (j^2*m - j^2*log(h))*d *e*g)*log(c)^2)*b^2)*x - 2*(a*b*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^2 + (a*b*d*e*g*j^2*m*n - (d*e*g*j^2*m*n^2 - d*e*g*j^2*m*n *log(c))*b^2)*x)*log(e*x + d) + 2*(b^2*d*e*g*i*j*log(c)*log(h) + a*b*d*e*g *i*j*log(h) - ((j^2*m - j^2*log(h))*a*b*e^2*g + ((j^2*m - j^2*log(h))*e...
\[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} \,d x } \]
Timed out. \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \]