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

Optimal result

Integrand size = 32, antiderivative size = 1174 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx =\text {Too large to display} \] Output:

-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*poly 
log(3,-j*(e*x+d)/(-d*j+e*i))/j^2-a*b*d^2*n*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g* 
ln(h*(j*x+i)^m))/e^2+1/2*x^2*(a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m)) 
-a*b*g*i*m*n*x/j-b^2*d^2*g*m*n^2*ln(e*x+d)/e^2-a*b*d*g*m*n*x/e-b^2*g*i*m*n 
*(e*x+d)*ln(c*(e*x+d)^n)/e/j-a*b*d^2*g*m*n*polylog(2,e*(j*x+i)/(-d*j+e*i)) 
/e^2-b^2*d*g*n^2*(j*x+i)*ln(h*(j*x+i)^m)/e/j+b^2*d^2*g*m*n*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))*po 
lylog(2,-j*(e*x+d)/(-d*j+e*i))/j^2-b^2*d*g*m*n*(e*x+d)*ln(c*(e*x+d)^n)/e^2 
+b^2*g*i*m*n^2*x/j-1/4*g*m*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^2/e^2-1/2*b^2*d 
^2*ln(c*(e*x+d)^n)^2*(f+g*ln(h*(j*x+i)^m))/e^2+1/4*b^2*n^2*(-e*x+d)^2*(f+g 
*ln(h*(j*x+i)^m))/e^2+2*b^2*d^2*g*m*n^2*polylog(2,e*(j*x+i)/(-d*j+e*i))/e^ 
2+1/4*b*g*m*n*(-e*x+d)^2*(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+1/2 
*b^2*d^2*g*m*ln(c*(e*x+d)^n)^2*ln(e*(j*x+i)/(-d*j+e*i))/e^2+5/2*b^2*d*g*m* 
n^2*x/e+1/2*b^2*g*(d*j+e*i)^2*m*n^2*polylog(2,-j*(e*x+d)/(-d*j+e*i))/e^2/j 
^2-1/4*b^2*g*(d*j+e*i)^2*m*n^2*ln(j*x+i)/e^2/j^2+3/4*b^2*g*(d*j+e*i)*m*n^2 
*x/e/j-1/4*b^2*g*m*n^2*(-e*x+d)^2/e^2-1/8*b^2*g*m*n^2*(e*x+d)^2/e^2+2*b^2* 
d^2*n^2*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+i)^m))/e^2+1/2*d*g*m*(e*x 
+d)*(a+b*ln(c*(e*x+d)^n))^2/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*n*(-e*x+d)^2*(a+b*ln(c*(e*x+d)^n))*(f+g*ln...
 

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 2067, normalized size of antiderivative = 1.76 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Result too large to show} \] Input:

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

Output:

(-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*Lo 
g[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...
 

Rubi [A] (verified)

Time = 6.17 (sec) , antiderivative size = 1273, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2889, 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.

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

\(\Big \downarrow \) 2889

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} x^2 \left (f+g \log \left (h (i+j x)^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-b e n \left (-\frac {b f n \log ^2(d+e x) d^2}{2 e^3}+\frac {b g m n \log (d+e x) d^2}{4 e^3}+\frac {f \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^2}{e^3}-\frac {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 ) d^2}{2 b e^3 n}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (h (i+j x)^m\right ) d^2}{2 b e^3 n}-\frac {3 b g n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \log \left (h (i+j x)^m\right ) d^2}{2 e^3}-\frac {g m \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 ) d^2}{e^3}-\frac {3 b g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right ) d^2}{2 e^3}+\frac {b g m n \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) d^2}{e^3}+\frac {a g m x d}{e^2}+\frac {2 b f n x d}{e^2}-\frac {11 b g m n x d}{4 e^2}+\frac {b g m (d+e x) \log \left (c (d+e x)^n\right ) d}{e^3}-\frac {2 f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d}{e^3}-\frac {g i m \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^2 j}+\frac {3 b g n (i+j x) \log \left (h (i+j x)^m\right ) d}{2 e^2 j}-\frac {g x \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right ) d}{e^2}-\frac {b g i m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) d}{e^2 j}+\frac {b g m n x^2}{4 e}-\frac {b f n (d+e x)^2}{4 e^3}+\frac {a g i m x}{2 e j}-\frac {3 b g i m n x}{4 e j}+\frac {b g i m (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2 j}-\frac {g m x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e}+\frac {f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac {b g i^2 m n \log (i+j x)}{4 e j^2}-\frac {g i^2 m \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 e j^2}-\frac {b g n x^2 \log \left (h (i+j x)^m\right )}{4 e}+\frac {g x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{2 e}-\frac {b g i^2 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 e j^2}\right )-\frac {1}{2} g j m \left (\frac {n^2 (d+e x)^2 b^2}{4 e^2 j}-\frac {2 d n^2 x b^2}{e j}-\frac {2 i n^2 x b^2}{j^2}+\frac {2 d n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{e^2 j}+\frac {2 i n (d+e x) \log \left (c (d+e x)^n\right ) b^2}{e j^2}-\frac {2 i^2 n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{j^3}+\frac {2 a d n x b}{e j}+\frac {2 a i n x b}{j^2}-\frac {n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b}{2 e^2 j}+\frac {2 i^2 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^3}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 j}-\frac {d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 j}-\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e j^2}+\frac {i^2 \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^3}\right )\)

Input:

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

Output:

(x^2*(a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]))/2 - b*e*n*(( 
a*d*g*m*x)/e^2 + (a*g*i*m*x)/(2*e*j) + (2*b*d*f*n*x)/e^2 - (11*b*d*g*m*n*x 
)/(4*e^2) - (3*b*g*i*m*n*x)/(4*e*j) + (b*g*m*n*x^2)/(4*e) - (b*f*n*(d + e* 
x)^2)/(4*e^3) + (b*d^2*g*m*n*Log[d + e*x])/(4*e^3) - (b*d^2*f*n*Log[d + e* 
x]^2)/(2*e^3) + (b*d*g*m*(d + e*x)*Log[c*(d + e*x)^n])/e^3 + (b*g*i*m*(d + 
 e*x)*Log[c*(d + e*x)^n])/(2*e^2*j) - (g*m*x^2*(a + b*Log[c*(d + e*x)^n])) 
/(4*e) - (2*d*f*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/e^3 + (f*(d + e*x)^2 
*(a + b*Log[c*(d + e*x)^n]))/(2*e^3) + (d^2*f*Log[d + e*x]*(a + b*Log[c*(d 
 + e*x)^n]))/e^3 + (b*g*i^2*m*n*Log[i + j*x])/(4*e*j^2) - (g*i^2*m*(a + b* 
Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(2*e*j^2) - (d*g*i*m*( 
a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(e^2*j) - (d^2*g 
*m*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(2*b*e^3*n 
) - (b*g*n*x^2*Log[h*(i + j*x)^m])/(4*e) + (3*b*d*g*n*(i + j*x)*Log[h*(i + 
 j*x)^m])/(2*e^2*j) - (3*b*d^2*g*n*Log[-((j*(d + e*x))/(e*i - d*j))]*Log[h 
*(i + j*x)^m])/(2*e^3) - (d*g*x*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x) 
^m])/e^2 + (g*x^2*(a + b*Log[c*(d + e*x)^n])*Log[h*(i + j*x)^m])/(2*e) + ( 
d^2*g*(a + b*Log[c*(d + e*x)^n])^2*Log[h*(i + j*x)^m])/(2*b*e^3*n) - (b*g* 
i^2*m*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(2*e*j^2) - (b*d*g*i*m*n 
*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/(e^2*j) - (d^2*g*m*(a + b*Log[c 
*(d + e*x)^n])*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/e^3 - (3*b*d^2...
 

Defintions of rubi rules used

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 7293

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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...
 

Giac [F]

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

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

Output:

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 12*int(log((d + e*x)**n*c)**2/(d*j**2*x**2 + d + e*j**2*x**3 + e*x),x) 
*b**2*d**3*g*i*j**3*m*n - 12*int(log((d + e*x)**n*c)**2/(d*j**2*x**2 + d + 
 e*j**2*x**3 + e*x),x)*b**2*d**2*e*g*j**2*m*n - 12*int(log((d + e*x)**n*c) 
**2/(d*j**2*x**2 + d + e*j**2*x**3 + e*x),x)*b**2*d*e**2*g*i*j*m*n - 12*in 
t(log((d + e*x)**n*c)**2/(d*j**2*x**2 + d + e*j**2*x**3 + e*x),x)*b**2*e** 
3*g*m*n - 24*int(log((d + e*x)**n*c)/(d*j**2*x**2 + d + e*j**2*x**3 + e*x) 
,x)*a*b*d**3*g*i*j**3*m*n - 24*int(log((d + e*x)**n*c)/(d*j**2*x**2 + d + 
e*j**2*x**3 + e*x),x)*a*b*d**2*e*g*j**2*m*n - 24*int(log((d + e*x)**n*c)/( 
d*j**2*x**2 + d + e*j**2*x**3 + e*x),x)*a*b*d*e**2*g*i*j*m*n - 24*int(log( 
(d + e*x)**n*c)/(d*j**2*x**2 + d + e*j**2*x**3 + e*x),x)*a*b*e**3*g*m*n + 
36*int(log((d + e*x)**n*c)/(d*j**2*x**2 + d + e*j**2*x**3 + e*x),x)*b**2*d 
**3*g*i*j**3*m*n**2 + 60*int(log((d + e*x)**n*c)/(d*j**2*x**2 + d + e*j**2 
*x**3 + e*x),x)*b**2*d**2*e*g*j**2*m*n**2 - 12*int(log((d + e*x)**n*c)/(d* 
j**2*x**2 + d + e*j**2*x**3 + e*x),x)*b**2*d*e**2*g*i*j*m*n**2 + 12*int(lo 
g((d + e*x)**n*c)/(d*j**2*x**2 + d + e*j**2*x**3 + e*x),x)*b**2*e**3*g*m*n 
**2 + 12*int((log((d + e*x)**n*c)**2*x)/(d*j**2*x**2 + d + e*j**2*x**3 + e 
*x),x)*b**2*d**3*g*j**4*m*n - 12*int((log((d + e*x)**n*c)**2*x)/(d*j**2*x* 
*2 + d + e*j**2*x**3 + e*x),x)*b**2*d**2*e*g*i*j**3*m*n + 12*int((log((d + 
 e*x)**n*c)**2*x)/(d*j**2*x**2 + d + e*j**2*x**3 + e*x),x)*b**2*d*e**2*g*j 
**2*m*n - 12*int((log((d + e*x)**n*c)**2*x)/(d*j**2*x**2 + d + e*j**2*x...