3.4.97 \(\int x (a+b \log (c (d+e x)^n))^3 (f+g \log (h (i+j x)^m)) \, dx\) [397]

3.4.97.1 Optimal result

Integrand size = 32, antiderivative size = 2050 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx =\text {Too large to display} \]

output

1/2*x^2*(a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m))+6*b^3*d*f*n^3*x/e+3/ 
8*b^3*g*m*n^3*(e*x+d)^2/e^2+1/2*d*g*m*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^3/e^2+ 
1/2*d^2*g*m*(a+b*ln(c*(e*x+d)^n))^3*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))^3*ln(e*(j*x+i)/(-d*j+e*i))/j^2+3/4*b^2*g*n^2*x^2* 
(a+b*ln(c*(e*x+d)^n))*ln(h*(j*x+i)^m)-3/4*b*g*n*x^2*(a+b*ln(c*(e*x+d)^n))^ 
2*ln(h*(j*x+i)^m)-3/8*b^2*g*m*n^2*x^2*(a+b*ln(c*(e*x+d)^n))+3/4*b^2*f*n^2* 
(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))/e^2-3/4*b*f*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^ 
n))^2/e^2-3/4*b^3*g*i^2*m*n^3*polylog(2,-j*(e*x+d)/(-d*j+e*i))/j^2-21/4*b^ 
3*d^2*g*m*n^3*polylog(2,e*(j*x+i)/(-d*j+e*i))/e^2+9/2*b^3*d^2*g*m*n^3*poly 
log(3,-j*(e*x+d)/(-d*j+e*i))/e^2-3/2*b^3*g*i^2*m*n^3*polylog(3,-j*(e*x+d)/ 
(-d*j+e*i))/j^2+3*b^3*d^2*g*m*n^3*polylog(4,-j*(e*x+d)/(-d*j+e*i))/e^2-3*b 
^3*g*i^2*m*n^3*polylog(4,-j*(e*x+d)/(-d*j+e*i))/j^2+3*b*d*f*n*(e*x+d)*(a+b 
*ln(c*(e*x+d)^n))^2/e^2+3/4*b*g*m*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^2/e^2+ 
1/2*g*i*m*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^3/e/j+3/8*b^3*g*i^2*m*n^3*ln(j*x+i 
)/j^2-21/4*b^3*d^2*g*n^3*ln(-j*(e*x+d)/(-d*j+e*i))*ln(h*(j*x+i)^m)/e^2+9/4 
*b*d^2*g*n*(a+b*ln(c*(e*x+d)^n))^2*ln(h*(j*x+i)^m)/e^2+12*a*b^2*d*g*m*n^2* 
x/e+21/4*a*b^2*g*i*m*n^2*x/j+12*b^3*d*g*m*n^2*(e*x+d)*ln(c*(e*x+d)^n)/e^2- 
15/4*b*d*g*m*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e^2-3/4*b^2*g*i^2*m*n^2*(a+ 
b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e*i))/j^2-9/4*b*d^2*g*m*n*(a+b*ln(c* 
(e*x+d)^n))^2*ln(e*(j*x+i)/(-d*j+e*i))/e^2+3/4*b*g*i^2*m*n*(a+b*ln(c*(e...
 

3.4.97.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4971\) vs. \(2(2050)=4100\).

Time = 1.32 (sec) , antiderivative size = 4971, normalized size of antiderivative = 2.42 \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Result too large to show} \]

input

Integrate[x*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]),x]
 

output

(-12*a^2*b*d*e*g*i*j*m*n + 12*a*b^2*d*e*g*i*j*m*n^2 + 24*a*b^2*d^2*g*j^2*m 
*n^2 - 6*b^3*d*e*g*i*j*m*n^3 - 36*b^3*d^2*g*j^2*m*n^3 + 4*a^3*e^2*g*i*j*m* 
x + 12*a^2*b*d*e*f*j^2*n*x - 18*a^2*b*e^2*g*i*j*m*n*x - 18*a^2*b*d*e*g*j^2 
*m*n*x - 36*a*b^2*d*e*f*j^2*n^2*x + 42*a*b^2*e^2*g*i*j*m*n^2*x + 84*a*b^2* 
d*e*g*j^2*m*n^2*x + 42*b^3*d*e*f*j^2*n^3*x - 45*b^3*e^2*g*i*j*m*n^3*x - 13 
5*b^3*d*e*g*j^2*m*n^3*x + 4*a^3*e^2*f*j^2*x^2 - 2*a^3*e^2*g*j^2*m*x^2 - 6* 
a^2*b*e^2*f*j^2*n*x^2 + 6*a^2*b*e^2*g*j^2*m*n*x^2 + 6*a*b^2*e^2*f*j^2*n^2* 
x^2 - 9*a*b^2*e^2*g*j^2*m*n^2*x^2 - 3*b^3*e^2*f*j^2*n^3*x^2 + 6*b^3*e^2*g* 
j^2*m*n^3*x^2 - 12*a^2*b*d^2*f*j^2*n*Log[d + e*x] + 12*a^2*b*d*e*g*i*j*m*n 
*Log[d + e*x] + 6*a^2*b*d^2*g*j^2*m*n*Log[d + e*x] + 36*a*b^2*d^2*f*j^2*n^ 
2*Log[d + e*x] - 12*a*b^2*d*e*g*i*j*m*n^2*Log[d + e*x] - 48*a*b^2*d^2*g*j^ 
2*m*n^2*Log[d + e*x] - 42*b^3*d^2*f*j^2*n^3*Log[d + e*x] + 30*b^3*d*e*g*i* 
j*m*n^3*Log[d + e*x] + 69*b^3*d^2*g*j^2*m*n^3*Log[d + e*x] + 12*a*b^2*d^2* 
f*j^2*n^2*Log[d + e*x]^2 - 12*a*b^2*d*e*g*i*j*m*n^2*Log[d + e*x]^2 - 6*a*b 
^2*d^2*g*j^2*m*n^2*Log[d + e*x]^2 - 18*b^3*d^2*f*j^2*n^3*Log[d + e*x]^2 + 
6*b^3*d*e*g*i*j*m*n^3*Log[d + e*x]^2 + 24*b^3*d^2*g*j^2*m*n^3*Log[d + e*x] 
^2 - 4*b^3*d^2*f*j^2*n^3*Log[d + e*x]^3 + 4*b^3*d*e*g*i*j*m*n^3*Log[d + e* 
x]^3 + 2*b^3*d^2*g*j^2*m*n^3*Log[d + e*x]^3 - 24*a*b^2*d*e*g*i*j*m*n*Log[c 
*(d + e*x)^n] + 12*b^3*d*e*g*i*j*m*n^2*Log[c*(d + e*x)^n] + 24*b^3*d^2*g*j 
^2*m*n^2*Log[c*(d + e*x)^n] + 12*a^2*b*e^2*g*i*j*m*x*Log[c*(d + e*x)^n]...
 

3.4.97.3 Rubi [A] (verified)

Time = 7.34 (sec) , antiderivative size = 2278, normalized size of antiderivative = 1.11, 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.

\(\displaystyle \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx\)

\(\Big \downarrow \) 2889

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3}{2} b e n \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{d+e x}dx-\frac {1}{2} g j m \left (\frac {3 b^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 j}-\frac {6 b^2 i^2 n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3}-\frac {6 a b^2 d n^2 x}{e j}-\frac {6 a b^2 i n^2 x}{j^2}-\frac {3 b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 j}+\frac {3 b d n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 j}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 j}-\frac {d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 j}+\frac {3 b i^2 n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^3}+\frac {i^2 \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{j^3}+\frac {3 b i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e j^2}-\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e j^2}-\frac {6 b^3 d n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^2 j}-\frac {6 b^3 i n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e j^2}-\frac {3 b^3 n^3 (d+e x)^2}{8 e^2 j}+\frac {6 b^3 i^2 n^3 \operatorname {PolyLog}\left (4,-\frac {j (d+e x)}{e i-d j}\right )}{j^3}+\frac {6 b^3 d n^3 x}{e j}+\frac {6 b^3 i n^3 x}{j^2}\right )+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {3}{2} b e n \int \left (\frac {g x^2 \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x}+\frac {f x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x}\right )dx-\frac {1}{2} g j m \left (\frac {3 b^2 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^2 j}-\frac {6 b^2 i^2 n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{j^3}-\frac {6 a b^2 d n^2 x}{e j}-\frac {6 a b^2 i n^2 x}{j^2}-\frac {3 b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^2 j}+\frac {3 b d n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 j}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{2 e^2 j}-\frac {d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 j}+\frac {3 b i^2 n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{j^3}+\frac {i^2 \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{j^3}+\frac {3 b i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e j^2}-\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e j^2}-\frac {6 b^3 d n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e^2 j}-\frac {6 b^3 i n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e j^2}-\frac {3 b^3 n^3 (d+e x)^2}{8 e^2 j}+\frac {6 b^3 i^2 n^3 \operatorname {PolyLog}\left (4,-\frac {j (d+e x)}{e i-d j}\right )}{j^3}+\frac {6 b^3 d n^3 x}{e j}+\frac {6 b^3 i n^3 x}{j^2}\right )+\frac {1}{2} x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )\)

\(\Big \downarrow \) 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 )^3-\frac {3}{2} b e n \left (-\frac {d^2 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 )^3}{3 b e^3 n}+\frac {d^2 g \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 b e^3 n}+\frac {d^2 f \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{3 b e^3 n}+\frac {f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac {g m (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 e^3}-\frac {2 d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^3}+\frac {3 d g m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}+\frac {g i m (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 j}+\frac {3 d^2 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 e^3}-\frac {d 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 )^2}{e^2 j}-\frac {g i^2 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 e j^2}+\frac {g x^2 \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e}-\frac {3 d^2 g \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^3}-\frac {d g x \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2}-\frac {d^2 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 )^2}{e^3}+\frac {b g m n x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e}-\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^3}+\frac {b g m n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 e^3}+\frac {3 b d g i m n \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^2 j}+\frac {b g i^2 m n \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 e j^2}-\frac {b g n x^2 \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e}+\frac {3 b d g n x \log \left (h (i+j x)^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^2}+\frac {3 b d^2 g m 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 )}{e^3}-\frac {2 b d g i m 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 )}{e^2 j}-\frac {b g i^2 m 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 )}{e j^2}+\frac {2 b d^2 g m n \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^3}-\frac {b^2 g m n^2 x^2}{4 e}+\frac {b^2 f n^2 (d+e x)^2}{4 e^3}-\frac {b^2 g m n^2 (d+e x)^2}{8 e^3}-\frac {4 b^2 d f n^2 x}{e^2}+\frac {39 b^2 d g m n^2 x}{4 e^2}+\frac {7 b^2 g i m n^2 x}{4 e j}+\frac {4 a b d f n x}{e^2}-\frac {6 a b d g m n x}{e^2}-\frac {3 a b g i m n x}{2 e j}-\frac {b^2 d^2 g m n^2 \log (d+e x)}{4 e^3}+\frac {4 b^2 d f n (d+e x) \log \left (c (d+e x)^n\right )}{e^3}-\frac {6 b^2 d g m n (d+e x) \log \left (c (d+e x)^n\right )}{e^3}-\frac {3 b^2 g i m n (d+e x) \log \left (c (d+e x)^n\right )}{2 e^2 j}-\frac {b^2 g i^2 m n^2 \log (i+j x)}{4 e j^2}+\frac {b^2 g n^2 x^2 \log \left (h (i+j x)^m\right )}{4 e}-\frac {7 b^2 d g n^2 (i+j x) \log \left (h (i+j x)^m\right )}{2 e^2 j}+\frac {7 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^3}+\frac {3 b^2 d g i m n^2 \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{e^2 j}+\frac {b^2 g i^2 m n^2 \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 e j^2}+\frac {7 b^2 d^2 g m n^2 \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{2 e^3}-\frac {3 b^2 d^2 g m n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{e^3}+\frac {2 b^2 d g i m n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{e^2 j}+\frac {b^2 g i^2 m n^2 \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right )}{e j^2}-\frac {2 b^2 d^2 g m n^2 \operatorname {PolyLog}\left (4,-\frac {j (d+e x)}{e i-d j}\right )}{e^3}\right )-\frac {1}{2} g j m \left (-\frac {3 n^3 (d+e x)^2 b^3}{8 e^2 j}+\frac {6 d n^3 x b^3}{e j}+\frac {6 i n^3 x b^3}{j^2}-\frac {6 d n^2 (d+e x) \log \left (c (d+e x)^n\right ) b^3}{e^2 j}-\frac {6 i n^2 (d+e x) \log \left (c (d+e x)^n\right ) b^3}{e j^2}+\frac {6 i^2 n^3 \operatorname {PolyLog}\left (4,-\frac {j (d+e x)}{e i-d j}\right ) b^3}{j^3}-\frac {6 a d n^2 x b^2}{e j}-\frac {6 a i n^2 x b^2}{j^2}+\frac {3 n^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) b^2}{4 e^2 j}-\frac {6 i^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {j (d+e x)}{e i-d j}\right ) b^2}{j^3}-\frac {3 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 b}{4 e^2 j}+\frac {3 d n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 b}{e^2 j}+\frac {3 i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 b}{e j^2}+\frac {3 i^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \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 )^3}{2 e^2 j}-\frac {d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2 j}-\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e j^2}+\frac {i^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \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])^3*(f + g*Log[h*(i + j*x)^m]),x]
 

output

(x^2*(a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]))/2 - (3*b*e*n 
*((4*a*b*d*f*n*x)/e^2 - (6*a*b*d*g*m*n*x)/e^2 - (3*a*b*g*i*m*n*x)/(2*e*j) 
- (4*b^2*d*f*n^2*x)/e^2 + (39*b^2*d*g*m*n^2*x)/(4*e^2) + (7*b^2*g*i*m*n^2* 
x)/(4*e*j) - (b^2*g*m*n^2*x^2)/(4*e) + (b^2*f*n^2*(d + e*x)^2)/(4*e^3) - ( 
b^2*g*m*n^2*(d + e*x)^2)/(8*e^3) - (b^2*d^2*g*m*n^2*Log[d + e*x])/(4*e^3) 
+ (4*b^2*d*f*n*(d + e*x)*Log[c*(d + e*x)^n])/e^3 - (6*b^2*d*g*m*n*(d + e*x 
)*Log[c*(d + e*x)^n])/e^3 - (3*b^2*g*i*m*n*(d + e*x)*Log[c*(d + e*x)^n])/( 
2*e^2*j) + (b*g*m*n*x^2*(a + b*Log[c*(d + e*x)^n]))/(4*e) - (b*f*n*(d + e* 
x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^3) + (b*g*m*n*(d + e*x)^2*(a + b*Log 
[c*(d + e*x)^n]))/(4*e^3) - (2*d*f*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2) 
/e^3 + (3*d*g*m*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^3) + (g*i*m*( 
d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2*j) + (f*(d + e*x)^2*(a + b*L 
og[c*(d + e*x)^n])^2)/(2*e^3) - (g*m*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n] 
)^2)/(4*e^3) + (d^2*f*(a + b*Log[c*(d + e*x)^n])^3)/(3*b*e^3*n) - (b^2*g*i 
^2*m*n^2*Log[i + j*x])/(4*e*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*e*j^2) + (3*b*d*g*i*m*n*(a + b*Log[c*( 
d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/(e^2*j) + (3*d^2*g*m*(a + b*L 
og[c*(d + e*x)^n])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(2*e^3) - (g*i^2*m*(a 
 + b*Log[c*(d + e*x)^n])^2*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])^2*Log[(e*(i + j*x))/(e*i - d*j)])/(e^2...
 

3.4.97.3.1 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] 
]
 

3.4.97.4 Maple [F]

\[\int x {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3} \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )d x\]

input

int(x*(a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m)),x)
 

output

int(x*(a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m)),x)
 

3.4.97.5 Fricas [F]

\[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x \,d x } \]

input

integrate(x*(a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m)),x, algorithm=" 
fricas")
 

output

integral(b^3*f*x*log((e*x + d)^n*c)^3 + 3*a*b^2*f*x*log((e*x + d)^n*c)^2 + 
 3*a^2*b*f*x*log((e*x + d)^n*c) + a^3*f*x + (b^3*g*x*log((e*x + d)^n*c)^3 
+ 3*a*b^2*g*x*log((e*x + d)^n*c)^2 + 3*a^2*b*g*x*log((e*x + d)^n*c) + a^3* 
g*x)*log((j*x + i)^m*h), x)
 

3.4.97.6 Sympy [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\text {Timed out} \]

input

integrate(x*(a+b*ln(c*(e*x+d)**n))**3*(f+g*ln(h*(j*x+i)**m)),x)
 

output

Timed out
 

3.4.97.7 Maxima [F]

\[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x \,d x } \]

input

integrate(x*(a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m)),x, algorithm=" 
maxima")
 

output

1/2*b^3*f*x^2*log((e*x + d)^n*c)^3 + 3/2*a*b^2*f*x^2*log((e*x + d)^n*c)^2 
- 3/4*a^2*b*e*f*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2) - 1/4*a^3 
*g*j*m*(2*i^2*log(j*x + i)/j^3 + (j*x^2 - 2*i*x)/j^2) + 3/2*a^2*b*f*x^2*lo 
g((e*x + d)^n*c) + 1/2*a^3*g*x^2*log((j*x + i)^m*h) + 1/2*a^3*f*x^2 - 3/4* 
(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)* 
a*b^2*f - 1/8*(6*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)^2 + e*n*((4*d^2*log(e*x + d)^3 + 3*e^2*x^2 + 18*d^2*log(e*x + 
 d)^2 - 42*d*e*x + 42*d^2*log(e*x + d))*n^2/e^3 - 6*(e^2*x^2 + 2*d^2*log(e 
*x + d)^2 - 6*d*e*x + 6*d^2*log(e*x + d))*n*log((e*x + d)^n*c)/e^3))*b^3*f 
 + 1/8*(2*(2*b^3*e^2*g*i*j*m*x - 2*b^3*e^2*g*i^2*m*log(j*x + i) - (j^2*m - 
 2*j^2*log(h))*b^3*e^2*g*x^2)*log((e*x + d)^n)^3 - (4*b^3*d^2*g*j^2*n^3*lo 
g(e*x + d)^3 - 4*b^3*e^2*g*j^2*x^2*log((e*x + d)^n)^3 + (6*(e^2*g*j^2*n - 
2*e^2*g*j^2*log(c))*a^2*b - 6*(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)*a*b^2 + (3*e^2*g*j^2*n^3 - 6*e^2*g*j^2*n^2*log(c) + 6*e^ 
2*g*j^2*n*log(c)^2 - 4*e^2*g*j^2*log(c)^3)*b^3)*x^2 - 6*(2*a*b^2*d^2*g*j^2 
*n^2 - (3*d^2*g*j^2*n^3 - 2*d^2*g*j^2*n^2*log(c))*b^3)*log(e*x + d)^2 - 6* 
(2*b^3*d*e*g*j^2*n*x - 2*b^3*d^2*g*j^2*n*log(e*x + d) + (2*a*b^2*e^2*g*j^2 
 - (e^2*g*j^2*n - 2*e^2*g*j^2*log(c))*b^3)*x^2)*log((e*x + d)^n)^2 - 6*(2* 
a^2*b*d*e*g*j^2*n - 2*(3*d*e*g*j^2*n^2 - 2*d*e*g*j^2*n*log(c))*a*b^2 + ...
 

3.4.97.8 Giac [F]

\[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )} x \,d x } \]

input

integrate(x*(a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m)),x, algorithm=" 
giac")
 

output

integrate((b*log((e*x + d)^n*c) + a)^3*(g*log((j*x + i)^m*h) + f)*x, x)
 

3.4.97.9 Mupad [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right ) \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right ) \,d x \]

input

int(x*(a + b*log(c*(d + e*x)^n))^3*(f + g*log(h*(i + j*x)^m)),x)
 

output

int(x*(a + b*log(c*(d + e*x)^n))^3*(f + g*log(h*(i + j*x)^m)), x)