Integrand size = 32, antiderivative size = 503 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\frac {13 B^2 (b c-a d)^4 g^4 x}{30 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{60 b d^3}+\frac {B^2 (b c-a d)^2 g^4 (a+b x)^3}{30 b d^2}-\frac {5 B^2 (b c-a d)^5 g^4 \log (a+b x)}{6 b d^5}-\frac {13 B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{30 b d^5}+\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 b d^3}-\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{10 b d}-\frac {2 B (b c-a d)^4 g^4 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{5 d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{5 b}-\frac {2 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}+\frac {2 B^2 (b c-a d)^5 g^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \] Output:
13/30*B^2*(-a*d+b*c)^4*g^4*x/d^4-7/60*B^2*(-a*d+b*c)^3*g^4*(b*x+a)^2/b/d^3 +1/30*B^2*(-a*d+b*c)^2*g^4*(b*x+a)^3/b/d^2-5/6*B^2*(-a*d+b*c)^5*g^4*ln(b*x +a)/b/d^5-13/30*B^2*(-a*d+b*c)^5*g^4*ln((d*x+c)/(b*x+a))/b/d^5+1/5*B*(-a*d +b*c)^3*g^4*(b*x+a)^2*(A+B*ln(e*(d*x+c)/(b*x+a)))/b/d^3-2/15*B*(-a*d+b*c)^ 2*g^4*(b*x+a)^3*(A+B*ln(e*(d*x+c)/(b*x+a)))/b/d^2+1/10*B*(-a*d+b*c)*g^4*(b *x+a)^4*(A+B*ln(e*(d*x+c)/(b*x+a)))/b/d-2/5*B*(-a*d+b*c)^4*g^4*(d*x+c)*(A+ B*ln(e*(d*x+c)/(b*x+a)))/d^5+1/5*g^4*(b*x+a)^5*(A+B*ln(e*(d*x+c)/(b*x+a))) ^2/b-2/5*B*(-a*d+b*c)^5*g^4*(A+B*ln(e*(d*x+c)/(b*x+a)))*ln(1-d*(b*x+a)/b/( d*x+c))/b/d^5+2/5*B^2*(-a*d+b*c)^5*g^4*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^ 5
Time = 0.53 (sec) , antiderivative size = 512, normalized size of antiderivative = 1.02 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2-\frac {B (b c-a d) \left (24 A b d (b c-a d)^3 x+24 B (b c-a d)^4 \log (a+b x)-4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )-B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )-12 B (b c-a d)^3 (b d x+(-b c+a d) \log (c+d x))+24 b B (b c-a d)^3 (c+d x) \log \left (\frac {e (c+d x)}{a+b x}\right )-12 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+8 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-6 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-24 (b c-a d)^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-12 B (b c-a d)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{12 d^5}\right )}{5 b} \] Input:
Integrate[(a*g + b*g*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2,x]
Output:
(g^4*((a + b*x)^5*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2 - (B*(b*c - a*d)* (24*A*b*d*(b*c - a*d)^3*x + 24*B*(b*c - a*d)^4*Log[a + b*x] - 4*B*(b*c - a *d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x ]) - B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x) ^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) - 12*B*(b*c - a*d)^ 3*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 24*b*B*(b*c - a*d)^3*(c + d*x)*L og[(e*(c + d*x))/(a + b*x)] - 12*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[ (e*(c + d*x))/(a + b*x)]) + 8*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]) - 6*d^4*(a + b*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x )]) - 24*(b*c - a*d)^4*Log[c + d*x]*(A + B*Log[(e*(c + d*x))/(a + b*x)]) - 12*B*(b*c - a*d)^4*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])* Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(12*d^5)))/(5*b)
Time = 1.94 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.39, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.594, Rules used = {2952, 2756, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
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.
| \(\displaystyle \int (a g+b g x)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2 \, dx\) |
| \(\Big \downarrow \) 2952 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{\left (d-\frac {b (c+d x)}{a+b x}\right )^6}d\frac {c+d x}{a+b x}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^5}d\frac {c+d x}{a+b x}}{5 b}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {b \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{\left (d-\frac {b (c+d x)}{a+b x}\right )^5}d\frac {c+d x}{a+b x}}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^4}d\frac {c+d x}{a+b x}}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \int \frac {a+b x}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^4}d\frac {c+d x}{a+b x}}{4 b}\right )}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^4}d\frac {c+d x}{a+b x}}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \int \left (\frac {b}{d^4 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {b}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {b}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^3}+\frac {b}{d \left (d-\frac {b (c+d x)}{a+b x}\right )^4}+\frac {a+b x}{d^4 (c+d x)}\right )d\frac {c+d x}{a+b x}}{4 b}\right )}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^4}d\frac {c+d x}{a+b x}}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^4}d\frac {c+d x}{a+b x}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {b \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{\left (d-\frac {b (c+d x)}{a+b x}\right )^4}d\frac {c+d x}{a+b x}}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^3}d\frac {c+d x}{a+b x}}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \int \frac {a+b x}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^3}d\frac {c+d x}{a+b x}}{3 b}\right )}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^3}d\frac {c+d x}{a+b x}}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \int \left (\frac {b}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {b}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {b}{d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}+\frac {a+b x}{d^3 (c+d x)}\right )d\frac {c+d x}{a+b x}}{3 b}\right )}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^3}d\frac {c+d x}{a+b x}}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^3}d\frac {c+d x}{a+b x}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\frac {b \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{\left (d-\frac {b (c+d x)}{a+b x}\right )^3}d\frac {c+d x}{a+b x}}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^2}d\frac {c+d x}{a+b x}}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {B \int \frac {a+b x}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^2}d\frac {c+d x}{a+b x}}{2 b}\right )}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^2}d\frac {c+d x}{a+b x}}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {B \int \left (\frac {b}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {b}{d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {a+b x}{d^2 (c+d x)}\right )d\frac {c+d x}{a+b x}}{2 b}\right )}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^2}d\frac {c+d x}{a+b x}}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )^2}d\frac {c+d x}{a+b x}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^2}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^2}+\frac {1}{d \left (d-\frac {b (c+d x)}{a+b x}\right )}\right )}{2 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\frac {\frac {b \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{\left (d-\frac {b (c+d x)}{a+b x}\right )^2}d\frac {c+d x}{a+b x}}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^2}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^2}+\frac {1}{d \left (d-\frac {b (c+d x)}{a+b x}\right )}\right )}{2 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\frac {\frac {b \left (\frac {(c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{d (a+b x) \left (d-\frac {b (c+d x)}{a+b x}\right )}-\frac {B \int \frac {1}{d-\frac {b (c+d x)}{a+b x}}d\frac {c+d x}{a+b x}}{d}\right )}{d}+\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^2}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^2}+\frac {1}{d \left (d-\frac {b (c+d x)}{a+b x}\right )}\right )}{2 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\frac {\frac {\int \frac {(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(c+d x) \left (d-\frac {b (c+d x)}{a+b x}\right )}d\frac {c+d x}{a+b x}}{d}+\frac {b \left (\frac {(c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{d (a+b x) \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {B \log \left (d-\frac {b (c+d x)}{a+b x}\right )}{b d}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^2}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^2}+\frac {1}{d \left (d-\frac {b (c+d x)}{a+b x}\right )}\right )}{2 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\frac {\frac {\frac {B \int \frac {(a+b x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{c+d x}d\frac {c+d x}{a+b x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{d}}{d}+\frac {b \left (\frac {(c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{d (a+b x) \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {B \log \left (d-\frac {b (c+d x)}{a+b x}\right )}{b d}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^2}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^2}+\frac {1}{d \left (d-\frac {b (c+d x)}{a+b x}\right )}\right )}{2 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle g^4 \left (-(b c-a d)^5\right ) \left (\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{5 b \left (d-\frac {b (c+d x)}{a+b x}\right )^5}-\frac {2 B \left (\frac {\frac {\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b \left (d-\frac {b (c+d x)}{a+b x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^2}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^2}+\frac {1}{d \left (d-\frac {b (c+d x)}{a+b x}\right )}\right )}{2 b}\right )}{d}+\frac {\frac {\frac {B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{d}}{d}+\frac {b \left (\frac {(c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{d (a+b x) \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {B \log \left (d-\frac {b (c+d x)}{a+b x}\right )}{b d}\right )}{d}}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b \left (d-\frac {b (c+d x)}{a+b x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^3}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^3}+\frac {1}{d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d \left (d-\frac {b (c+d x)}{a+b x}\right )^2}\right )}{3 b}\right )}{d}}{d}+\frac {b \left (\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b \left (d-\frac {b (c+d x)}{a+b x}\right )^4}-\frac {B \left (\frac {\log \left (\frac {c+d x}{a+b x}\right )}{d^4}-\frac {\log \left (d-\frac {b (c+d x)}{a+b x}\right )}{d^4}+\frac {1}{d^3 \left (d-\frac {b (c+d x)}{a+b x}\right )}+\frac {1}{2 d^2 \left (d-\frac {b (c+d x)}{a+b x}\right )^2}+\frac {1}{3 d \left (d-\frac {b (c+d x)}{a+b x}\right )^3}\right )}{4 b}\right )}{d}\right )}{5 b}\right )\) |
Input:
Int[(a*g + b*g*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2,x]
Output:
-((b*c - a*d)^5*g^4*((A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(5*b*(d - (b*( c + d*x))/(a + b*x))^5) - (2*B*((b*((A + B*Log[(e*(c + d*x))/(a + b*x)])/( 4*b*(d - (b*(c + d*x))/(a + b*x))^4) - (B*(1/(3*d*(d - (b*(c + d*x))/(a + b*x))^3) + 1/(2*d^2*(d - (b*(c + d*x))/(a + b*x))^2) + 1/(d^3*(d - (b*(c + d*x))/(a + b*x))) + Log[(c + d*x)/(a + b*x)]/d^4 - Log[d - (b*(c + d*x))/ (a + b*x)]/d^4))/(4*b)))/d + ((b*((A + B*Log[(e*(c + d*x))/(a + b*x)])/(3* b*(d - (b*(c + d*x))/(a + b*x))^3) - (B*(1/(2*d*(d - (b*(c + d*x))/(a + b* x))^2) + 1/(d^2*(d - (b*(c + d*x))/(a + b*x))) + Log[(c + d*x)/(a + b*x)]/ d^3 - Log[d - (b*(c + d*x))/(a + b*x)]/d^3))/(3*b)))/d + ((b*((A + B*Log[( e*(c + d*x))/(a + b*x)])/(2*b*(d - (b*(c + d*x))/(a + b*x))^2) - (B*(1/(d* (d - (b*(c + d*x))/(a + b*x))) + Log[(c + d*x)/(a + b*x)]/d^2 - Log[d - (b *(c + d*x))/(a + b*x)]/d^2))/(2*b)))/d + ((b*(((c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(d*(a + b*x)*(d - (b*(c + d*x))/(a + b*x))) + (B*Log[ d - (b*(c + d*x))/(a + b*x)])/(b*d)))/d + (-(((A + B*Log[(e*(c + d*x))/(a + b*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d) + (B*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d)/d)/d)/d)/d))/(5*b)))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( m + 1)*(g/d)^m Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, ( a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[ n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
\[\int \left (b g x +a g \right )^{4} \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )^{2}d x\]
Input:
int((b*g*x+a*g)^4*(A+B*ln(e*(d*x+c)/(b*x+a)))^2,x)
Output:
int((b*g*x+a*g)^4*(A+B*ln(e*(d*x+c)/(b*x+a)))^2,x)
\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2} \,d x } \] Input:
integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)/(b*x+a)))^2,x, algorithm="frica s")
Output:
integral(A^2*b^4*g^4*x^4 + 4*A^2*a*b^3*g^4*x^3 + 6*A^2*a^2*b^2*g^4*x^2 + 4 *A^2*a^3*b*g^4*x + A^2*a^4*g^4 + (B^2*b^4*g^4*x^4 + 4*B^2*a*b^3*g^4*x^3 + 6*B^2*a^2*b^2*g^4*x^2 + 4*B^2*a^3*b*g^4*x + B^2*a^4*g^4)*log((d*e*x + c*e) /(b*x + a))^2 + 2*(A*B*b^4*g^4*x^4 + 4*A*B*a*b^3*g^4*x^3 + 6*A*B*a^2*b^2*g ^4*x^2 + 4*A*B*a^3*b*g^4*x + A*B*a^4*g^4)*log((d*e*x + c*e)/(b*x + a)), x)
Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\text {Timed out} \] Input:
integrate((b*g*x+a*g)**4*(A+B*ln(e*(d*x+c)/(b*x+a)))**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 2395 vs. \(2 (478) = 956\).
Time = 0.17 (sec) , antiderivative size = 2395, normalized size of antiderivative = 4.76 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)/(b*x+a)))^2,x, algorithm="maxim a")
Output:
1/5*A^2*b^4*g^4*x^5 + A^2*a*b^3*g^4*x^4 + 2*A^2*a^2*b^2*g^4*x^3 + 2*A^2*a^ 3*b*g^4*x^2 + 2*(x*log(d*e*x/(b*x + a) + c*e/(b*x + a)) - a*log(b*x + a)/b + c*log(d*x + c)/d)*A*B*a^4*g^4 + 4*(x^2*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d))* A*B*a^3*b*g^4 + 2*(2*x^3*log(d*e*x/(b*x + a) + c*e/(b*x + a)) - 2*a^3*log( b*x + a)/b^3 + 2*c^3*log(d*x + c)/d^3 + ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2* c^2 - a^2*d^2)*x)/(b^2*d^2))*A*B*a^2*b^2*g^4 + 1/3*(6*x^4*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + 6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + ( 2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B*a*b^3*g^4 + 1/30*(12*x^5*log(d*e*x/(b*x + a) + c*e/(b*x + a)) - 12*a^5*log(b*x + a)/b^5 + 12*c^5*log(d*x + c)/d^5 + (3 *(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4* c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*A*B*b^4*g^4 + A^2*a^4*g^4*x + 1/30*((12*g^4*log(e) - 25*g^4)*b^4*c^5 - (60*g^4*log(e) - 113*g^4)*a*b^3*c^4*d + 4*(30*g^4*log(e) - 49*g^4)*a^2*b^2*c^3*d^2 - 12*( 10*g^4*log(e) - 13*g^4)*a^3*b*c^2*d^3 + 12*(5*g^4*log(e) - 4*g^4)*a^4*c*d^ 4)*B^2*log(d*x + c)/d^5 - 2/5*(b^5*c^5*g^4 - 5*a*b^4*c^4*d*g^4 + 10*a^2*b^ 3*c^3*d^2*g^4 - 10*a^3*b^2*c^2*d^3*g^4 + 5*a^4*b*c*d^4*g^4 - a^5*d^5*g^4)* (log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b *c - a*d)))*B^2/(b*d^5) + 1/60*(12*B^2*b^5*d^5*g^4*x^5*log(e)^2 + 6*(b^...
\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2} \,d x } \] Input:
integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)/(b*x+a)))^2,x, algorithm="giac" )
Output:
integrate((b*g*x + a*g)^4*(B*log((d*x + c)*e/(b*x + a)) + A)^2, x)
Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^4\,{\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )}^2 \,d x \] Input:
int((a*g + b*g*x)^4*(A + B*log((e*(c + d*x))/(a + b*x)))^2,x)
Output:
int((a*g + b*g*x)^4*(A + B*log((e*(c + d*x))/(a + b*x)))^2, x)
\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\text {too large to display} \] Input:
int((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)/(b*x+a)))^2,x)
Output:
(g**4*( - 24*int((log((c*e + d*e*x)/(a + b*x))*x)/(a*c + a*d*x + b*c*x + b *d*x**2),x)*a**5*b**2*d**6 + 120*int((log((c*e + d*e*x)/(a + b*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**4*b**3*c*d**5 - 240*int((log((c*e + d*e *x)/(a + b*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**3*b**4*c**2*d**4 + 240*int((log((c*e + d*e*x)/(a + b*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2 ),x)*a**2*b**5*c**3*d**3 - 120*int((log((c*e + d*e*x)/(a + b*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b**6*c**4*d**2 + 24*int((log((c*e + d*e*x) /(a + b*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**7*c**5*d - 24*log(c + d*x)*a**6*d**5 + 120*log(c + d*x)*a**5*b*c*d**4 + 50*log(c + d*x)*a**5*b *d**5 - 240*log(c + d*x)*a**4*b**2*c**2*d**3 - 250*log(c + d*x)*a**4*b**2* c*d**4 + 240*log(c + d*x)*a**3*b**3*c**3*d**2 + 500*log(c + d*x)*a**3*b**3 *c**2*d**3 - 120*log(c + d*x)*a**2*b**4*c**4*d - 500*log(c + d*x)*a**2*b** 4*c**3*d**2 + 24*log(c + d*x)*a*b**5*c**5 + 250*log(c + d*x)*a*b**5*c**4*d - 50*log(c + d*x)*b**6*c**5 + 48*log((c*e + d*e*x)/(a + b*x))**2*a**4*b** 2*c*d**4 + 60*log((c*e + d*e*x)/(a + b*x))**2*a**4*b**2*d**5*x - 72*log((c *e + d*e*x)/(a + b*x))**2*a**3*b**3*c**2*d**3 + 120*log((c*e + d*e*x)/(a + b*x))**2*a**3*b**3*d**5*x**2 + 48*log((c*e + d*e*x)/(a + b*x))**2*a**2*b* *4*c**3*d**2 + 120*log((c*e + d*e*x)/(a + b*x))**2*a**2*b**4*d**5*x**3 - 1 2*log((c*e + d*e*x)/(a + b*x))**2*a*b**5*c**4*d + 60*log((c*e + d*e*x)/(a + b*x))**2*a*b**5*d**5*x**4 + 12*log((c*e + d*e*x)/(a + b*x))**2*b**6*d...