Integrand size = 40, antiderivative size = 356 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=-\frac {5 B d (b c-a d)^2 i^3 x}{6 b^3 g}-\frac {B (b c-a d) i^3 (c+d x)^2}{6 b^2 g}-\frac {5 B (b c-a d)^3 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{6 b^4 g}+\frac {d (b c-a d)^2 i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g}+\frac {(b c-a d) i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g}+\frac {i^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b g}-\frac {11 B (b c-a d)^3 i^3 \log (c+d x)}{6 b^4 g}-\frac {(b c-a d)^3 i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g}+\frac {B (b c-a d)^3 i^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g} \]
-5/6*B*d*(-a*d+b*c)^2*i^3*x/b^3/g-1/6*B*(-a*d+b*c)*i^3*(d*x+c)^2/b^2/g-5/6 *B*(-a*d+b*c)^3*i^3*ln((b*x+a)/(d*x+c))/b^4/g+d*(-a*d+b*c)^2*i^3*(b*x+a)*( A+B*ln(e*(b*x+a)/(d*x+c)))/b^4/g+1/2*(-a*d+b*c)*i^3*(d*x+c)^2*(A+B*ln(e*(b *x+a)/(d*x+c)))/b^2/g+1/3*i^3*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/g-11 /6*B*(-a*d+b*c)^3*i^3*ln(d*x+c)/b^4/g-(-a*d+b*c)^3*i^3*(A+B*ln(e*(b*x+a)/( d*x+c)))*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/g+B*(-a*d+b*c)^3*i^3*polylog(2,b*(d *x+c)/d/(b*x+a))/b^4/g
Time = 0.19 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.99 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\frac {i^3 \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 (b d x+(b c-a d) \log (a+b x))-B (b c-a d) \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 b^2 (b c-a d) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 (b c-a d)^3 \log (g (a+b x)) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)-3 B (b c-a d)^3 \left (\log (g (a+b x)) \left (\log (g (a+b x))-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{6 b^4 g} \]
(i^3*(6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c - a*d)^2*(b*d*x + (b*c - a*d)*Log [a + b*x]) - B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]) + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/ (c + d*x)] + 3*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d *x)]) + 2*b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*(b*c - a*d)^3*Log[g*(a + b*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*B*(b*c - a*d)^3*Log[c + d*x] - 3*B*(b*c - a*d)^3*(Log[g*(a + b*x)]*(Log[g*(a + b*x) ] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])))/(6*b^4*g)
Time = 1.22 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.30, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2962, 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.
| \(\displaystyle \int \frac {(c i+d i x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a g+b g x} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{g}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \int \frac {c+d x}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}\right )}{g}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \int \left (\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {c+d x}{b^3 (a+b x)}\right )d\frac {a+b x}{c+d x}}{3 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\frac {d \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \int \frac {c+d x}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \int \left (\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {c+d x}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{2 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\frac {\frac {d \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\frac {\frac {d \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {B \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{b}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\frac {\frac {\int \frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\frac {\frac {\frac {B \int \frac {(c+d x) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}+\frac {\frac {\frac {B \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}}{b}+\frac {d \left (\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{g}\) |
((b*c - a*d)^3*i^3*((d*((A + B*Log[(e*(a + b*x))/(c + d*x)])/(3*d*(b - (d* (a + b*x))/(c + d*x))^3) - (B*(1/(2*b*(b - (d*(a + b*x))/(c + d*x))^2) + 1 /(b^2*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + d*x)]/b^3 - Log[ b - (d*(a + b*x))/(c + d*x)]/b^3))/(3*d)))/b + ((d*((A + B*Log[(e*(a + b*x ))/(c + d*x)])/(2*d*(b - (d*(a + b*x))/(c + d*x))^2) - (B*(1/(b*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + d*x)]/b^2 - Log[b - (d*(a + b*x) )/(c + d*x)]/b^2))/(2*d)))/b + ((d*(((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) + (B*Log[b - (d*(a + b*x))/(c + d*x)])/(b*d)))/b + (-(((A + B*Log[(e*(a + b*x))/(c + d*x)])*L og[1 - (b*(c + d*x))/(d*(a + b*x))])/b) + (B*PolyLog[2, (b*(c + d*x))/(d*( a + b*x))])/b)/b)/b)/b))/g
3.1.24.3.1 Defintions of rubi rules used
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_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I ntegersQ[m, q]
Leaf count of result is larger than twice the leaf count of optimal. \(1085\) vs. \(2(344)=688\).
Time = 1.81 (sec) , antiderivative size = 1086, normalized size of antiderivative = 3.05
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1086\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1178\) |
| default | \(\text {Expression too large to display}\) | \(1178\) |
| risch | \(\text {Expression too large to display}\) | \(5361\) |
i^3*A/g*(d/b^3*(1/3*b^2*d^2*x^3-1/2*a*b*d^2*x^2+3/2*b^2*c*d*x^2+a^2*d^2*x- 3*a*b*c*d*x+3*b^2*c^2*x)+(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+b^3*c^3)/b^ 4*ln(b*x+a))-i^3*B/g/d^5*(a*d-b*c)^4*e^4*(1/(a*d-b*c)*d^6/b^3/e^3*(1/b/e/d *ln((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)-ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*( b*e/d+(a*d-b*c)*e/d/(d*x+c))/b/e/((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e))-1/ (a*d-b*c)*d^6/b^2/e^2*(-1/2/e^2/b^2/d*ln((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b *e)-1/2/e/b/d/((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)+1/2*ln(b*e/d+(a*d-b*c) *e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))*((b*e/d+(a*d-b*c)*e/d/(d*x+c)) *d-2*b*e)/e^2/b^2/((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)^2)+1/2/(a*d-b*c)*d ^5/b^4/e^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2+1/(a*d-b*c)*d^6/b/e*(-1/6/b/e /d/((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)^2+1/3/b^2/e^2/d/((b*e/d+(a*d-b*c) *e/d/(d*x+c))*d-b*e)+1/3/b^3/e^3/d*ln((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e) -1/3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(3*e^2* b^2-3*(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d*b*e+d^2*(b*e/d+(a*d-b*c)*e/d/(d*x+c) )^2)/b^3/e^3/((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)^3)-1/(a*d-b*c)*d^6/b^4/ e^4*(dilog(-((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/b/e)/d+ln(b*e/d+(a*d-b*c )*e/d/(d*x+c))*ln(-((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/b/e)/d))
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{b g x + a g} \,d x } \]
integral((A*d^3*i^3*x^3 + 3*A*c*d^2*i^3*x^2 + 3*A*c^2*d*i^3*x + A*c^3*i^3 + (B*d^3*i^3*x^3 + 3*B*c*d^2*i^3*x^2 + 3*B*c^2*d*i^3*x + B*c^3*i^3)*log((b *e*x + a*e)/(d*x + c)))/(b*g*x + a*g), x)
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\frac {i^{3} \left (\int \frac {A c^{3}}{a + b x}\, dx + \int \frac {A d^{3} x^{3}}{a + b x}\, dx + \int \frac {B c^{3} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a + b x}\, dx + \int \frac {3 A c d^{2} x^{2}}{a + b x}\, dx + \int \frac {3 A c^{2} d x}{a + b x}\, dx + \int \frac {B d^{3} x^{3} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a + b x}\, dx + \int \frac {3 B c d^{2} x^{2} \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a + b x}\, dx + \int \frac {3 B c^{2} d x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a + b x}\, dx\right )}{g} \]
i**3*(Integral(A*c**3/(a + b*x), x) + Integral(A*d**3*x**3/(a + b*x), x) + Integral(B*c**3*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a + b*x), x) + Inte gral(3*A*c*d**2*x**2/(a + b*x), x) + Integral(3*A*c**2*d*x/(a + b*x), x) + Integral(B*d**3*x**3*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a + b*x), x) + Integral(3*B*c*d**2*x**2*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a + b*x), x) + Integral(3*B*c**2*d*x*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a + b*x), x))/g
Leaf count of result is larger than twice the leaf count of optimal. 850 vs. \(2 (343) = 686\).
Time = 0.31 (sec) , antiderivative size = 850, normalized size of antiderivative = 2.39 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=3 \, A c^{2} d i^{3} {\left (\frac {x}{b g} - \frac {a \log \left (b x + a\right )}{b^{2} g}\right )} - \frac {1}{6} \, A d^{3} i^{3} {\left (\frac {6 \, a^{3} \log \left (b x + a\right )}{b^{4} g} - \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{b^{3} g}\right )} + \frac {3}{2} \, A c d^{2} i^{3} {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{3} g} + \frac {b x^{2} - 2 \, a x}{b^{2} g}\right )} + \frac {A c^{3} i^{3} \log \left (b g x + a g\right )}{b g} - \frac {{\left (11 \, b^{2} c^{3} i^{3} - 15 \, a b c^{2} d i^{3} + 6 \, a^{2} c d^{2} i^{3}\right )} B \log \left (d x + c\right )}{6 \, b^{3} g} + \frac {{\left (b^{3} c^{3} i^{3} - 3 \, a b^{2} c^{2} d i^{3} + 3 \, a^{2} b c d^{2} i^{3} - a^{3} d^{3} i^{3}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{b^{4} g} + \frac {2 \, B b^{3} d^{3} i^{3} x^{3} \log \left (e\right ) + {\left ({\left (9 \, i^{3} \log \left (e\right ) - i^{3}\right )} b^{3} c d^{2} - {\left (3 \, i^{3} \log \left (e\right ) - i^{3}\right )} a b^{2} d^{3}\right )} B x^{2} + 3 \, {\left (b^{3} c^{3} i^{3} - 3 \, a b^{2} c^{2} d i^{3} + 3 \, a^{2} b c d^{2} i^{3} - a^{3} d^{3} i^{3}\right )} B \log \left (b x + a\right )^{2} + {\left ({\left (18 \, i^{3} \log \left (e\right ) - 7 \, i^{3}\right )} b^{3} c^{2} d - 6 \, {\left (3 \, i^{3} \log \left (e\right ) - 2 \, i^{3}\right )} a b^{2} c d^{2} + {\left (6 \, i^{3} \log \left (e\right ) - 5 \, i^{3}\right )} a^{2} b d^{3}\right )} B x + {\left (2 \, B b^{3} d^{3} i^{3} x^{3} + 3 \, {\left (3 \, b^{3} c d^{2} i^{3} - a b^{2} d^{3} i^{3}\right )} B x^{2} + 6 \, {\left (3 \, b^{3} c^{2} d i^{3} - 3 \, a b^{2} c d^{2} i^{3} + a^{2} b d^{3} i^{3}\right )} B x + {\left (6 \, b^{3} c^{3} i^{3} \log \left (e\right ) - 18 \, {\left (i^{3} \log \left (e\right ) - i^{3}\right )} a b^{2} c^{2} d + 9 \, {\left (2 \, i^{3} \log \left (e\right ) - 3 \, i^{3}\right )} a^{2} b c d^{2} - {\left (6 \, i^{3} \log \left (e\right ) - 11 \, i^{3}\right )} a^{3} d^{3}\right )} B\right )} \log \left (b x + a\right ) - {\left (2 \, B b^{3} d^{3} i^{3} x^{3} + 3 \, {\left (3 \, b^{3} c d^{2} i^{3} - a b^{2} d^{3} i^{3}\right )} B x^{2} + 6 \, {\left (3 \, b^{3} c^{2} d i^{3} - 3 \, a b^{2} c d^{2} i^{3} + a^{2} b d^{3} i^{3}\right )} B x + 6 \, {\left (b^{3} c^{3} i^{3} - 3 \, a b^{2} c^{2} d i^{3} + 3 \, a^{2} b c d^{2} i^{3} - a^{3} d^{3} i^{3}\right )} B \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{6 \, b^{4} g} \]
3*A*c^2*d*i^3*(x/(b*g) - a*log(b*x + a)/(b^2*g)) - 1/6*A*d^3*i^3*(6*a^3*lo g(b*x + a)/(b^4*g) - (2*b^2*x^3 - 3*a*b*x^2 + 6*a^2*x)/(b^3*g)) + 3/2*A*c* d^2*i^3*(2*a^2*log(b*x + a)/(b^3*g) + (b*x^2 - 2*a*x)/(b^2*g)) + A*c^3*i^3 *log(b*g*x + a*g)/(b*g) - 1/6*(11*b^2*c^3*i^3 - 15*a*b*c^2*d*i^3 + 6*a^2*c *d^2*i^3)*B*log(d*x + c)/(b^3*g) + (b^3*c^3*i^3 - 3*a*b^2*c^2*d*i^3 + 3*a^ 2*b*c*d^2*i^3 - a^3*d^3*i^3)*(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/(b^4*g) + 1/6*(2*B*b^3*d^3*i^3* x^3*log(e) + ((9*i^3*log(e) - i^3)*b^3*c*d^2 - (3*i^3*log(e) - i^3)*a*b^2* d^3)*B*x^2 + 3*(b^3*c^3*i^3 - 3*a*b^2*c^2*d*i^3 + 3*a^2*b*c*d^2*i^3 - a^3* d^3*i^3)*B*log(b*x + a)^2 + ((18*i^3*log(e) - 7*i^3)*b^3*c^2*d - 6*(3*i^3* log(e) - 2*i^3)*a*b^2*c*d^2 + (6*i^3*log(e) - 5*i^3)*a^2*b*d^3)*B*x + (2*B *b^3*d^3*i^3*x^3 + 3*(3*b^3*c*d^2*i^3 - a*b^2*d^3*i^3)*B*x^2 + 6*(3*b^3*c^ 2*d*i^3 - 3*a*b^2*c*d^2*i^3 + a^2*b*d^3*i^3)*B*x + (6*b^3*c^3*i^3*log(e) - 18*(i^3*log(e) - i^3)*a*b^2*c^2*d + 9*(2*i^3*log(e) - 3*i^3)*a^2*b*c*d^2 - (6*i^3*log(e) - 11*i^3)*a^3*d^3)*B)*log(b*x + a) - (2*B*b^3*d^3*i^3*x^3 + 3*(3*b^3*c*d^2*i^3 - a*b^2*d^3*i^3)*B*x^2 + 6*(3*b^3*c^2*d*i^3 - 3*a*b^2 *c*d^2*i^3 + a^2*b*d^3*i^3)*B*x + 6*(b^3*c^3*i^3 - 3*a*b^2*c^2*d*i^3 + 3*a ^2*b*c*d^2*i^3 - a^3*d^3*i^3)*B*log(b*x + a))*log(d*x + c))/(b^4*g)
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{b g x + a g} \,d x } \]
Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^3\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{a\,g+b\,g\,x} \,d x \]