Integrand size = 38, antiderivative size = 269 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {B d^2 i (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b B d i (c+d x)^3}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i (c+d x)^4}{16 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b c-a d)^3 g^5 (a+b x)^4} \] Output:
-1/4*B*d^2*i*(d*x+c)^2/(-a*d+b*c)^3/g^5/(b*x+a)^2+2/9*b*B*d*i*(d*x+c)^3/(- a*d+b*c)^3/g^5/(b*x+a)^3-1/16*b^2*B*i*(d*x+c)^4/(-a*d+b*c)^3/g^5/(b*x+a)^4 -1/2*d^2*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^5/(b*x+a)^ 2+2/3*b*d*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^5/(b*x+a) ^3-1/4*b^2*i*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^5/(b*x+a )^4
Time = 0.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.78 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {i \left (\frac {36 A b c}{(a+b x)^4}+\frac {9 b B c}{(a+b x)^4}-\frac {36 a A d}{(a+b x)^4}-\frac {9 a B d}{(a+b x)^4}+\frac {48 A d}{(a+b x)^3}+\frac {4 B d}{(a+b x)^3}-\frac {6 B d^2}{(b c-a d) (a+b x)^2}+\frac {12 B d^3}{(b c-a d)^2 (a+b x)}+\frac {12 B d^4 \log (a+b x)}{(b c-a d)^3}+\frac {12 B (3 b c+a d+4 b d x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4}-\frac {12 B d^4 \log (c+d x)}{(b c-a d)^3}\right )}{144 b^2 g^5} \] Input:
Integrate[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g* x)^5,x]
Output:
-1/144*(i*((36*A*b*c)/(a + b*x)^4 + (9*b*B*c)/(a + b*x)^4 - (36*a*A*d)/(a + b*x)^4 - (9*a*B*d)/(a + b*x)^4 + (48*A*d)/(a + b*x)^3 + (4*B*d)/(a + b*x )^3 - (6*B*d^2)/((b*c - a*d)*(a + b*x)^2) + (12*B*d^3)/((b*c - a*d)^2*(a + b*x)) + (12*B*d^4*Log[a + b*x])/(b*c - a*d)^3 + (12*B*(3*b*c + a*d + 4*b* d*x)*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x)^4 - (12*B*d^4*Log[c + d*x])/( b*c - a*d)^3))/(b^2*g^5)
Time = 0.42 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.75, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2962, 2772, 27, 1140, 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.
| \(\displaystyle \int \frac {(c i+d i x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a g+b g x)^5} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i \int \frac {(c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^5}d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {i \left (-B \int -\frac {(c+d x)^5 \left (3 b^2-\frac {8 d (a+b x) b}{c+d x}+\frac {6 d^2 (a+b x)^2}{(c+d x)^2}\right )}{12 (a+b x)^5}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}\right )}{g^5 (b c-a d)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i \left (\frac {1}{12} B \int \frac {(c+d x)^5 \left (3 b^2-\frac {8 d (a+b x) b}{c+d x}+\frac {6 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^5}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}\right )}{g^5 (b c-a d)^3}\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \frac {i \left (\frac {1}{12} B \int \left (\frac {3 b^2 (c+d x)^5}{(a+b x)^5}-\frac {8 b d (c+d x)^4}{(a+b x)^4}+\frac {6 d^2 (c+d x)^3}{(a+b x)^3}\right )d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}\right )}{g^5 (b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i \left (-\frac {b^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {1}{12} B \left (-\frac {3 b^2 (c+d x)^4}{4 (a+b x)^4}-\frac {3 d^2 (c+d x)^2}{(a+b x)^2}+\frac {8 b d (c+d x)^3}{3 (a+b x)^3}\right )\right )}{g^5 (b c-a d)^3}\) |
Input:
Int[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^5,x ]
Output:
(i*((B*((-3*d^2*(c + d*x)^2)/(a + b*x)^2 + (8*b*d*(c + d*x)^3)/(3*(a + b*x )^3) - (3*b^2*(c + d*x)^4)/(4*(a + b*x)^4)))/12 - (d^2*(c + d*x)^2*(A + B* Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x)^2) + (2*b*d*(c + d*x)^3*(A + B *Log[(e*(a + b*x))/(c + d*x)]))/(3*(a + b*x)^3) - (b^2*(c + d*x)^4*(A + B* Log[(e*(a + b*x))/(c + d*x)]))/(4*(a + b*x)^4)))/((b*c - a*d)^3*g^5)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -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]
Time = 2.09 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.45
| method | result | size |
| parts | \(\frac {i A \left (-\frac {d}{3 b^{2} \left (b x +a \right )^{3}}-\frac {-d a +b c}{4 b^{2} \left (b x +a \right )^{4}}\right )}{g^{5}}-\frac {i B \left (d a -b c \right )^{2} e^{2} \left (\frac {d^{5} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{5}}-\frac {2 d^{4} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{5}}+\frac {d^{3} b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{5}}\right )}{g^{5} d^{3}}\) | \(390\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i \,d^{2} e^{3} A \,b^{2}}{4 \left (d a -b c \right )^{4} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}+\frac {2 i \,d^{3} e^{2} A b}{3 \left (d a -b c \right )^{4} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{4} e A}{2 \left (d a -b c \right )^{4} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,d^{2} e^{3} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{4} g^{5}}-\frac {2 i \,d^{3} e^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{4} g^{5}}+\frac {i \,d^{4} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{4} g^{5}}\right )}{d^{2}}\) | \(516\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {i \,d^{2} e^{3} A \,b^{2}}{4 \left (d a -b c \right )^{4} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}+\frac {2 i \,d^{3} e^{2} A b}{3 \left (d a -b c \right )^{4} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{4} e A}{2 \left (d a -b c \right )^{4} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {i \,d^{2} e^{3} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{4} g^{5}}-\frac {2 i \,d^{3} e^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{4} g^{5}}+\frac {i \,d^{4} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{4} g^{5}}\right )}{d^{2}}\) | \(516\) |
| orering | \(-\frac {\left (b x +a \right ) \left (-60 b^{4} d^{4} x^{4}-204 a \,b^{3} d^{4} x^{3}-36 b^{4} c \,d^{3} x^{3}-238 a^{2} b^{2} d^{4} x^{2}-136 a \,b^{3} c \,d^{3} x^{2}+14 b^{4} c^{2} d^{2} x^{2}-476 a^{2} b^{2} c \,d^{3} x +340 a \,b^{3} c^{2} d^{2} x -104 b^{4} c^{3} d x +13 a^{4} d^{4}-52 a^{3} b c \,d^{3}-160 a^{2} b^{2} c^{2} d^{2}+220 a \,b^{3} c^{3} d -81 b^{4} c^{4}\right ) \left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{144 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b g x +a g \right )^{5}}+\frac {\left (12 b^{3} d^{3} x^{3}+42 a \,b^{2} d^{3} x^{2}-6 b^{3} c \,d^{2} x^{2}+52 a^{2} b \,d^{3} x -20 a \,b^{2} c \,d^{2} x +4 b^{3} c^{2} d x +13 a^{3} d^{3}+13 a^{2} b c \,d^{2}-23 a \,b^{2} c^{2} d +9 b^{3} c^{3}\right ) \left (b x +a \right )^{2} \left (\frac {d i \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (b g x +a g \right )^{5}}+\frac {\left (d i x +c i \right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{5}}-\frac {5 \left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{6}}\right )}{144 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(546\) |
| risch | \(-\frac {i B \left (4 b d x +d a +3 b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{12 \left (b x +a \right )^{4} b^{2} g^{5}}-\frac {\left (-36 B \,a^{2} b^{2} c^{2} d^{2}+6 B \,b^{4} c^{2} d^{2} x^{2}+12 B a \,b^{3} d^{4} x^{3}-12 B \,b^{4} c \,d^{3} x^{3}+52 B \,a^{3} b \,d^{4} x -4 B \,b^{4} c^{3} d x +12 B \ln \left (d x +c \right ) a^{4} d^{4}+24 B a \,b^{3} c^{2} d^{2} x -12 B \ln \left (-b x -a \right ) a^{4} d^{4}+12 B \ln \left (d x +c \right ) b^{4} d^{4} x^{4}-12 B \ln \left (-b x -a \right ) b^{4} d^{4} x^{4}+42 B \,a^{2} b^{2} d^{4} x^{2}-48 B a \,b^{3} c \,d^{3} x^{2}-72 B \,a^{2} b^{2} c \,d^{3} x +48 A \,a^{3} b \,d^{4} x +32 B a \,b^{3} c^{3} d -144 A \,a^{2} b^{2} c \,d^{3} x -72 A \,a^{2} b^{2} c^{2} d^{2}+96 A a \,b^{3} c^{3} d +48 B \ln \left (d x +c \right ) a \,b^{3} d^{4} x^{3}-48 B \ln \left (-b x -a \right ) a \,b^{3} d^{4} x^{3}+72 B \ln \left (d x +c \right ) a^{2} b^{2} d^{4} x^{2}-72 B \ln \left (-b x -a \right ) a^{2} b^{2} d^{4} x^{2}+48 B \ln \left (d x +c \right ) a^{3} b \,d^{4} x -48 B \ln \left (-b x -a \right ) a^{3} b \,d^{4} x +12 A \,a^{4} d^{4}-48 A \,b^{4} c^{3} d x +13 B \,a^{4} d^{4}-9 B \,b^{4} c^{4}-36 A \,b^{4} c^{4}+144 A a \,b^{3} c^{2} d^{2} x \right ) i}{144 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (b x +a \right )^{4} b^{2} g^{5}}\) | \(555\) |
| norman | \(\frac {\frac {\left (12 A \,a^{2} c \,d^{2} i -24 A a b \,c^{2} d i +12 A \,b^{2} c^{3} i +6 B \,a^{2} c \,d^{2} i -8 B a b \,c^{2} d i +3 B \,b^{2} c^{3} i \right ) x}{12 g a \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {\left (12 A \,a^{3} d^{3} i +12 A \,a^{2} b c \,d^{2} i -60 A a \,b^{2} c^{2} d i +36 A \,b^{3} c^{3} i +6 B \,a^{3} d^{3} i +14 B \,a^{2} b c \,d^{2} i -23 B a \,b^{2} c^{2} d i +9 B \,b^{3} c^{3} i \right ) x^{2}}{24 g \,a^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {\left (12 A \,a^{3} d^{3} i +12 A \,a^{2} b c \,d^{2} i -60 A a \,b^{2} c^{2} d i +36 A \,b^{3} c^{3} i +10 B \,a^{3} d^{3} i +13 B \,a^{2} b c \,d^{2} i -23 B a \,b^{2} c^{2} d i +9 B \,b^{3} c^{3} i \right ) b \,x^{3}}{36 g \,a^{3} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {\left (12 A \,a^{3} d^{3} i +12 A \,a^{2} b c \,d^{2} i -60 A a \,b^{2} c^{2} d i +36 A \,b^{3} c^{3} i +13 B \,a^{3} d^{3} i +13 B \,a^{2} b c \,d^{2} i -23 B a \,b^{2} c^{2} d i +9 B \,b^{3} c^{3} i \right ) b^{2} x^{4}}{144 a^{4} g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {i B \,c^{2} \left (6 a^{2} d^{2}-8 a c d b +3 c^{2} b^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{12 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {B \,a^{2} d^{4} i \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}+\frac {b^{2} d^{4} B i \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{12 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}+\frac {a b \,d^{4} B i \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}+\frac {i B c d \left (3 a^{2} d^{2}-3 a c d b +c^{2} b^{2}\right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}}{g^{4} \left (b x +a \right )^{4}}\) | \(858\) |
| parallelrisch | \(\frac {48 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b c \,d^{4} i -144 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b \,c^{3} d^{2} i +12 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c \,d^{4} i +72 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{8} c^{3} d^{2} i +36 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{5} i +48 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{4} d i +72 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{8} c \,d^{4} i -432 A \,x^{2} a^{6} b^{2} c^{3} d^{2} i +576 A \,x^{2} a^{5} b^{3} c^{4} d i -36 A \,x^{4} a^{2} b^{6} c^{5} i -9 B \,x^{4} a^{2} b^{6} c^{5} i -144 A \,x^{3} a^{3} b^{5} c^{5} i -36 B \,x^{3} a^{3} b^{5} c^{5} i +72 A \,x^{2} a^{8} c \,d^{4} i -216 A \,x^{2} a^{4} b^{4} c^{5} i +36 B \,x^{2} a^{8} c \,d^{4} i -54 B \,x^{2} a^{4} b^{4} c^{5} i +144 A x \,a^{8} c^{2} d^{3} i -144 A x \,a^{5} b^{3} c^{5} i +72 B x \,a^{8} c^{2} d^{3} i -36 B x \,a^{5} b^{3} c^{5} i +48 A \,x^{3} a^{7} b c \,d^{4} i -288 A \,x^{3} a^{5} b^{3} c^{3} d^{2} i +384 A \,x^{3} a^{4} b^{4} c^{4} d i +40 B \,x^{3} a^{7} b c \,d^{4} i +12 B \,x^{3} a^{6} b^{2} c^{2} d^{3} i +12 A \,x^{4} a^{6} b^{2} c \,d^{4} i -72 A \,x^{4} a^{4} b^{4} c^{3} d^{2} i +96 A \,x^{4} a^{3} b^{5} c^{4} d i +13 B \,x^{4} a^{6} b^{2} c \,d^{4} i -36 B \,x^{4} a^{4} b^{4} c^{3} d^{2} i +32 B \,x^{4} a^{3} b^{5} c^{4} d i +48 B \,x^{2} a^{7} b \,c^{2} d^{3} i -222 B \,x^{2} a^{6} b^{2} c^{3} d^{2} i +192 B \,x^{2} a^{5} b^{3} c^{4} d i +144 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{8} c^{2} d^{3} i -144 B \,x^{3} a^{5} b^{3} c^{3} d^{2} i +128 B \,x^{3} a^{4} b^{4} c^{4} d i -432 A x \,a^{7} b \,c^{3} d^{2} i +432 A x \,a^{6} b^{2} c^{4} d i -168 B x \,a^{7} b \,c^{3} d^{2} i +132 B x \,a^{6} b^{2} c^{4} d i -96 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b \,c^{4} d i}{144 g^{5} \left (b x +a \right )^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a^{6} c}\) | \(877\) |
Input:
int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x,method=_RETURN VERBOSE)
Output:
i*A/g^5*(-1/3*d/b^2/(b*x+a)^3-1/4*(-a*d+b*c)/b^2/(b*x+a)^4)-i*B/g^5/d^3*(a *d-b*c)^2*e^2*(d^5/(a*d-b*c)^5*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b* e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)-2*d^4/(a*d -b*c)^5*b*e*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/( d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3)+d^3/(a*d-b*c)^5*b^2*e^2*(-1/4 /(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/16/(b*e /d+(a*d-b*c)*e/d/(d*x+c))^4))
Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (257) = 514\).
Time = 0.09 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.24 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} i x^{2} + 4 \, {\left ({\left (12 \, A + B\right )} b^{4} c^{3} d - 6 \, {\left (6 \, A + B\right )} a b^{3} c^{2} d^{2} + 18 \, {\left (2 \, A + B\right )} a^{2} b^{2} c d^{3} - {\left (12 \, A + 13 \, B\right )} a^{3} b d^{4}\right )} i x + {\left (9 \, {\left (4 \, A + B\right )} b^{4} c^{4} - 32 \, {\left (3 \, A + B\right )} a b^{3} c^{3} d + 36 \, {\left (2 \, A + B\right )} a^{2} b^{2} c^{2} d^{2} - {\left (12 \, A + 13 \, B\right )} a^{4} d^{4}\right )} i + 12 \, {\left (B b^{4} d^{4} i x^{4} + 4 \, B a b^{3} d^{4} i x^{3} + 6 \, B a^{2} b^{2} d^{4} i x^{2} + 4 \, {\left (B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + 3 \, B a^{2} b^{2} c d^{3}\right )} i x + {\left (3 \, B b^{4} c^{4} - 8 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{144 \, {\left ({\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} g^{5} x + {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} g^{5}\right )}} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algori thm="fricas")
Output:
-1/144*(12*(B*b^4*c*d^3 - B*a*b^3*d^4)*i*x^3 - 6*(B*b^4*c^2*d^2 - 8*B*a*b^ 3*c*d^3 + 7*B*a^2*b^2*d^4)*i*x^2 + 4*((12*A + B)*b^4*c^3*d - 6*(6*A + B)*a *b^3*c^2*d^2 + 18*(2*A + B)*a^2*b^2*c*d^3 - (12*A + 13*B)*a^3*b*d^4)*i*x + (9*(4*A + B)*b^4*c^4 - 32*(3*A + B)*a*b^3*c^3*d + 36*(2*A + B)*a^2*b^2*c^ 2*d^2 - (12*A + 13*B)*a^4*d^4)*i + 12*(B*b^4*d^4*i*x^4 + 4*B*a*b^3*d^4*i*x ^3 + 6*B*a^2*b^2*d^4*i*x^2 + 4*(B*b^4*c^3*d - 3*B*a*b^3*c^2*d^2 + 3*B*a^2* b^2*c*d^3)*i*x + (3*B*b^4*c^4 - 8*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2)*i)* log((b*e*x + a*e)/(d*x + c)))/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^ 2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c* d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^ 2 - a^7*b^2*d^3)*g^5)
Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (252) = 504\).
Time = 6.90 (sec) , antiderivative size = 928, normalized size of antiderivative = 3.45 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:
integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**5,x)
Output:
-B*d**4*i*log(x + (-B*a**4*d**8*i/(a*d - b*c)**3 + 4*B*a**3*b*c*d**7*i/(a* d - b*c)**3 - 6*B*a**2*b**2*c**2*d**6*i/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d **5*i/(a*d - b*c)**3 + B*a*d**5*i - B*b**4*c**4*d**4*i/(a*d - b*c)**3 + B* b*c*d**4*i)/(2*B*b*d**5*i))/(12*b**2*g**5*(a*d - b*c)**3) + B*d**4*i*log(x + (B*a**4*d**8*i/(a*d - b*c)**3 - 4*B*a**3*b*c*d**7*i/(a*d - b*c)**3 + 6* B*a**2*b**2*c**2*d**6*i/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d**5*i/(a*d - b*c )**3 + B*a*d**5*i + B*b**4*c**4*d**4*i/(a*d - b*c)**3 + B*b*c*d**4*i)/(2*B *b*d**5*i))/(12*b**2*g**5*(a*d - b*c)**3) + (-B*a*d*i - 3*B*b*c*i - 4*B*b* d*i*x)*log(e*(a + b*x)/(c + d*x))/(12*a**4*b**2*g**5 + 48*a**3*b**3*g**5*x + 72*a**2*b**4*g**5*x**2 + 48*a*b**5*g**5*x**3 + 12*b**6*g**5*x**4) + (-1 2*A*a**3*d**3*i - 12*A*a**2*b*c*d**2*i + 60*A*a*b**2*c**2*d*i - 36*A*b**3* c**3*i - 13*B*a**3*d**3*i - 13*B*a**2*b*c*d**2*i + 23*B*a*b**2*c**2*d*i - 9*B*b**3*c**3*i - 12*B*b**3*d**3*i*x**3 + x**2*(-42*B*a*b**2*d**3*i + 6*B* b**3*c*d**2*i) + x*(-48*A*a**2*b*d**3*i + 96*A*a*b**2*c*d**2*i - 48*A*b**3 *c**2*d*i - 52*B*a**2*b*d**3*i + 20*B*a*b**2*c*d**2*i - 4*B*b**3*c**2*d*i) )/(144*a**6*b**2*d**2*g**5 - 288*a**5*b**3*c*d*g**5 + 144*a**4*b**4*c**2*g **5 + x**4*(144*a**2*b**6*d**2*g**5 - 288*a*b**7*c*d*g**5 + 144*b**8*c**2* g**5) + x**3*(576*a**3*b**5*d**2*g**5 - 1152*a**2*b**6*c*d*g**5 + 576*a*b* *7*c**2*g**5) + x**2*(864*a**4*b**4*d**2*g**5 - 1728*a**3*b**5*c*d*g**5 + 864*a**2*b**6*c**2*g**5) + x*(576*a**5*b**3*d**2*g**5 - 1152*a**4*b**4*...
Leaf count of result is larger than twice the leaf count of optimal. 1386 vs. \(2 (257) = 514\).
Time = 0.10 (sec) , antiderivative size = 1386, normalized size of antiderivative = 5.15 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algori thm="maxima")
Output:
-1/144*B*d*i*(12*(4*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^6*g^5 *x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5 ) + (7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4* b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d ^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3) *x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4 *(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6 *(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12 *(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^ 2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d *x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)) + 1/48*B*c*i*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^ 2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b ^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3* a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3* a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d...
Time = 0.34 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.58 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {1}{144} \, {\left (\frac {12 \, {\left (3 \, B b^{2} e^{5} i - \frac {8 \, {\left (b e x + a e\right )} B b d e^{4} i}{d x + c} + \frac {6 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{3} i}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b e x + a e\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b e x + a e\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {36 \, A b^{2} e^{5} i + 9 \, B b^{2} e^{5} i - \frac {96 \, {\left (b e x + a e\right )} A b d e^{4} i}{d x + c} - \frac {32 \, {\left (b e x + a e\right )} B b d e^{4} i}{d x + c} + \frac {72 \, {\left (b e x + a e\right )}^{2} A d^{2} e^{3} i}{{\left (d x + c\right )}^{2}} + \frac {36 \, {\left (b e x + a e\right )}^{2} B d^{2} e^{3} i}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b e x + a e\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b e x + a e\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b e x + a e\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algori thm="giac")
Output:
-1/144*(12*(3*B*b^2*e^5*i - 8*(b*e*x + a*e)*B*b*d*e^4*i/(d*x + c) + 6*(b*e *x + a*e)^2*B*d^2*e^3*i/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*e*x + a*e)^4*a*b*c*d*g^5/(d*x + c) ^4 + (b*e*x + a*e)^4*a^2*d^2*g^5/(d*x + c)^4) + (36*A*b^2*e^5*i + 9*B*b^2* e^5*i - 96*(b*e*x + a*e)*A*b*d*e^4*i/(d*x + c) - 32*(b*e*x + a*e)*B*b*d*e^ 4*i/(d*x + c) + 72*(b*e*x + a*e)^2*A*d^2*e^3*i/(d*x + c)^2 + 36*(b*e*x + a *e)^2*B*d^2*e^3*i/(d*x + c)^2)/((b*e*x + a*e)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*e*x + a*e)^4*a*b*c*d*g^5/(d*x + c)^4 + (b*e*x + a*e)^4*a^2*d^2*g^5/(d *x + c)^4))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
Time = 28.15 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.19 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=\frac {B\,d^4\,i\,\mathrm {atanh}\left (\frac {12\,a^3\,b^2\,d^3\,g^5-12\,a^2\,b^3\,c\,d^2\,g^5-12\,a\,b^4\,c^2\,d\,g^5+12\,b^5\,c^3\,g^5}{12\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{6\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{4\,b^2\,g^5}+\frac {B\,a\,d\,i}{12\,b^3\,g^5}+\frac {B\,d\,i\,x}{3\,b^2\,g^5}\right )}{4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3}-\frac {\frac {12\,A\,a^3\,d^3\,i+36\,A\,b^3\,c^3\,i+13\,B\,a^3\,d^3\,i+9\,B\,b^3\,c^3\,i-60\,A\,a\,b^2\,c^2\,d\,i+12\,A\,a^2\,b\,c\,d^2\,i-23\,B\,a\,b^2\,c^2\,d\,i+13\,B\,a^2\,b\,c\,d^2\,i}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (12\,A\,a^2\,b\,d^3\,i+13\,B\,a^2\,b\,d^3\,i+12\,A\,b^3\,c^2\,d\,i+B\,b^3\,c^2\,d\,i-24\,A\,a\,b^2\,c\,d^2\,i-5\,B\,a\,b^2\,c\,d^2\,i\right )}{3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {d\,x^2\,\left (B\,b^3\,c\,d\,i-7\,B\,a\,b^2\,d^2\,i\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^3\,d^3\,i\,x^3}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{12\,a^4\,b^2\,g^5+48\,a^3\,b^3\,g^5\,x+72\,a^2\,b^4\,g^5\,x^2+48\,a\,b^5\,g^5\,x^3+12\,b^6\,g^5\,x^4} \] Input:
int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^5,x )
Output:
(B*d^4*i*atanh((12*b^5*c^3*g^5 + 12*a^3*b^2*d^3*g^5 - 12*a*b^4*c^2*d*g^5 - 12*a^2*b^3*c*d^2*g^5)/(12*b^2*g^5*(a*d - b*c)^3) + (2*b*d*x*(a^2*d^2 + b^ 2*c^2 - 2*a*b*c*d))/(a*d - b*c)^3))/(6*b^2*g^5*(a*d - b*c)^3) - (log((e*(a + b*x))/(c + d*x))*((B*c*i)/(4*b^2*g^5) + (B*a*d*i)/(12*b^3*g^5) + (B*d*i *x)/(3*b^2*g^5)))/(4*a^3*x + a^4/b + b^3*x^4 + 6*a^2*b*x^2 + 4*a*b^2*x^3) - ((12*A*a^3*d^3*i + 36*A*b^3*c^3*i + 13*B*a^3*d^3*i + 9*B*b^3*c^3*i - 60* A*a*b^2*c^2*d*i + 12*A*a^2*b*c*d^2*i - 23*B*a*b^2*c^2*d*i + 13*B*a^2*b*c*d ^2*i)/(12*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(12*A*a^2*b*d^3*i + 13*B*a ^2*b*d^3*i + 12*A*b^3*c^2*d*i + B*b^3*c^2*d*i - 24*A*a*b^2*c*d^2*i - 5*B*a *b^2*c*d^2*i))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (d*x^2*(B*b^3*c*d*i - 7*B*a*b^2*d^2*i))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B*b^3*d^3*i*x^3) /(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(12*a^4*b^2*g^5 + 12*b^6*g^5*x^4 + 48*a^ 3*b^3*g^5*x + 48*a*b^5*g^5*x^3 + 72*a^2*b^4*g^5*x^2)
Time = 0.20 (sec) , antiderivative size = 954, normalized size of antiderivative = 3.55 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:
int((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x)
Output:
(i*(12*log(a + b*x)*a**5*b*d**4 + 48*log(a + b*x)*a**4*b**2*d**4*x + 72*lo g(a + b*x)*a**3*b**3*d**4*x**2 + 48*log(a + b*x)*a**2*b**4*d**4*x**3 + 12* log(a + b*x)*a*b**5*d**4*x**4 - 12*log(c + d*x)*a**5*b*d**4 - 48*log(c + d *x)*a**4*b**2*d**4*x - 72*log(c + d*x)*a**3*b**3*d**4*x**2 - 48*log(c + d* x)*a**2*b**4*d**4*x**3 - 12*log(c + d*x)*a*b**5*d**4*x**4 - 12*log((a*e + b*e*x)/(c + d*x))*a**5*b*d**4 - 48*log((a*e + b*e*x)/(c + d*x))*a**4*b**2* d**4*x + 72*log((a*e + b*e*x)/(c + d*x))*a**3*b**3*c**2*d**2 + 144*log((a* e + b*e*x)/(c + d*x))*a**3*b**3*c*d**3*x - 96*log((a*e + b*e*x)/(c + d*x)) *a**2*b**4*c**3*d - 144*log((a*e + b*e*x)/(c + d*x))*a**2*b**4*c**2*d**2*x + 36*log((a*e + b*e*x)/(c + d*x))*a*b**5*c**4 + 48*log((a*e + b*e*x)/(c + d*x))*a*b**5*c**3*d*x - 12*a**6*d**4 - 48*a**5*b*d**4*x - 10*a**5*b*d**4 + 72*a**4*b**2*c**2*d**2 + 144*a**4*b**2*c*d**3*x - 3*a**4*b**2*c*d**3 - 4 0*a**4*b**2*d**4*x - 96*a**3*b**3*c**3*d - 144*a**3*b**3*c**2*d**2*x + 36* a**3*b**3*c**2*d**2 + 60*a**3*b**3*c*d**3*x - 24*a**3*b**3*d**4*x**2 + 36* a**2*b**4*c**4 + 48*a**2*b**4*c**3*d*x - 32*a**2*b**4*c**3*d - 24*a**2*b** 4*c**2*d**2*x + 30*a**2*b**4*c*d**3*x**2 + 9*a*b**5*c**4 + 4*a*b**5*c**3*d *x - 6*a*b**5*c**2*d**2*x**2 + 3*a*b**5*d**4*x**4 - 3*b**6*c*d**3*x**4))/( 144*a*b**2*g**5*(a**7*d**3 - 3*a**6*b*c*d**2 + 4*a**6*b*d**3*x + 3*a**5*b* *2*c**2*d - 12*a**5*b**2*c*d**2*x + 6*a**5*b**2*d**3*x**2 - a**4*b**3*c**3 + 12*a**4*b**3*c**2*d*x - 18*a**4*b**3*c*d**2*x**2 + 4*a**4*b**3*d**3*...