Integrand size = 40, antiderivative size = 181 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=\frac {B d i^2 (c+d x)^3}{9 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b B i^2 (c+d x)^4}{16 (b c-a d)^2 g^5 (a+b x)^4}+\frac {d i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b c-a d)^2 g^5 (a+b x)^4} \] Output:
1/9*B*d*i^2*(d*x+c)^3/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/16*b*B*i^2*(d*x+c)^4/(- a*d+b*c)^2/g^5/(b*x+a)^4+1/3*d*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/( -a*d+b*c)^2/g^5/(b*x+a)^3-1/4*b*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/ (-a*d+b*c)^2/g^5/(b*x+a)^4
Leaf count is larger than twice the leaf count of optimal. \(454\) vs. \(2(181)=362\).
Time = 0.42 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.51 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {i^2 \left (36 A b^4 c^4+9 b^4 B c^4-48 a A b^3 c^3 d-16 a b^3 B c^3 d+12 a^4 A d^4+7 a^4 B d^4+96 A b^4 c^3 d x+20 b^4 B c^3 d x-144 a A b^3 c^2 d^2 x-48 a b^3 B c^2 d^2 x+48 a^3 A b d^4 x+28 a^3 b B d^4 x+72 A b^4 c^2 d^2 x^2+6 b^4 B c^2 d^2 x^2-144 a A b^3 c d^3 x^2-48 a b^3 B c d^3 x^2+72 a^2 A b^2 d^4 x^2+42 a^2 b^2 B d^4 x^2-12 b^4 B c d^3 x^3+12 a b^3 B d^4 x^3-12 B d^4 (a+b x)^4 \log (a+b x)+12 B (b c-a d)^2 \left (a^2 d^2+2 a b d (c+2 d x)+b^2 \left (3 c^2+8 c d x+6 d^2 x^2\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+12 a^4 B d^4 \log (c+d x)+48 a^3 b B d^4 x \log (c+d x)+72 a^2 b^2 B d^4 x^2 \log (c+d x)+48 a b^3 B d^4 x^3 \log (c+d x)+12 b^4 B d^4 x^4 \log (c+d x)\right )}{144 b^3 (b c-a d)^2 g^5 (a+b x)^4} \] Input:
Integrate[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b* g*x)^5,x]
Output:
-1/144*(i^2*(36*A*b^4*c^4 + 9*b^4*B*c^4 - 48*a*A*b^3*c^3*d - 16*a*b^3*B*c^ 3*d + 12*a^4*A*d^4 + 7*a^4*B*d^4 + 96*A*b^4*c^3*d*x + 20*b^4*B*c^3*d*x - 1 44*a*A*b^3*c^2*d^2*x - 48*a*b^3*B*c^2*d^2*x + 48*a^3*A*b*d^4*x + 28*a^3*b* B*d^4*x + 72*A*b^4*c^2*d^2*x^2 + 6*b^4*B*c^2*d^2*x^2 - 144*a*A*b^3*c*d^3*x ^2 - 48*a*b^3*B*c*d^3*x^2 + 72*a^2*A*b^2*d^4*x^2 + 42*a^2*b^2*B*d^4*x^2 - 12*b^4*B*c*d^3*x^3 + 12*a*b^3*B*d^4*x^3 - 12*B*d^4*(a + b*x)^4*Log[a + b*x ] + 12*B*(b*c - a*d)^2*(a^2*d^2 + 2*a*b*d*(c + 2*d*x) + b^2*(3*c^2 + 8*c*d *x + 6*d^2*x^2))*Log[(e*(a + b*x))/(c + d*x)] + 12*a^4*B*d^4*Log[c + d*x] + 48*a^3*b*B*d^4*x*Log[c + d*x] + 72*a^2*b^2*B*d^4*x^2*Log[c + d*x] + 48*a *b^3*B*d^4*x^3*Log[c + d*x] + 12*b^4*B*d^4*x^4*Log[c + d*x]))/(b^3*(b*c - a*d)^2*g^5*(a + b*x)^4)
Time = 0.37 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.76, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2962, 2772, 27, 53, 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.
| \(\displaystyle \int \frac {(c i+d i x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a g+b g x)^5} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i^2 \int \frac {(c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^5}d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {i^2 \left (-B \int -\frac {(c+d x)^5 \left (3 b-\frac {4 d (a+b x)}{c+d x}\right )}{12 (a+b x)^5}d\frac {a+b x}{c+d x}-\frac {b (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}+\frac {d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}\right )}{g^5 (b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i^2 \left (\frac {1}{12} B \int \frac {(c+d x)^5 \left (3 b-\frac {4 d (a+b x)}{c+d x}\right )}{(a+b x)^5}d\frac {a+b x}{c+d x}-\frac {b (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}+\frac {d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}\right )}{g^5 (b c-a d)^2}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {i^2 \left (\frac {1}{12} B \int \left (\frac {3 b (c+d x)^5}{(a+b x)^5}-\frac {4 d (c+d x)^4}{(a+b x)^4}\right )d\frac {a+b x}{c+d x}-\frac {b (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}+\frac {d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}\right )}{g^5 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i^2 \left (-\frac {b (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}+\frac {d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {1}{12} B \left (\frac {4 d (c+d x)^3}{3 (a+b x)^3}-\frac {3 b (c+d x)^4}{4 (a+b x)^4}\right )\right )}{g^5 (b c-a d)^2}\) |
Input:
Int[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^5 ,x]
Output:
(i^2*((B*((4*d*(c + d*x)^3)/(3*(a + b*x)^3) - (3*b*(c + d*x)^4)/(4*(a + b* x)^4)))/12 + (d*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(a + b*x)^3) - (b*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(a + b*x )^4)))/((b*c - a*d)^2*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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 = 2.21 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.76
| method | result | size |
| parts | \(\frac {i^{2} A \left (-\frac {d^{2}}{2 b^{3} \left (b x +a \right )^{2}}+\frac {2 d \left (d a -b c \right )}{3 b^{3} \left (b x +a \right )^{3}}-\frac {a^{2} d^{2}-2 a c d b +c^{2} b^{2}}{4 b^{3} \left (b x +a \right )^{4}}\right )}{g^{5}}-\frac {i^{2} B \left (d a -b c \right )^{3} e^{3} \left (\frac {d^{5} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{5}}-\frac {d^{4} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{5}}\right )}{g^{5} d^{4}}\) | \(319\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (\frac {i^{2} d^{2} e^{3} A b}{4 \left (d a -b c \right )^{3} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{2} d^{3} e^{2} A}{3 \left (d a -b c \right )^{3} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i^{2} d^{2} e^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{3} g^{5}}+\frac {i^{2} d^{3} e^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{3} g^{5}}\right )}{d^{2}}\) | \(357\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (\frac {i^{2} d^{2} e^{3} A b}{4 \left (d a -b c \right )^{3} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{2} d^{3} e^{2} A}{3 \left (d a -b c \right )^{3} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i^{2} d^{2} e^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{3} g^{5}}+\frac {i^{2} d^{3} e^{2} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{3} g^{5}}\right )}{d^{2}}\) | \(357\) |
| orering | \(-\frac {\left (b x +a \right ) \left (-48 b^{4} d^{4} x^{4}-150 a \,b^{3} d^{4} x^{3}-42 b^{4} c \,d^{3} x^{3}-450 a \,b^{3} c \,d^{3} x^{2}+162 b^{4} c^{2} d^{2} x^{2}+35 a^{3} b \,d^{4} x -105 a^{2} b^{2} c \,d^{3} x -345 a \,b^{3} c^{2} d^{2} x +223 b^{4} c^{3} d x +14 a^{4} d^{4}-21 a^{3} b c \,d^{3}-21 a^{2} b^{2} c^{2} d^{2}-101 a \,b^{3} c^{3} d +81 b^{4} c^{4}\right ) \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{144 b^{3} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (d x +c \right )^{2} \left (b g x +a g \right )^{5}}+\frac {\left (12 b^{3} d^{3} x^{3}+42 a \,b^{2} d^{3} x^{2}-6 b^{3} c \,d^{2} x^{2}+28 a^{2} b \,d^{3} x +28 a \,b^{2} c \,d^{2} x -20 b^{3} c^{2} d x +7 a^{3} d^{3}+7 a^{2} b c \,d^{2}+7 a \,b^{2} c^{2} d -9 b^{3} c^{3}\right ) \left (b x +a \right )^{2} \left (\frac {2 \left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) d i}{\left (b g x +a g \right )^{5}}+\frac {\left (d i x +c i \right )^{2} B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{5}}-\frac {5 \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{6}}\right )}{144 b^{3} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (d x +c \right )}\) | \(534\) |
| risch | \(-\frac {i^{2} B \left (6 d^{2} b^{2} x^{2}+4 a b \,d^{2} x +8 b^{2} c d x +a^{2} d^{2}+2 a c d b +3 c^{2} b^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{12 \left (b x +a \right )^{4} g^{5} b^{3}}-\frac {\left (6 B \,b^{4} c^{2} d^{2} x^{2}+12 B a \,b^{3} d^{4} x^{3}-12 B \,b^{4} c \,d^{3} x^{3}+72 A \,a^{2} b^{2} d^{4} x^{2}+28 B \,a^{3} b \,d^{4} x +20 B \,b^{4} c^{3} d x +12 B \ln \left (d x +c \right ) a^{4} d^{4}-48 B a \,b^{3} c^{2} d^{2} x -12 B \ln \left (-b x -a \right ) a^{4} d^{4}+12 B \ln \left (d x +c \right ) b^{4} d^{4} x^{4}-12 B \ln \left (-b x -a \right ) b^{4} d^{4} x^{4}+42 B \,a^{2} b^{2} d^{4} x^{2}-48 B a \,b^{3} c \,d^{3} x^{2}+48 A \,a^{3} b \,d^{4} x -16 B a \,b^{3} c^{3} d +72 A \,b^{4} c^{2} d^{2} x^{2}-144 A a \,b^{3} c \,d^{3} x^{2}-48 A a \,b^{3} c^{3} d +48 B \ln \left (d x +c \right ) a \,b^{3} d^{4} x^{3}-48 B \ln \left (-b x -a \right ) a \,b^{3} d^{4} x^{3}+72 B \ln \left (d x +c \right ) a^{2} b^{2} d^{4} x^{2}-72 B \ln \left (-b x -a \right ) a^{2} b^{2} d^{4} x^{2}+48 B \ln \left (d x +c \right ) a^{3} b \,d^{4} x -48 B \ln \left (-b x -a \right ) a^{3} b \,d^{4} x +12 A \,a^{4} d^{4}+96 A \,b^{4} c^{3} d x +7 B \,a^{4} d^{4}+9 B \,b^{4} c^{4}+36 A \,b^{4} c^{4}-144 A a \,b^{3} c^{2} d^{2} x \right ) i^{2}}{144 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (b x +a \right )^{4} g^{5} b^{3}}\) | \(566\) |
| norman | \(\frac {\frac {\left (12 A a \,c^{2} d \,i^{2}-12 A b \,c^{3} i^{2}+4 B a \,c^{2} d \,i^{2}-3 B b \,c^{3} i^{2}\right ) x}{12 g a \left (d a -b c \right )}+\frac {\left (24 A \,a^{2} c \,d^{2} i^{2}+12 A a b \,c^{2} d \,i^{2}-36 A \,b^{2} c^{3} i^{2}+8 B \,a^{2} c \,d^{2} i^{2}+7 B a b \,c^{2} d \,i^{2}-9 B \,b^{2} c^{3} i^{2}\right ) x^{2}}{24 g \,a^{2} \left (d a -b c \right )}+\frac {\left (12 A \,a^{3} d^{3} i^{2}+12 A \,a^{2} b c \,d^{2} i^{2}+12 A a \,b^{2} c^{2} d \,i^{2}-36 A \,b^{3} c^{3} i^{2}+4 B \,a^{3} d^{3} i^{2}+7 B \,a^{2} b c \,d^{2} i^{2}+7 B a \,b^{2} c^{2} d \,i^{2}-9 B \,b^{3} c^{3} i^{2}\right ) x^{3}}{36 g \,a^{3} \left (d a -b c \right )}+\frac {\left (12 A \,a^{3} d^{3} i^{2}+12 A \,a^{2} b c \,d^{2} i^{2}+12 A a \,b^{2} c^{2} d \,i^{2}-36 A \,b^{3} c^{3} i^{2}+7 B \,a^{3} d^{3} i^{2}+7 B \,a^{2} b c \,d^{2} i^{2}+7 B a \,b^{2} c^{2} d \,i^{2}-9 B \,b^{3} c^{3} i^{2}\right ) b \,x^{4}}{144 a^{4} g \left (d a -b c \right )}+\frac {i^{2} B \,c^{3} \left (4 d a -3 b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{12 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B a \,d^{4} i^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {B \,d^{4} i^{2} b \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{12 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {i^{2} B c \,d^{2} \left (2 d a -b c \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {i^{2} B \,c^{2} d \left (3 d a -2 b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}}{g^{4} \left (b x +a \right )^{4}}\) | \(715\) |
| parallelrisch | \(\frac {12 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b c \,d^{4} i^{2}-72 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b \,c^{3} d^{2} i^{2}-96 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b \,c^{4} d \,i^{2}+36 A \,x^{4} a^{2} b^{5} c^{5} i^{2}+9 B \,x^{4} a^{2} b^{5} c^{5} i^{2}+48 A \,x^{3} a^{7} c \,d^{4} i^{2}+144 A \,x^{3} a^{3} b^{4} c^{5} i^{2}+16 B \,x^{3} a^{7} c \,d^{4} i^{2}+36 B \,x^{3} a^{3} b^{4} c^{5} i^{2}+144 A \,x^{2} a^{7} c^{2} d^{3} i^{2}+216 A \,x^{2} a^{4} b^{3} c^{5} i^{2}+48 B \,x^{2} a^{7} c^{2} d^{3} i^{2}+54 B \,x^{2} a^{4} b^{3} c^{5} i^{2}+144 A x \,a^{7} c^{3} d^{2} i^{2}+144 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} c^{2} d^{3} i^{2}-72 A \,x^{2} a^{6} b \,c^{3} d^{2} i^{2}-288 A \,x^{2} a^{5} b^{2} c^{4} d \,i^{2}-6 B \,x^{2} a^{6} b \,c^{3} d^{2} i^{2}-96 B \,x^{2} a^{5} b^{2} c^{4} d \,i^{2}+144 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} c^{3} d^{2} i^{2}-288 A x \,a^{6} b \,c^{4} d \,i^{2}-84 B x \,a^{6} b \,c^{4} d \,i^{2}+144 A x \,a^{5} b^{2} c^{5} i^{2}+48 B x \,a^{7} c^{3} d^{2} i^{2}+36 B x \,a^{5} b^{2} c^{5} i^{2}+48 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} c^{4} d \,i^{2}-36 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b \,c^{5} i^{2}+12 A \,x^{4} a^{6} b c \,d^{4} i^{2}-48 A \,x^{4} a^{3} b^{4} c^{4} d \,i^{2}+7 B \,x^{4} a^{6} b c \,d^{4} i^{2}-16 B \,x^{4} a^{3} b^{4} c^{4} d \,i^{2}+48 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} c \,d^{4} i^{2}-192 A \,x^{3} a^{4} b^{3} c^{4} d \,i^{2}+12 B \,x^{3} a^{6} b \,c^{2} d^{3} i^{2}-64 B \,x^{3} a^{4} b^{3} c^{4} d \,i^{2}}{144 g^{5} \left (b x +a \right )^{4} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) a^{6} c}\) | \(765\) |
Input:
int((d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x,method=_RETU RNVERBOSE)
Output:
i^2*A/g^5*(-1/2*d^2/b^3/(b*x+a)^2+2/3*d*(a*d-b*c)/b^3/(b*x+a)^3-1/4*(a^2*d ^2-2*a*b*c*d+b^2*c^2)/b^3/(b*x+a)^4)-i^2*B/g^5/d^4*(a*d-b*c)^3*e^3*(d^5/(a *d-b*c)^5*(-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^4/(a*d-b*c)^5*b*e*(-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. 510 vs. \(2 (173) = 346\).
Time = 0.09 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.82 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=\frac {12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i^{2} x^{3} - 6 \, {\left ({\left (12 \, A + B\right )} b^{4} c^{2} d^{2} - 8 \, {\left (3 \, A + B\right )} a b^{3} c d^{3} + {\left (12 \, A + 7 \, B\right )} a^{2} b^{2} d^{4}\right )} i^{2} x^{2} - 4 \, {\left ({\left (24 \, A + 5 \, B\right )} b^{4} c^{3} d - 12 \, {\left (3 \, A + B\right )} a b^{3} c^{2} d^{2} + {\left (12 \, A + 7 \, B\right )} a^{3} b d^{4}\right )} i^{2} x - {\left (9 \, {\left (4 \, A + B\right )} b^{4} c^{4} - 16 \, {\left (3 \, A + B\right )} a b^{3} c^{3} d + {\left (12 \, A + 7 \, B\right )} a^{4} d^{4}\right )} i^{2} + 12 \, {\left (B b^{4} d^{4} i^{2} x^{4} + 4 \, B a b^{3} d^{4} i^{2} x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 2 \, B a b^{3} c d^{3}\right )} i^{2} x^{2} - 4 \, {\left (2 \, B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2}\right )} i^{2} x - {\left (3 \, B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{144 \, {\left ({\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{2} - 2 \, a^{2} b^{7} c d + a^{3} b^{6} d^{2}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{2} - 2 \, a^{3} b^{6} c d + a^{4} b^{5} d^{2}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{2} - 2 \, a^{4} b^{5} c d + a^{5} b^{4} d^{2}\right )} g^{5} x + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} g^{5}\right )}} \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algo rithm="fricas")
Output:
1/144*(12*(B*b^4*c*d^3 - B*a*b^3*d^4)*i^2*x^3 - 6*((12*A + B)*b^4*c^2*d^2 - 8*(3*A + B)*a*b^3*c*d^3 + (12*A + 7*B)*a^2*b^2*d^4)*i^2*x^2 - 4*((24*A + 5*B)*b^4*c^3*d - 12*(3*A + B)*a*b^3*c^2*d^2 + (12*A + 7*B)*a^3*b*d^4)*i^2 *x - (9*(4*A + B)*b^4*c^4 - 16*(3*A + B)*a*b^3*c^3*d + (12*A + 7*B)*a^4*d^ 4)*i^2 + 12*(B*b^4*d^4*i^2*x^4 + 4*B*a*b^3*d^4*i^2*x^3 - 6*(B*b^4*c^2*d^2 - 2*B*a*b^3*c*d^3)*i^2*x^2 - 4*(2*B*b^4*c^3*d - 3*B*a*b^3*c^2*d^2)*i^2*x - (3*B*b^4*c^4 - 4*B*a*b^3*c^3*d)*i^2)*log((b*e*x + a*e)/(d*x + c)))/((b^9* c^2 - 2*a*b^8*c*d + a^2*b^7*d^2)*g^5*x^4 + 4*(a*b^8*c^2 - 2*a^2*b^7*c*d + a^3*b^6*d^2)*g^5*x^3 + 6*(a^2*b^7*c^2 - 2*a^3*b^6*c*d + a^4*b^5*d^2)*g^5*x ^2 + 4*(a^3*b^6*c^2 - 2*a^4*b^5*c*d + a^5*b^4*d^2)*g^5*x + (a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*g^5)
Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (165) = 330\).
Time = 18.93 (sec) , antiderivative size = 928, normalized size of antiderivative = 5.13 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:
integrate((d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**5,x)
Output:
-B*d**4*i**2*log(x + (-B*a**3*d**7*i**2/(a*d - b*c)**2 + 3*B*a**2*b*c*d**6 *i**2/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**5*i**2/(a*d - b*c)**2 + B*a*d**5 *i**2 + B*b**3*c**3*d**4*i**2/(a*d - b*c)**2 + B*b*c*d**4*i**2)/(2*B*b*d** 5*i**2))/(12*b**3*g**5*(a*d - b*c)**2) + B*d**4*i**2*log(x + (B*a**3*d**7* i**2/(a*d - b*c)**2 - 3*B*a**2*b*c*d**6*i**2/(a*d - b*c)**2 + 3*B*a*b**2*c **2*d**5*i**2/(a*d - b*c)**2 + B*a*d**5*i**2 - B*b**3*c**3*d**4*i**2/(a*d - b*c)**2 + B*b*c*d**4*i**2)/(2*B*b*d**5*i**2))/(12*b**3*g**5*(a*d - b*c)* *2) + (-12*A*a**3*d**3*i**2 - 12*A*a**2*b*c*d**2*i**2 - 12*A*a*b**2*c**2*d *i**2 + 36*A*b**3*c**3*i**2 - 7*B*a**3*d**3*i**2 - 7*B*a**2*b*c*d**2*i**2 - 7*B*a*b**2*c**2*d*i**2 + 9*B*b**3*c**3*i**2 - 12*B*b**3*d**3*i**2*x**3 + x**2*(-72*A*a*b**2*d**3*i**2 + 72*A*b**3*c*d**2*i**2 - 42*B*a*b**2*d**3*i **2 + 6*B*b**3*c*d**2*i**2) + x*(-48*A*a**2*b*d**3*i**2 - 48*A*a*b**2*c*d* *2*i**2 + 96*A*b**3*c**2*d*i**2 - 28*B*a**2*b*d**3*i**2 - 28*B*a*b**2*c*d* *2*i**2 + 20*B*b**3*c**2*d*i**2))/(144*a**5*b**3*d*g**5 - 144*a**4*b**4*c* g**5 + x**4*(144*a*b**7*d*g**5 - 144*b**8*c*g**5) + x**3*(576*a**2*b**6*d* g**5 - 576*a*b**7*c*g**5) + x**2*(864*a**3*b**5*d*g**5 - 864*a**2*b**6*c*g **5) + x*(576*a**4*b**4*d*g**5 - 576*a**3*b**5*c*g**5)) + (-B*a**2*d**2*i* *2 - 2*B*a*b*c*d*i**2 - 4*B*a*b*d**2*i**2*x - 3*B*b**2*c**2*i**2 - 8*B*b** 2*c*d*i**2*x - 6*B*b**2*d**2*i**2*x**2)*log(e*(a + b*x)/(c + d*x))/(12*a** 4*b**3*g**5 + 48*a**3*b**4*g**5*x + 72*a**2*b**5*g**5*x**2 + 48*a*b**6*...
Leaf count of result is larger than twice the leaf count of optimal. 2218 vs. \(2 (173) = 346\).
Time = 0.14 (sec) , antiderivative size = 2218, normalized size of antiderivative = 12.25 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algo rithm="maxima")
Output:
-1/144*B*d^2*i^2*(12*(6*b^2*x^2 + 4*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e /(d*x + c))/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4 *g^5*x + a^4*b^3*g^5) + (13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*d^ 2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 6*(6* b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 + 4*(10*a *b^4*c^3 - 63*a^2*b^3*c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^10*c^ 3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5*x^4 + 4*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a^2*b^8*c^ 3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^3*b^7* c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*g^5*x + (a^4*b^6*c^ 3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*(6*b^2*c^2* d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^ 2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5) + 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(d*x + c)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5* c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5)) - 1/72*B*c*d*i^2*(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*...
Time = 0.35 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.56 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {1}{144} \, {\left (\frac {12 \, {\left (3 \, B b e^{5} i^{2} - \frac {4 \, {\left (b e x + a e\right )} B d e^{4} i^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b e x + a e\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {36 \, A b e^{5} i^{2} + 9 \, B b e^{5} i^{2} - \frac {48 \, {\left (b e x + a e\right )} A d e^{4} i^{2}}{d x + c} - \frac {16 \, {\left (b e x + a e\right )} B d e^{4} i^{2}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b e x + a e\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x, algo rithm="giac")
Output:
-1/144*(12*(3*B*b*e^5*i^2 - 4*(b*e*x + a*e)*B*d*e^4*i^2/(d*x + c))*log((b* e*x + a*e)/(d*x + c))/((b*e*x + a*e)^4*b*c*g^5/(d*x + c)^4 - (b*e*x + a*e) ^4*a*d*g^5/(d*x + c)^4) + (36*A*b*e^5*i^2 + 9*B*b*e^5*i^2 - 48*(b*e*x + a* e)*A*d*e^4*i^2/(d*x + c) - 16*(b*e*x + a*e)*B*d*e^4*i^2/(d*x + c))/((b*e*x + a*e)^4*b*c*g^5/(d*x + c)^4 - (b*e*x + a*e)^4*a*d*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 = 28.89 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.57 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx=-\frac {\frac {12\,A\,a^3\,d^3\,i^2-36\,A\,b^3\,c^3\,i^2+7\,B\,a^3\,d^3\,i^2-9\,B\,b^3\,c^3\,i^2+12\,A\,a\,b^2\,c^2\,d\,i^2+12\,A\,a^2\,b\,c\,d^2\,i^2+7\,B\,a\,b^2\,c^2\,d\,i^2+7\,B\,a^2\,b\,c\,d^2\,i^2}{12\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (12\,A\,a\,b^2\,d^3\,i^2+7\,B\,a\,b^2\,d^3\,i^2-12\,A\,b^3\,c\,d^2\,i^2-B\,b^3\,c\,d^2\,i^2\right )}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (12\,A\,a^2\,b\,d^3\,i^2+7\,B\,a^2\,b\,d^3\,i^2-24\,A\,b^3\,c^2\,d\,i^2-5\,B\,b^3\,c^2\,d\,i^2+12\,A\,a\,b^2\,c\,d^2\,i^2+7\,B\,a\,b^2\,c\,d^2\,i^2\right )}{3\,\left (a\,d-b\,c\right )}+\frac {B\,b^3\,d^3\,i^2\,x^3}{a\,d-b\,c}}{12\,a^4\,b^3\,g^5+48\,a^3\,b^4\,g^5\,x+72\,a^2\,b^5\,g^5\,x^2+48\,a\,b^6\,g^5\,x^3+12\,b^7\,g^5\,x^4}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{12\,b^4\,g^5}+\frac {B\,c\,d\,i^2}{6\,b^3\,g^5}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{12\,b^4\,g^5}+\frac {B\,c\,d\,i^2}{6\,b^3\,g^5}\right )+\frac {B\,a\,d^2\,i^2}{4\,b^3\,g^5}+\frac {B\,c\,d\,i^2}{2\,b^2\,g^5}\right )+\frac {B\,c^2\,i^2}{4\,b^2\,g^5}+\frac {B\,d^2\,i^2\,x^2}{2\,b^2\,g^5}\right )}{4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3}-\frac {B\,d^4\,i^2\,\mathrm {atanh}\left (\frac {12\,b^5\,c^2\,g^5-12\,a^2\,b^3\,d^2\,g^5}{12\,b^3\,g^5\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{6\,b^3\,g^5\,{\left (a\,d-b\,c\right )}^2} \] Input:
int(((c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^5 ,x)
Output:
- ((12*A*a^3*d^3*i^2 - 36*A*b^3*c^3*i^2 + 7*B*a^3*d^3*i^2 - 9*B*b^3*c^3*i^ 2 + 12*A*a*b^2*c^2*d*i^2 + 12*A*a^2*b*c*d^2*i^2 + 7*B*a*b^2*c^2*d*i^2 + 7* B*a^2*b*c*d^2*i^2)/(12*(a*d - b*c)) + (x^2*(12*A*a*b^2*d^3*i^2 + 7*B*a*b^2 *d^3*i^2 - 12*A*b^3*c*d^2*i^2 - B*b^3*c*d^2*i^2))/(2*(a*d - b*c)) + (x*(12 *A*a^2*b*d^3*i^2 + 7*B*a^2*b*d^3*i^2 - 24*A*b^3*c^2*d*i^2 - 5*B*b^3*c^2*d* i^2 + 12*A*a*b^2*c*d^2*i^2 + 7*B*a*b^2*c*d^2*i^2))/(3*(a*d - b*c)) + (B*b^ 3*d^3*i^2*x^3)/(a*d - b*c))/(12*a^4*b^3*g^5 + 12*b^7*g^5*x^4 + 48*a^3*b^4* g^5*x + 48*a*b^6*g^5*x^3 + 72*a^2*b^5*g^5*x^2) - (log((e*(a + b*x))/(c + d *x))*(a*((B*a*d^2*i^2)/(12*b^4*g^5) + (B*c*d*i^2)/(6*b^3*g^5)) + x*(b*((B* a*d^2*i^2)/(12*b^4*g^5) + (B*c*d*i^2)/(6*b^3*g^5)) + (B*a*d^2*i^2)/(4*b^3* g^5) + (B*c*d*i^2)/(2*b^2*g^5)) + (B*c^2*i^2)/(4*b^2*g^5) + (B*d^2*i^2*x^2 )/(2*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) - (B*d^4*i^2*atanh((12*b^5*c^2*g^5 - 12*a^2*b^3*d^2*g^5)/(12*b^3*g^5*(a*d - b*c)^2) - (2*b*d*x)/(a*d - b*c)))/(6*b^3*g^5*(a*d - b*c)^2)
Time = 0.22 (sec) , antiderivative size = 1274, normalized size of antiderivative = 7.04 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:
int((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^5,x)
Output:
( - 24*log(a + b*x)*a**5*b**2*c*d**3 + 12*log(a + b*x)*a**4*b**3*c**2*d**2 - 96*log(a + b*x)*a**4*b**3*c*d**3*x + 48*log(a + b*x)*a**3*b**4*c**2*d** 2*x - 144*log(a + b*x)*a**3*b**4*c*d**3*x**2 + 72*log(a + b*x)*a**2*b**5*c **2*d**2*x**2 - 96*log(a + b*x)*a**2*b**5*c*d**3*x**3 + 48*log(a + b*x)*a* b**6*c**2*d**2*x**3 - 24*log(a + b*x)*a*b**6*c*d**3*x**4 + 12*log(a + b*x) *b**7*c**2*d**2*x**4 + 24*log(c + d*x)*a**5*b**2*c*d**3 - 12*log(c + d*x)* a**4*b**3*c**2*d**2 + 96*log(c + d*x)*a**4*b**3*c*d**3*x - 48*log(c + d*x) *a**3*b**4*c**2*d**2*x + 144*log(c + d*x)*a**3*b**4*c*d**3*x**2 - 72*log(c + d*x)*a**2*b**5*c**2*d**2*x**2 + 96*log(c + d*x)*a**2*b**5*c*d**3*x**3 - 48*log(c + d*x)*a*b**6*c**2*d**2*x**3 + 24*log(c + d*x)*a*b**6*c*d**3*x** 4 - 12*log(c + d*x)*b**7*c**2*d**2*x**4 + 24*log((a*e + b*e*x)/(c + d*x))* a**5*b**2*c*d**3 - 12*log((a*e + b*e*x)/(c + d*x))*a**4*b**3*c**2*d**2 + 9 6*log((a*e + b*e*x)/(c + d*x))*a**4*b**3*c*d**3*x - 48*log((a*e + b*e*x)/( c + d*x))*a**3*b**4*c**3*d - 192*log((a*e + b*e*x)/(c + d*x))*a**3*b**4*c* *2*d**2*x - 48*log((a*e + b*e*x)/(c + d*x))*a**3*b**4*d**4*x**3 + 36*log(( a*e + b*e*x)/(c + d*x))*a**2*b**5*c**4 + 96*log((a*e + b*e*x)/(c + d*x))*a **2*b**5*c**3*d*x + 96*log((a*e + b*e*x)/(c + d*x))*a**2*b**5*c*d**3*x**3 - 12*log((a*e + b*e*x)/(c + d*x))*a**2*b**5*d**4*x**4 - 48*log((a*e + b*e* x)/(c + d*x))*a*b**6*c**2*d**2*x**3 + 24*log((a*e + b*e*x)/(c + d*x))*a*b* *6*c*d**3*x**4 - 12*log((a*e + b*e*x)/(c + d*x))*b**7*c**2*d**2*x**4 + ...