Integrand size = 38, antiderivative size = 173 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\frac {B d i (c+d x)^2}{4 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b B i (c+d x)^3}{9 (b c-a d)^2 g^4 (a+b x)^3}+\frac {d i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^2 g^4 (a+b x)^3} \] Output:
1/4*B*d*i*(d*x+c)^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-1/9*b*B*i*(d*x+c)^3/(-a*d+b *c)^2/g^4/(b*x+a)^3+1/2*d*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b* c)^2/g^4/(b*x+a)^2-1/3*b*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c )^2/g^4/(b*x+a)^3
Time = 0.41 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.08 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=-\frac {i \left (\frac {12 A b c}{(a+b x)^3}+\frac {4 b B c}{(a+b x)^3}-\frac {12 a A d}{(a+b x)^3}-\frac {4 a B d}{(a+b x)^3}+\frac {18 A d}{(a+b x)^2}+\frac {3 B d}{(a+b x)^2}-\frac {6 B d^2}{(b c-a d) (a+b x)}-\frac {6 B d^3 \log (a+b x)}{(b c-a d)^2}+\frac {6 B (2 b c+a d+3 b d x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3}+\frac {6 B d^3 \log (c+d x)}{(b c-a d)^2}\right )}{36 b^2 g^4} \] Input:
Integrate[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g* x)^4,x]
Output:
-1/36*(i*((12*A*b*c)/(a + b*x)^3 + (4*b*B*c)/(a + b*x)^3 - (12*a*A*d)/(a + b*x)^3 - (4*a*B*d)/(a + b*x)^3 + (18*A*d)/(a + b*x)^2 + (3*B*d)/(a + b*x) ^2 - (6*B*d^2)/((b*c - a*d)*(a + b*x)) - (6*B*d^3*Log[a + b*x])/(b*c - a*d )^2 + (6*B*(2*b*c + a*d + 3*b*d*x)*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x) ^3 + (6*B*d^3*Log[c + d*x])/(b*c - a*d)^2))/(b^2*g^4)
Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.79, 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, 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.
| \(\displaystyle \int \frac {(c i+d i x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a g+b g x)^4} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i \int \frac {(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {i \left (-B \int -\frac {(c+d x)^4 \left (2 b-\frac {3 d (a+b x)}{c+d x}\right )}{6 (a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}\right )}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i \left (\frac {1}{6} B \int \frac {(c+d x)^4 \left (2 b-\frac {3 d (a+b x)}{c+d x}\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}\right )}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {i \left (\frac {1}{6} B \int \left (\frac {2 b (c+d x)^4}{(a+b x)^4}-\frac {3 d (c+d x)^3}{(a+b x)^3}\right )d\frac {a+b x}{c+d x}-\frac {b (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}\right )}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i \left (-\frac {b (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {1}{6} B \left (\frac {3 d (c+d x)^2}{2 (a+b x)^2}-\frac {2 b (c+d x)^3}{3 (a+b x)^3}\right )\right )}{g^4 (b c-a d)^2}\) |
Input:
Int[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^4,x ]
Output:
(i*((B*((3*d*(c + d*x)^2)/(2*(a + b*x)^2) - (2*b*(c + d*x)^3)/(3*(a + b*x) ^3)))/6 + (d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x )^2) - (b*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(a + b*x)^3 )))/((b*c - a*d)^2*g^4)
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 = 1.50 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.61
| method | result | size |
| parts | \(\frac {i A \left (-\frac {d}{2 b^{2} \left (b x +a \right )^{2}}-\frac {-d a +b c}{3 b^{2} \left (b x +a \right )^{3}}\right )}{g^{4}}-\frac {i B \left (d a -b c \right )^{2} e^{2} \left (\frac {d^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{4}}-\frac {d^{3} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{4}}\right )}{g^{4} d^{3}}\) | \(279\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (\frac {i \,d^{2} e^{2} A b}{3 \left (d a -b c \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{3} e A}{2 \left (d a -b c \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i \,d^{2} e^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{3} g^{4}}+\frac {i \,d^{3} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{3} g^{4}}\right )}{d^{2}}\) | \(345\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (\frac {i \,d^{2} e^{2} A b}{3 \left (d a -b c \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{3} e A}{2 \left (d a -b c \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i \,d^{2} e^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{3} g^{4}}+\frac {i \,d^{3} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{3} g^{4}}\right )}{d^{2}}\) | \(345\) |
| norman | \(\frac {-\frac {6 A \,a^{2} d^{2} i +6 A a b c d i -12 A \,b^{2} c^{2} i +3 B \,a^{2} d^{2} i +5 B a b c d i -4 B \,b^{2} c^{2} i}{36 b^{2} g \left (d a -b c \right )}-\frac {\left (6 A a \,d^{2} i -6 A b c d i +3 B a \,d^{2} i -B b c d i \right ) x}{12 g \left (d a -b c \right ) b}+\frac {B \,d^{2} i b \,x^{3}}{18 a \left (d a -b c \right ) g}+\frac {B i \,c^{2} \left (3 d a -2 b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {B a i \,d^{3} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B i \,d^{3} b \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B i c d \left (2 d a -b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}}{\left (b x +a \right )^{3} g^{3}}\) | \(383\) |
| risch | \(-\frac {B i \left (3 b d x +d a +2 b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 \left (b x +a \right )^{3} b^{2} g^{4}}-\frac {\left (-6 B \ln \left (-b x -a \right ) b^{3} d^{3} x^{3}+6 B \ln \left (d x +c \right ) b^{3} d^{3} x^{3}-18 B \ln \left (-b x -a \right ) a \,b^{2} d^{3} x^{2}+18 B \ln \left (d x +c \right ) a \,b^{2} d^{3} x^{2}-18 B \ln \left (-b x -a \right ) a^{2} b \,d^{3} x +18 B \ln \left (d x +c \right ) a^{2} b \,d^{3} x +6 B a \,b^{2} d^{3} x^{2}-6 B \,b^{3} c \,d^{2} x^{2}+18 A \,a^{2} b \,d^{3} x -36 A a \,b^{2} c \,d^{2} x +18 A \,b^{3} c^{2} d x -6 B \ln \left (-b x -a \right ) a^{3} d^{3}+6 B \ln \left (d x +c \right ) a^{3} d^{3}+15 B \,a^{2} b \,d^{3} x -18 B a \,b^{2} c \,d^{2} x +3 B \,b^{3} c^{2} d x +6 A \,a^{3} d^{3}-18 A a \,b^{2} c^{2} d +12 A \,b^{3} c^{3}+5 B \,a^{3} d^{3}-9 B a \,b^{2} c^{2} d +4 B \,c^{3} b^{3}\right ) i}{36 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (b x +a \right )^{3} b^{2} g^{4}}\) | \(389\) |
| orering | \(-\frac {\left (b x +a \right ) \left (-24 b^{3} d^{3} x^{3}-57 a \,b^{2} d^{3} x^{2}-15 b^{3} c \,d^{2} x^{2}-114 a \,b^{2} c \,d^{2} x +42 b^{3} c^{2} d x +5 a^{3} d^{3}-15 a^{2} b c \,d^{2}-42 a \,b^{2} c^{2} d +28 b^{3} c^{3}\right ) \left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{36 b^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (d x +c \right ) \left (b g x +a g \right )^{4}}+\frac {\left (6 d^{2} b^{2} x^{2}+15 a b \,d^{2} x -3 b^{2} c d x +5 a^{2} d^{2}+5 a c d b -4 c^{2} b^{2}\right ) \left (b x +a \right )^{2} \left (\frac {d i \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (b g x +a g \right )^{4}}+\frac {\left (d i x +c i \right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{4}}-\frac {4 \left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{5}}\right )}{36 b^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}\) | \(393\) |
| parallelrisch | \(-\frac {-36 A x a \,b^{5} c \,d^{3} i -18 B x a \,b^{5} c \,d^{3} i -18 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c^{2} d^{2} i +4 B \,b^{6} c^{3} d i -6 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} d^{4} i +6 A \,a^{3} b^{3} d^{4} i +12 A \,b^{6} c^{3} d i +5 B \,a^{3} b^{3} d^{4} i +6 B \,x^{2} a \,b^{5} d^{4} i -6 B \,x^{2} b^{6} c \,d^{3} i +18 A x \,a^{2} b^{4} d^{4} i +18 A x \,b^{6} c^{2} d^{2} i +15 B x \,a^{2} b^{4} d^{4} i +3 B x \,b^{6} c^{2} d^{2} i +12 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{3} d i -18 A a \,b^{5} c^{2} d^{2} i -9 B a \,b^{5} c^{2} d^{2} i -36 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c \,d^{3} i -18 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} d^{4} i +18 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{2} d^{2} i}{36 g^{4} \left (b x +a \right )^{3} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b^{5} d}\) | \(398\) |
Input:
int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x,method=_RETURN VERBOSE)
Output:
i*A/g^4*(-1/2*d/b^2/(b*x+a)^2-1/3*(-a*d+b*c)/b^2/(b*x+a)^3)-i*B/g^4/d^3*(a *d-b*c)^2*e^2*(d^4/(a*d-b*c)^4*(-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)-d^3/(a*d-b *c)^4*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))
Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (165) = 330\).
Time = 0.08 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.10 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=\frac {6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i x^{2} - 3 \, {\left ({\left (6 \, A + B\right )} b^{3} c^{2} d - 6 \, {\left (2 \, A + B\right )} a b^{2} c d^{2} + {\left (6 \, A + 5 \, B\right )} a^{2} b d^{3}\right )} i x - {\left (4 \, {\left (3 \, A + B\right )} b^{3} c^{3} - 9 \, {\left (2 \, A + B\right )} a b^{2} c^{2} d + {\left (6 \, A + 5 \, B\right )} a^{3} d^{3}\right )} i + 6 \, {\left (B b^{3} d^{3} i x^{3} + 3 \, B a b^{2} d^{3} i x^{2} - 3 \, {\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2}\right )} i x - {\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algori thm="fricas")
Output:
1/36*(6*(B*b^3*c*d^2 - B*a*b^2*d^3)*i*x^2 - 3*((6*A + B)*b^3*c^2*d - 6*(2* A + B)*a*b^2*c*d^2 + (6*A + 5*B)*a^2*b*d^3)*i*x - (4*(3*A + B)*b^3*c^3 - 9 *(2*A + B)*a*b^2*c^2*d + (6*A + 5*B)*a^3*d^3)*i + 6*(B*b^3*d^3*i*x^3 + 3*B *a*b^2*d^3*i*x^2 - 3*(B*b^3*c^2*d - 2*B*a*b^2*c*d^2)*i*x - (2*B*b^3*c^3 - 3*B*a*b^2*c^2*d)*i)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x ^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4)
Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (158) = 316\).
Time = 4.15 (sec) , antiderivative size = 629, normalized size of antiderivative = 3.64 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=- \frac {B d^{3} i \log {\left (x + \frac {- \frac {B a^{3} d^{6} i}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{5} i}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{4} i}{\left (a d - b c\right )^{2}} + B a d^{4} i + \frac {B b^{3} c^{3} d^{3} i}{\left (a d - b c\right )^{2}} + B b c d^{3} i}{2 B b d^{4} i} \right )}}{6 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {B d^{3} i \log {\left (x + \frac {\frac {B a^{3} d^{6} i}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{5} i}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{4} i}{\left (a d - b c\right )^{2}} + B a d^{4} i - \frac {B b^{3} c^{3} d^{3} i}{\left (a d - b c\right )^{2}} + B b c d^{3} i}{2 B b d^{4} i} \right )}}{6 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {\left (- B a d i - 2 B b c i - 3 B b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{6 a^{3} b^{2} g^{4} + 18 a^{2} b^{3} g^{4} x + 18 a b^{4} g^{4} x^{2} + 6 b^{5} g^{4} x^{3}} + \frac {- 6 A a^{2} d^{2} i - 6 A a b c d i + 12 A b^{2} c^{2} i - 5 B a^{2} d^{2} i - 5 B a b c d i + 4 B b^{2} c^{2} i - 6 B b^{2} d^{2} i x^{2} + x \left (- 18 A a b d^{2} i + 18 A b^{2} c d i - 15 B a b d^{2} i + 3 B b^{2} c d i\right )}{36 a^{4} b^{2} d g^{4} - 36 a^{3} b^{3} c g^{4} + x^{3} \cdot \left (36 a b^{5} d g^{4} - 36 b^{6} c g^{4}\right ) + x^{2} \cdot \left (108 a^{2} b^{4} d g^{4} - 108 a b^{5} c g^{4}\right ) + x \left (108 a^{3} b^{3} d g^{4} - 108 a^{2} b^{4} c g^{4}\right )} \] Input:
integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**4,x)
Output:
-B*d**3*i*log(x + (-B*a**3*d**6*i/(a*d - b*c)**2 + 3*B*a**2*b*c*d**5*i/(a* d - b*c)**2 - 3*B*a*b**2*c**2*d**4*i/(a*d - b*c)**2 + B*a*d**4*i + B*b**3* c**3*d**3*i/(a*d - b*c)**2 + B*b*c*d**3*i)/(2*B*b*d**4*i))/(6*b**2*g**4*(a *d - b*c)**2) + B*d**3*i*log(x + (B*a**3*d**6*i/(a*d - b*c)**2 - 3*B*a**2* b*c*d**5*i/(a*d - b*c)**2 + 3*B*a*b**2*c**2*d**4*i/(a*d - b*c)**2 + B*a*d* *4*i - B*b**3*c**3*d**3*i/(a*d - b*c)**2 + B*b*c*d**3*i)/(2*B*b*d**4*i))/( 6*b**2*g**4*(a*d - b*c)**2) + (-B*a*d*i - 2*B*b*c*i - 3*B*b*d*i*x)*log(e*( a + b*x)/(c + d*x))/(6*a**3*b**2*g**4 + 18*a**2*b**3*g**4*x + 18*a*b**4*g* *4*x**2 + 6*b**5*g**4*x**3) + (-6*A*a**2*d**2*i - 6*A*a*b*c*d*i + 12*A*b** 2*c**2*i - 5*B*a**2*d**2*i - 5*B*a*b*c*d*i + 4*B*b**2*c**2*i - 6*B*b**2*d* *2*i*x**2 + x*(-18*A*a*b*d**2*i + 18*A*b**2*c*d*i - 15*B*a*b*d**2*i + 3*B* b**2*c*d*i))/(36*a**4*b**2*d*g**4 - 36*a**3*b**3*c*g**4 + x**3*(36*a*b**5* d*g**4 - 36*b**6*c*g**4) + x**2*(108*a**2*b**4*d*g**4 - 108*a*b**5*c*g**4) + x*(108*a**3*b**3*d*g**4 - 108*a**2*b**4*c*g**4))
Leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (165) = 330\).
Time = 0.07 (sec) , antiderivative size = 933, normalized size of antiderivative = 5.39 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algori thm="maxima")
Output:
-1/36*B*d*i*(6*(3*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^4*x ^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) + (5*a*b^2*c^2 - 22* a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16* a*b^2*c*d + 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^ 2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c )/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) - 1/18* B*c*i*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^ 2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6* log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^ 2*b^2*g^4*x + a^3*b*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3 *c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/6*(3*b*x + a)*A*d*i/(b^5*g ^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/3*A*c*i/(b^4 *g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)
Time = 0.27 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.56 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=-\frac {1}{36} \, {\left (\frac {6 \, {\left (2 \, B b e^{4} i - \frac {3 \, {\left (b e x + a e\right )} B d e^{3} i}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {12 \, A b e^{4} i + 4 \, B b e^{4} i - \frac {18 \, {\left (b e x + a e\right )} A d e^{3} i}{d x + c} - \frac {9 \, {\left (b e x + a e\right )} B d e^{3} i}{d x + c}}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algori thm="giac")
Output:
-1/36*(6*(2*B*b*e^4*i - 3*(b*e*x + a*e)*B*d*e^3*i/(d*x + c))*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^3*b*c*g^4/(d*x + c)^3 - (b*e*x + a*e)^3*a*d *g^4/(d*x + c)^3) + (12*A*b*e^4*i + 4*B*b*e^4*i - 18*(b*e*x + a*e)*A*d*e^3 *i/(d*x + c) - 9*(b*e*x + a*e)*B*d*e^3*i/(d*x + c))/((b*e*x + a*e)^3*b*c*g ^4/(d*x + c)^3 - (b*e*x + a*e)^3*a*d*g^4/(d*x + c)^3))*(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 = 27.82 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.09 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx=-\frac {\frac {6\,A\,a^2\,d^2\,i-12\,A\,b^2\,c^2\,i+5\,B\,a^2\,d^2\,i-4\,B\,b^2\,c^2\,i+6\,A\,a\,b\,c\,d\,i+5\,B\,a\,b\,c\,d\,i}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (6\,A\,a\,b\,d^2\,i+5\,B\,a\,b\,d^2\,i-6\,A\,b^2\,c\,d\,i-B\,b^2\,c\,d\,i\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^2\,d^2\,i\,x^2}{a\,d-b\,c}}{6\,a^3\,b^2\,g^4+18\,a^2\,b^3\,g^4\,x+18\,a\,b^4\,g^4\,x^2+6\,b^5\,g^4\,x^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{3\,b^2\,g^4}+\frac {B\,a\,d\,i}{6\,b^3\,g^4}+\frac {B\,d\,i\,x}{2\,b^2\,g^4}\right )}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B\,d^3\,i\,\mathrm {atanh}\left (\frac {6\,b^4\,c^2\,g^4-6\,a^2\,b^2\,d^2\,g^4}{6\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{3\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2} \] Input:
int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^4,x )
Output:
- ((6*A*a^2*d^2*i - 12*A*b^2*c^2*i + 5*B*a^2*d^2*i - 4*B*b^2*c^2*i + 6*A*a *b*c*d*i + 5*B*a*b*c*d*i)/(6*(a*d - b*c)) + (x*(6*A*a*b*d^2*i + 5*B*a*b*d^ 2*i - 6*A*b^2*c*d*i - B*b^2*c*d*i))/(2*(a*d - b*c)) + (B*b^2*d^2*i*x^2)/(a *d - b*c))/(6*a^3*b^2*g^4 + 6*b^5*g^4*x^3 + 18*a^2*b^3*g^4*x + 18*a*b^4*g^ 4*x^2) - (log((e*(a + b*x))/(c + d*x))*((B*c*i)/(3*b^2*g^4) + (B*a*d*i)/(6 *b^3*g^4) + (B*d*i*x)/(2*b^2*g^4)))/(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b*x^2 ) - (B*d^3*i*atanh((6*b^4*c^2*g^4 - 6*a^2*b^2*d^2*g^4)/(6*b^2*g^4*(a*d - b *c)^2) - (2*b*d*x)/(a*d - b*c)))/(3*b^2*g^4*(a*d - b*c)^2)
Time = 0.20 (sec) , antiderivative size = 894, normalized size of antiderivative = 5.17 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:
int((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x)
Output:
(i*(12*log(a + b*x)*a**4*b**2*c*d**2 - 6*log(a + b*x)*a**3*b**3*c**2*d + 3 6*log(a + b*x)*a**3*b**3*c*d**2*x - 18*log(a + b*x)*a**2*b**4*c**2*d*x + 3 6*log(a + b*x)*a**2*b**4*c*d**2*x**2 - 18*log(a + b*x)*a*b**5*c**2*d*x**2 + 12*log(a + b*x)*a*b**5*c*d**2*x**3 - 6*log(a + b*x)*b**6*c**2*d*x**3 - 1 2*log(c + d*x)*a**4*b**2*c*d**2 + 6*log(c + d*x)*a**3*b**3*c**2*d - 36*log (c + d*x)*a**3*b**3*c*d**2*x + 18*log(c + d*x)*a**2*b**4*c**2*d*x - 36*log (c + d*x)*a**2*b**4*c*d**2*x**2 + 18*log(c + d*x)*a*b**5*c**2*d*x**2 - 12* log(c + d*x)*a*b**5*c*d**2*x**3 + 6*log(c + d*x)*b**6*c**2*d*x**3 - 12*log ((a*e + b*e*x)/(c + d*x))*a**4*b**2*c*d**2 + 24*log((a*e + b*e*x)/(c + d*x ))*a**3*b**3*c**2*d + 18*log((a*e + b*e*x)/(c + d*x))*a**3*b**3*d**3*x**2 - 12*log((a*e + b*e*x)/(c + d*x))*a**2*b**4*c**3 - 36*log((a*e + b*e*x)/(c + d*x))*a**2*b**4*c*d**2*x**2 + 6*log((a*e + b*e*x)/(c + d*x))*a**2*b**4* d**3*x**3 + 18*log((a*e + b*e*x)/(c + d*x))*a*b**5*c**2*d*x**2 - 12*log((a *e + b*e*x)/(c + d*x))*a*b**5*c*d**2*x**3 + 6*log((a*e + b*e*x)/(c + d*x)) *b**6*c**2*d*x**3 - 6*a**6*d**3 - 18*a**5*b*d**3*x - 3*a**5*b*d**3 + 18*a* *4*b**2*c**2*d + 36*a**4*b**2*c*d**2*x - 2*a**4*b**2*c*d**2 - 9*a**4*b**2* d**3*x - 12*a**3*b**3*c**3 - 18*a**3*b**3*c**2*d*x + 9*a**3*b**3*c**2*d + 12*a**3*b**3*c*d**2*x - 4*a**2*b**4*c**3 - 3*a**2*b**4*c**2*d*x + 2*a**2*b **4*d**3*x**3 - 2*a*b**5*c*d**2*x**3))/(36*a**2*b**2*g**4*(a**5*d**2 - 2*a **4*b*c*d + 3*a**4*b*d**2*x + a**3*b**2*c**2 - 6*a**3*b**2*c*d*x + 3*a*...