Integrand size = 38, antiderivative size = 160 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=-\frac {A g (a+b x)}{d i^2 (c+d x)}+\frac {B g (a+b x)}{d i^2 (c+d x)}-\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d i^2 (c+d x)}-\frac {b g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i^2}-\frac {b B g \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i^2} \] Output:
-A*g*(b*x+a)/d/i^2/(d*x+c)+B*g*(b*x+a)/d/i^2/(d*x+c)-B*g*(b*x+a)*ln(e*(b*x +a)/(d*x+c))/d/i^2/(d*x+c)-b*g*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/ (d*x+c)))/d^2/i^2-b*B*g*polylog(2,d*(b*x+a)/b/(d*x+c))/d^2/i^2
Time = 0.20 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\frac {g \left (\frac {2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}+2 b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 B \left (\frac {b c-a d}{c+d x}+b \log (a+b x)-b \log (c+d x)\right )-b B \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^2 i^2} \] Input:
Integrate[((a*g + b*g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c*i + d*i* x)^2,x]
Output:
(g*((2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) + 2*b*( A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 2*B*((b*c - a*d)/(c + d *x) + b*Log[a + b*x] - b*Log[c + d*x]) - b*B*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(2*d^2*i^2)
Time = 0.38 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2962, 2793, 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.
| \(\displaystyle \int \frac {(a g+b g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(c i+d i x)^2} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {g \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{i^2}\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \frac {g \int \left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{d}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d \left (\frac {d (a+b x)}{c+d x}-b\right )}\right )d\frac {a+b x}{c+d x}}{i^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {g \left (-\frac {b \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^2}-\frac {A (a+b x)}{d (c+d x)}-\frac {b B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2}-\frac {B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d (c+d x)}+\frac {B (a+b x)}{d (c+d x)}\right )}{i^2}\) |
Input:
Int[((a*g + b*g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c*i + d*i*x)^2,x ]
Output:
(g*(-((A*(a + b*x))/(d*(c + d*x))) + (B*(a + b*x))/(d*(c + d*x)) - (B*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d*(c + d*x)) - (b*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^2 - (b*B*PolyLog [2, (d*(a + b*x))/(b*(c + d*x))])/d^2))/i^2
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer Q[r]))
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 = 2.92 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.84
| method | result | size |
| parts | \(\frac {g A \left (\frac {b \ln \left (d x +c \right )}{d^{2}}-\frac {d a -b c}{d^{2} \left (d x +c \right )}\right )}{i^{2}}-\frac {g B \left (\frac {\left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right ) \left (d a -b c \right )}{d}+\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) b e \left (d a -b c \right )}{d}\right )}{i^{2} \left (d a -b c \right ) e}\) | \(295\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (-\frac {g \,d^{2} A \left (-\frac {\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}{d}-\frac {b e \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d^{2}}\right )}{\left (d a -b c \right ) e^{2} i^{2}}-\frac {g \,d^{2} B \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}}{d}-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) b e}{d}\right )}{\left (d a -b c \right ) e^{2} i^{2}}\right )}{d^{2}}\) | \(358\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (-\frac {g \,d^{2} A \left (-\frac {\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}{d}-\frac {b e \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d^{2}}\right )}{\left (d a -b c \right ) e^{2} i^{2}}-\frac {g \,d^{2} B \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}}{d}-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) b e}{d}\right )}{\left (d a -b c \right ) e^{2} i^{2}}\right )}{d^{2}}\) | \(358\) |
| risch | \(\frac {g A b \ln \left (d x +c \right )}{i^{2} d^{2}}-\frac {g A a}{i^{2} d \left (d x +c \right )}+\frac {g A b c}{i^{2} d^{2} \left (d x +c \right )}-\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b a}{i^{2} \left (d a -b c \right ) d}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b^{2} c}{i^{2} \left (d a -b c \right ) d^{2}}-\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a^{2}}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}+\frac {2 g B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a b c}{i^{2} \left (d a -b c \right ) d \left (d x +c \right )}-\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b^{2} c^{2}}{i^{2} \left (d a -b c \right ) d^{2} \left (d x +c \right )}+\frac {g B \,a^{2}}{i^{2} \left (d a -b c \right ) \left (d x +c \right )}-\frac {2 g B a b c}{i^{2} \left (d a -b c \right ) d \left (d x +c \right )}+\frac {g B \,b^{2} c^{2}}{i^{2} \left (d a -b c \right ) d^{2} \left (d x +c \right )}+\frac {g B a b}{i^{2} \left (d a -b c \right ) d}-\frac {g B \,b^{2} c}{i^{2} \left (d a -b c \right ) d^{2}}-\frac {g B b \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) a}{i^{2} \left (d a -b c \right ) d}+\frac {g B \,b^{2} \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) c}{i^{2} \left (d a -b c \right ) d^{2}}-\frac {g B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) a}{i^{2} \left (d a -b c \right ) d}+\frac {g B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) c}{i^{2} \left (d a -b c \right ) d^{2}}\) | \(780\) |
Input:
int((b*g*x+a*g)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x,method=_RETURN VERBOSE)
Output:
g*A/i^2*(b/d^2*ln(d*x+c)-(a*d-b*c)/d^2/(d*x+c))-g*B/i^2/(a*d-b*c)/e*(((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))-(a*d-b*c)*e/d/(d *x+c)-b*e/d)*(a*d-b*c)/d+(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)*b*e*(a*d-b*c)/d)
\[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\int { \frac {{\left (b g x + a g\right )} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (d i x + c i\right )}^{2}} \,d x } \] Input:
integrate((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x, algori thm="fricas")
Output:
integral((A*b*g*x + A*a*g + (B*b*g*x + B*a*g)*log((b*e*x + a*e)/(d*x + c)) )/(d^2*i^2*x^2 + 2*c*d*i^2*x + c^2*i^2), x)
Timed out. \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\text {Timed out} \] Input:
integrate((b*g*x+a*g)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)**2,x)
Output:
Timed out
\[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\int { \frac {{\left (b g x + a g\right )} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (d i x + c i\right )}^{2}} \,d x } \] Input:
integrate((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x, algori thm="maxima")
Output:
-1/2*B*b*g*(((d*x + c)*log(d*x + c)^2 + 2*c*log(d*x + c))/(d^3*i^2*x + c*d ^2*i^2) - 2*integrate((d*x*log(b*x + a) + d*x*log(e) + c)/(d^3*i^2*x^2 + 2 *c*d^2*i^2*x + c^2*d*i^2), x)) + A*b*g*(c/(d^3*i^2*x + c*d^2*i^2) + log(d* x + c)/(d^2*i^2)) - B*a*g*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^2*i^2*x + c*d*i^2) - 1/(d^2*i^2*x + c*d*i^2) - b*log(b*x + a)/((b*c*d - a*d^2)*i^ 2) + b*log(d*x + c)/((b*c*d - a*d^2)*i^2)) - A*a*g/(d^2*i^2*x + c*d*i^2)
Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (159) = 318\).
Time = 46.76 (sec) , antiderivative size = 895, normalized size of antiderivative = 5.59 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx =\text {Too large to display} \] Input:
integrate((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x, algori thm="giac")
Output:
-1/2*((B*b^4*c^3*e^3*g - 3*B*a*b^3*c^2*d*e^3*g + 3*B*a^2*b^2*c*d^2*e^3*g - B*a^3*b*d^3*e^3*g - 2*(b*e*x + a*e)*B*b^3*c^3*d*e^2*g/(d*x + c) + 6*(b*e* x + a*e)*B*a*b^2*c^2*d^2*e^2*g/(d*x + c) - 6*(b*e*x + a*e)*B*a^2*b*c*d^3*e ^2*g/(d*x + c) + 2*(b*e*x + a*e)*B*a^3*d^4*e^2*g/(d*x + c))*log((b*e*x + a *e)/(d*x + c))/(b^2*d^2*e^2*i^2 - 2*(b*e*x + a*e)*b*d^3*e*i^2/(d*x + c) + (b*e*x + a*e)^2*d^4*i^2/(d*x + c)^2) + (A*b^4*c^3*e^3*g + B*b^4*c^3*e^3*g - 3*A*a*b^3*c^2*d*e^3*g - 3*B*a*b^3*c^2*d*e^3*g + 3*A*a^2*b^2*c*d^2*e^3*g + 3*B*a^2*b^2*c*d^2*e^3*g - A*a^3*b*d^3*e^3*g - B*a^3*b*d^3*e^3*g - 2*(b*e *x + a*e)*A*b^3*c^3*d*e^2*g/(d*x + c) - (b*e*x + a*e)*B*b^3*c^3*d*e^2*g/(d *x + c) + 6*(b*e*x + a*e)*A*a*b^2*c^2*d^2*e^2*g/(d*x + c) + 3*(b*e*x + a*e )*B*a*b^2*c^2*d^2*e^2*g/(d*x + c) - 6*(b*e*x + a*e)*A*a^2*b*c*d^3*e^2*g/(d *x + c) - 3*(b*e*x + a*e)*B*a^2*b*c*d^3*e^2*g/(d*x + c) + 2*(b*e*x + a*e)* A*a^3*d^4*e^2*g/(d*x + c) + (b*e*x + a*e)*B*a^3*d^4*e^2*g/(d*x + c))/(b^2* d^2*e^2*i^2 - 2*(b*e*x + a*e)*b*d^3*e*i^2/(d*x + c) + (b*e*x + a*e)^2*d^4* i^2/(d*x + c)^2) + (B*b^3*c^3*e*g - 3*B*a*b^2*c^2*d*e*g + 3*B*a^2*b*c*d^2* e*g - B*a^3*d^3*e*g)*log(-b*e + (b*e*x + a*e)*d/(d*x + c))/(b*d^2*i^2) - ( B*b^3*c^3*e*g - 3*B*a*b^2*c^2*d*e*g + 3*B*a^2*b*c*d^2*e*g - B*a^3*d^3*e*g) *log((b*e*x + a*e)/(d*x + c))/(b*d^2*i^2))*(b*c/((b*c*e - a*d*e)*(b*c - a* d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2
Timed out. \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\int \frac {\left (a\,g+b\,g\,x\right )\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (c\,i+d\,i\,x\right )}^2} \,d x \] Input:
int(((a*g + b*g*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(c*i + d*i*x)^2,x )
Output:
int(((a*g + b*g*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(c*i + d*i*x)^2, x)
\[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\frac {g \left (-\left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a \,b^{2} c^{2} d^{3}-\left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) a \,b^{2} c \,d^{4} x +\left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) b^{3} c^{3} d^{2}+\left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) b^{3} c^{2} d^{3} x +\mathrm {log}\left (b x +a \right ) a^{2} b c \,d^{2}+\mathrm {log}\left (b x +a \right ) a^{2} b \,d^{3} x -\mathrm {log}\left (d x +c \right ) a^{2} b \,c^{2} d -\mathrm {log}\left (d x +c \right ) a^{2} b c \,d^{2} x -\mathrm {log}\left (d x +c \right ) a^{2} b c \,d^{2}-\mathrm {log}\left (d x +c \right ) a^{2} b \,d^{3} x +\mathrm {log}\left (d x +c \right ) a \,b^{2} c^{3}+\mathrm {log}\left (d x +c \right ) a \,b^{2} c^{2} d x -\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b \,d^{3} x +\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} c \,d^{2} x -a^{3} d^{3} x +2 a^{2} b c \,d^{2} x +a^{2} b \,d^{3} x -a \,b^{2} c^{2} d x -a \,b^{2} c \,d^{2} x \right )}{c \,d^{2} \left (a \,d^{2} x -b c d x +a c d -b \,c^{2}\right )} \] Input:
int((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^2,x)
Output:
(g*( - int((log((a*e + b*e*x)/(c + d*x))*x)/(c**2 + 2*c*d*x + d**2*x**2),x )*a*b**2*c**2*d**3 - int((log((a*e + b*e*x)/(c + d*x))*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a*b**2*c*d**4*x + int((log((a*e + b*e*x)/(c + d*x))*x)/(c* *2 + 2*c*d*x + d**2*x**2),x)*b**3*c**3*d**2 + int((log((a*e + b*e*x)/(c + d*x))*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*b**3*c**2*d**3*x + log(a + b*x)*a **2*b*c*d**2 + log(a + b*x)*a**2*b*d**3*x - log(c + d*x)*a**2*b*c**2*d - l og(c + d*x)*a**2*b*c*d**2*x - log(c + d*x)*a**2*b*c*d**2 - log(c + d*x)*a* *2*b*d**3*x + log(c + d*x)*a*b**2*c**3 + log(c + d*x)*a*b**2*c**2*d*x - lo g((a*e + b*e*x)/(c + d*x))*a**2*b*d**3*x + log((a*e + b*e*x)/(c + d*x))*a* b**2*c*d**2*x - a**3*d**3*x + 2*a**2*b*c*d**2*x + a**2*b*d**3*x - a*b**2*c **2*d*x - a*b**2*c*d**2*x))/(c*d**2*(a*c*d + a*d**2*x - b*c**2 - b*c*d*x))