Integrand size = 38, antiderivative size = 133 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\frac {i (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}-\frac {(b c-a d) i \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {B (b c-a d) i \operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g} \] Output:
i*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/g-(-a*d+b*c)*i*ln(-(-a*d+b*c)/d/(b *x+a))*(A-B+B*ln(e*(b*x+a)/(d*x+c)))/b^2/g+B*(-a*d+b*c)*i*polylog(2,1+(-a* d+b*c)/d/(b*x+a))/b^2/g
Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.23 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\frac {i \left ((-b B c+a B d) \log ^2(a+b x)+2 \left (A b d x+B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+(-b B c+a B d) \log (c+d x)\right )+2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )+B \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 B (b c-a d) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )}{2 b^2 g} \] Input:
Integrate[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g* x),x]
Output:
(i*((-(b*B*c) + a*B*d)*Log[a + b*x]^2 + 2*(A*b*d*x + B*d*(a + b*x)*Log[(e* (a + b*x))/(c + d*x)] + (-(b*B*c) + a*B*d)*Log[c + d*x]) + 2*(b*c - a*d)*L og[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)] + B*Log[(b*(c + d*x))/(b*c - a*d)]) + 2*B*(b*c - a*d)*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]))/(2* b^2*g)
Time = 0.67 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2960, 27, 2942, 2858, 27, 2778, 2005, 2752}
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.
| \(\displaystyle \int \frac {(c i+d i x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a g+b g x} \, dx\) |
| \(\Big \downarrow \) 2960 |
| \(\displaystyle \frac {i (b c-a d) \int \frac {A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{g (a+b x)}dx}{b}+\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i (b c-a d) \int \frac {A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x}dx}{b g}+\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2942 |
| \(\displaystyle \frac {i (b c-a d) \left (\frac {B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)}dx}{b}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{b}\right )}{b g}+\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {i (b c-a d) \left (\frac {B (b c-a d) \int \frac {b \log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}d(a+b x)}{b^2}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{b}\right )}{b g}+\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i (b c-a d) \left (\frac {B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (b c-a d+d (a+b x))}d(a+b x)}{b}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{b}\right )}{b g}+\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2778 |
| \(\displaystyle \frac {i (b c-a d) \left (-\frac {B (b c-a d) \int \frac {(a+b x) \log \left (-\frac {b c-a d}{d (a+b x)}\right )}{b c-a d+d (a+b x)}d\frac {1}{a+b x}}{b}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{b}\right )}{b g}+\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \frac {i (b c-a d) \left (-\frac {B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{d+\frac {b c-a d}{a+b x}}d\frac {1}{a+b x}}{b}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{b}\right )}{b g}+\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {i (b c-a d) \left (\frac {B \operatorname {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{b}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{b}\right )}{b g}+\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
Input:
Int[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x),x]
Output:
(i*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*g) + ((b*c - a*d)*i* (-((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A - B + B*Log[(e*(a + b*x))/(c + d* x)]))/b) + (B*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/b))/(b*g)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[1/n Subst[Int[(a + b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(-Log[-(b*c - a*d)/(d*(a + b*x))])*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/g), x] + Simp[B*n*((b*c - a*d)/g) Int[Log[-(b*c - a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && NeQ[b* c - a*d, 0] && EqQ[b*f - a*g, 0]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_)), x_Symbol] :> Sim p[(f + g*x)^(m + 1)*(h + i*x)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g* (m + 2))), x] + Simp[i*((b*c - a*d)/(b*d*(m + 2))) Int[(f + g*x)^m*(A - B *n + B*Log[e*((a + b*x)^n/(c + d*x)^n)]), x], x] /; FreeQ[{a, b, c, d, e, f , g, h, i, A, B, m, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d , 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IGtQ[m, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(416\) vs. \(2(133)=266\).
Time = 2.94 (sec) , antiderivative size = 417, normalized size of antiderivative = 3.14
| method | result | size |
| parts | \(\frac {i A \left (\frac {x d}{b}+\frac {\left (-d a +b c \right ) \ln \left (b x +a \right )}{b^{2}}\right )}{g}-\frac {i B \left (d a -b c \right )^{2} e^{2} \left (\frac {\left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right ) d^{4}}{\left (d a -b c \right ) b e}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d^{3}}{2 \left (d a -b c \right ) b^{2} e^{2}}-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{4}}{\left (d a -b c \right ) b^{2} e^{2}}\right )}{g \,d^{3}}\) | \(417\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (\frac {i \,d^{2} e A \left (\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b^{2} e^{2}}+\frac {1}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{2} e^{2}}\right )}{g}+\frac {i \,d^{2} e B \left (-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d}{b^{2} e^{2}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 b^{2} e^{2}}+\frac {\left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right ) d}{b e}\right )}{g}\right )}{d^{2}}\) | \(481\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (\frac {i \,d^{2} e A \left (\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b^{2} e^{2}}+\frac {1}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{2} e^{2}}\right )}{g}+\frac {i \,d^{2} e B \left (-\frac {\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d}{b^{2} e^{2}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 b^{2} e^{2}}+\frac {\left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right ) d}{b e}\right )}{g}\right )}{d^{2}}\) | \(481\) |
| risch | \(\text {Expression too large to display}\) | \(1428\) |
Input:
int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g),x,method=_RETURNVE RBOSE)
Output:
i*A/g*(x*d/b+(-a*d+b*c)/b^2*ln(b*x+a))-i*B/g/d^3*(a*d-b*c)^2*e^2*((1/b/e*l n((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/d-ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*( b*e/d+(a*d-b*c)*e/d/(d*x+c))/b/e/((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e))/(a *d-b*c)*d^4/b/e+1/2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(a*d-b*c)*d^3/b^2/e^ 2-(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)/(a*d-b*c)*d ^4/b^2/e^2)
\[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{b g x + a g} \,d x } \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g),x, algorith m="fricas")
Output:
integral((A*d*i*x + A*c*i + (B*d*i*x + B*c*i)*log((b*e*x + a*e)/(d*x + c)) )/(b*g*x + a*g), x)
\[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\frac {i \left (\int \frac {A c}{a + b x}\, dx + \int \frac {A d x}{a + b x}\, dx + \int \frac {B c \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a + b x}\, dx + \int \frac {B d x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a + b x}\, dx\right )}{g} \] Input:
integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g),x)
Output:
i*(Integral(A*c/(a + b*x), x) + Integral(A*d*x/(a + b*x), x) + Integral(B* c*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a + b*x), x) + Integral(B*d*x*log( a*e/(c + d*x) + b*e*x/(c + d*x))/(a + b*x), x))/g
Time = 0.10 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.81 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=A d i {\left (\frac {x}{b g} - \frac {a \log \left (b x + a\right )}{b^{2} g}\right )} + \frac {A c i \log \left (b g x + a g\right )}{b g} - \frac {B c i \log \left (d x + c\right )}{b g} + \frac {{\left (b c i - a d i\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{b^{2} g} + \frac {2 \, B b d i x \log \left (e\right ) + {\left (b c i - a d i\right )} B \log \left (b x + a\right )^{2} + 2 \, {\left (B b d i x + {\left (b c i \log \left (e\right ) - {\left (i \log \left (e\right ) - i\right )} a d\right )} B\right )} \log \left (b x + a\right ) - 2 \, {\left (B b d i x + {\left (b c i - a d i\right )} B \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{2 \, b^{2} g} \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g),x, algorith m="maxima")
Output:
A*d*i*(x/(b*g) - a*log(b*x + a)/(b^2*g)) + A*c*i*log(b*g*x + a*g)/(b*g) - B*c*i*log(d*x + c)/(b*g) + (b*c*i - a*d*i)*(log(b*x + a)*log((b*d*x + a*d) /(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B/(b^2*g) + 1/2*(2* B*b*d*i*x*log(e) + (b*c*i - a*d*i)*B*log(b*x + a)^2 + 2*(B*b*d*i*x + (b*c* i*log(e) - (i*log(e) - i)*a*d)*B)*log(b*x + a) - 2*(B*b*d*i*x + (b*c*i - a *d*i)*B*log(b*x + a))*log(d*x + c))/(b^2*g)
\[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{b g x + a g} \,d x } \] Input:
integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g),x, algorith m="giac")
Output:
integrate((d*i*x + c*i)*(B*log((b*x + a)*e/(d*x + c)) + A)/(b*g*x + a*g), x)
Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\int \frac {\left (c\,i+d\,i\,x\right )\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{a\,g+b\,g\,x} \,d x \] Input:
int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x),x)
Output:
int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x), x)
\[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a g+b g x} \, dx=\frac {i \left (\left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )}{b x +a}d x \right ) b^{3} c +\left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x}{b x +a}d x \right ) b^{3} d -\mathrm {log}\left (b x +a \right ) a^{2} d +\mathrm {log}\left (b x +a \right ) a b c +a b d x \right )}{b^{2} g} \] Input:
int((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g),x)
Output:
(i*(int(log((a*e + b*e*x)/(c + d*x))/(a + b*x),x)*b**3*c + int((log((a*e + b*e*x)/(c + d*x))*x)/(a + b*x),x)*b**3*d - log(a + b*x)*a**2*d + log(a + b*x)*a*b*c + a*b*d*x))/(b**2*g)