Integrand size = 38, antiderivative size = 125 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}+\frac {(b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}+\frac {B (b c-a d) g \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \] Output:
g*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/d/i+(-a*d+b*c)*g*ln((-a*d+b*c)/b/(d* x+c))*(A+B+B*ln(e*(b*x+a)/(d*x+c)))/d^2/i+B*(-a*d+b*c)*g*polylog(2,d*(b*x+ a)/b/(d*x+c))/d^2/i
Time = 0.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.30 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g \left (2 A b d x+2 B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 B (b c-a d) \log (c+d x)-2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^2 i} \] Input:
Integrate[((a*g + b*g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c*i + d*i* x),x]
Output:
(g*(2*A*b*d*x + 2*B*d*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] - 2*B*(b*c - a*d)*Log[c + d*x] - 2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log [c + d*x] + B*(b*c - a*d)*((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)
Time = 0.43 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2962, 2784, 2754, 2838}
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.
| \(\displaystyle \int \frac {(a g+b g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c i+d i x} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {g (b c-a d) \int \frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{i}\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle \frac {g (b c-a d) \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}\right )}{i}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {g (b c-a d) \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {B \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{d}}{d}\right )}{i}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {g (b c-a d) \left (\frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{d}-\frac {B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}\right )}{i}\) |
Input:
Int[((a*g + b*g*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c*i + d*i*x),x]
Output:
((b*c - a*d)*g*(((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*(c + d *x)*(b - (d*(a + b*x))/(c + d*x))) - (-(((A + B + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d) - (B*PolyLog[2, (d*(a + b *x))/(b*(c + d*x))])/d)/d))/i
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )/(e*(q + 1))), x] - Simp[f/(e*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(331\) vs. \(2(125)=250\).
Time = 2.96 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.66
| method | result | size |
| parts | \(\frac {g A \left (\frac {b x}{d}+\frac {\left (d a -b c \right ) \ln \left (d x +c \right )}{d^{2}}\right )}{i}-\frac {g B \left (\left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right ) b e \left (d a -b c \right )+\left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) \left (d a -b c \right )\right )}{i d}\) | \(332\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (\frac {g A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g A b}{i \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}+\frac {g B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g d B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d^{2}}\) | \(380\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (\frac {g A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g A b}{i \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{e i}+\frac {g B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g d B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d^{2}}\) | \(380\) |
| risch | \(\frac {g A b x}{i d}+\frac {g A \ln \left (d x +c \right ) a}{i d}-\frac {g A \ln \left (d x +c \right ) b c}{i \,d^{2}}-\frac {g B \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 ) a}{i d}+\frac {g B b \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 ) c}{i \,d^{2}}+\frac {g B b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a}{i d \left (\frac {e d a}{d x +c}-\frac {e b c}{d x +c}\right )}-\frac {g B \,b^{2} e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) c}{i \,d^{2} \left (\frac {e d a}{d x +c}-\frac {e b c}{d x +c}\right )}+\frac {g B e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a^{2}}{i \left (\frac {e d a}{d x +c}-\frac {e b c}{d x +c}\right ) \left (d x +c \right )}-\frac {2 g B b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a c}{i d \left (\frac {e d a}{d x +c}-\frac {e b c}{d x +c}\right ) \left (d x +c \right )}+\frac {g B \,b^{2} e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) c^{2}}{i \,d^{2} \left (\frac {e d a}{d x +c}-\frac {e b c}{d x +c}\right ) \left (d x +c \right )}-\frac {g 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 d}+\frac {g 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 ) b c}{i \,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 ) \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 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 ) \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 ) b c}{i \,d^{2}}\) | \(769\) |
Input:
int((b*g*x+a*g)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x,method=_RETURNVE RBOSE)
Output:
g*A/i*(b*x/d+(a*d-b*c)/d^2*ln(d*x+c))-g*B/i/d*((1/b/e*ln((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))*b*e*(a*d-b*c)+(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))
\[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int { \frac {{\left (b g x + a g\right )} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{d i x + c i} \,d x } \] Input:
integrate((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x, algorith m="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*i*x + c*i), x)
\[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g \left (\int \frac {A a}{c + d x}\, dx + \int \frac {A b x}{c + d x}\, dx + \int \frac {B a \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {B b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx\right )}{i} \] Input:
integrate((b*g*x+a*g)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x)
Output:
g*(Integral(A*a/(c + d*x), x) + Integral(A*b*x/(c + d*x), x) + Integral(B* a*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*x), x) + Integral(B*b*x*log( a*e/(c + d*x) + b*e*x/(c + d*x))/(c + d*x), x))/i
Time = 0.10 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.77 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=A b g {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {A a g \log \left (d i x + c i\right )}{d i} - \frac {{\left (b c g - a d g\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{2} i} + \frac {{\left (a d g \log \left (e\right ) - {\left (g \log \left (e\right ) + g\right )} b c\right )} B \log \left (d x + c\right )}{d^{2} i} - \frac {2 \, B b d g x \log \left (d x + c\right ) - 2 \, B b d g x \log \left (e\right ) - {\left (b c g - a d g\right )} B \log \left (d x + c\right )^{2} - 2 \, {\left (B b d g x + B a d g\right )} \log \left (b x + a\right )}{2 \, d^{2} i} \] Input:
integrate((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x, algorith m="maxima")
Output:
A*b*g*(x/(d*i) - c*log(d*x + c)/(d^2*i)) + A*a*g*log(d*i*x + c*i)/(d*i) - (b*c*g - a*d*g)*(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/(d^2*i) + (a*d*g*log(e) - (g*log(e) + g)*b*c )*B*log(d*x + c)/(d^2*i) - 1/2*(2*B*b*d*g*x*log(d*x + c) - 2*B*b*d*g*x*log (e) - (b*c*g - a*d*g)*B*log(d*x + c)^2 - 2*(B*b*d*g*x + B*a*d*g)*log(b*x + a))/(d^2*i)
Leaf count of result is larger than twice the leaf count of optimal. 1230 vs. \(2 (124) = 248\).
Time = 43.19 (sec) , antiderivative size = 1230, normalized size of antiderivative = 9.84 \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\text {Too large to display} \] Input:
integrate((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x, algorith m="giac")
Output:
-1/6*((B*b^5*c^4*e^4*g - 4*B*a*b^4*c^3*d*e^4*g + 6*B*a^2*b^3*c^2*d^2*e^4*g - 4*B*a^3*b^2*c*d^3*e^4*g + B*a^4*b*d^4*e^4*g - 3*(b*e*x + a*e)*B*b^4*c^4 *d*e^3*g/(d*x + c) + 12*(b*e*x + a*e)*B*a*b^3*c^3*d^2*e^3*g/(d*x + c) - 18 *(b*e*x + a*e)*B*a^2*b^2*c^2*d^3*e^3*g/(d*x + c) + 12*(b*e*x + a*e)*B*a^3* b*c*d^4*e^3*g/(d*x + c) - 3*(b*e*x + a*e)*B*a^4*d^5*e^3*g/(d*x + c))*log(( b*e*x + a*e)/(d*x + c))/(b^3*d^2*e^3*i - 3*(b*e*x + a*e)*b^2*d^3*e^2*i/(d* x + c) + 3*(b*e*x + a*e)^2*b*d^4*e*i/(d*x + c)^2 - (b*e*x + a*e)^3*d^5*i/( d*x + c)^3) + (A*b^6*c^4*e^4*g - 4*A*a*b^5*c^3*d*e^4*g + 6*A*a^2*b^4*c^2*d ^2*e^4*g - 4*A*a^3*b^3*c*d^3*e^4*g + A*a^4*b^2*d^4*e^4*g - 3*(b*e*x + a*e) *A*b^5*c^4*d*e^3*g/(d*x + c) + (b*e*x + a*e)*B*b^5*c^4*d*e^3*g/(d*x + c) + 12*(b*e*x + a*e)*A*a*b^4*c^3*d^2*e^3*g/(d*x + c) - 4*(b*e*x + a*e)*B*a*b^ 4*c^3*d^2*e^3*g/(d*x + c) - 18*(b*e*x + a*e)*A*a^2*b^3*c^2*d^3*e^3*g/(d*x + c) + 6*(b*e*x + a*e)*B*a^2*b^3*c^2*d^3*e^3*g/(d*x + c) + 12*(b*e*x + a*e )*A*a^3*b^2*c*d^4*e^3*g/(d*x + c) - 4*(b*e*x + a*e)*B*a^3*b^2*c*d^4*e^3*g/ (d*x + c) - 3*(b*e*x + a*e)*A*a^4*b*d^5*e^3*g/(d*x + c) + (b*e*x + a*e)*B* a^4*b*d^5*e^3*g/(d*x + c) - (b*e*x + a*e)^2*B*b^4*c^4*d^2*e^2*g/(d*x + c)^ 2 + 4*(b*e*x + a*e)^2*B*a*b^3*c^3*d^3*e^2*g/(d*x + c)^2 - 6*(b*e*x + a*e)^ 2*B*a^2*b^2*c^2*d^4*e^2*g/(d*x + c)^2 + 4*(b*e*x + a*e)^2*B*a^3*b*c*d^5*e^ 2*g/(d*x + c)^2 - (b*e*x + a*e)^2*B*a^4*d^6*e^2*g/(d*x + c)^2)/(b^4*d^2*e^ 3*i - 3*(b*e*x + a*e)*b^3*d^3*e^2*i/(d*x + c) + 3*(b*e*x + a*e)^2*b^2*d...
Timed out. \[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\int \frac {\left (a\,g+b\,g\,x\right )\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{c\,i+d\,i\,x} \,d x \] Input:
int(((a*g + b*g*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(c*i + d*i*x),x)
Output:
int(((a*g + b*g*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(c*i + d*i*x), x)
\[ \int \frac {(a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx=\frac {g i \left (-\left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )}{d x +c}d x \right ) a b \,d^{2}-\left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x}{d x +c}d x \right ) b^{2} d^{2}-\mathrm {log}\left (d x +c \right ) a^{2} d +\mathrm {log}\left (d x +c \right ) a b c -a b d x \right )}{d^{2}} \] Input:
int((b*g*x+a*g)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x)
Output:
(g*i*( - int(log((a*e + b*e*x)/(c + d*x))/(c + d*x),x)*a*b*d**2 - int((log ((a*e + b*e*x)/(c + d*x))*x)/(c + d*x),x)*b**2*d**2 - log(c + d*x)*a**2*d + log(c + d*x)*a*b*c - a*b*d*x))/d**2