Integrand size = 29, antiderivative size = 277 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{g}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{g}-\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g} \]
-ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/g+(A+B*ln(e*(b*x+a )/(d*x+c)))^2*ln(1-(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/g-2*B*(A+B*ln(e* (b*x+a)/(d*x+c)))*polylog(2,d*(b*x+a)/b/(d*x+c))/g+2*B*(A+B*ln(e*(b*x+a)/( d*x+c)))*polylog(2,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/g+2*B^2*polylog( 3,d*(b*x+a)/b/(d*x+c))/g-2*B^2*polylog(3,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d* x+c))/g
Leaf count is larger than twice the leaf count of optimal. \(1348\) vs. \(2(277)=554\).
Time = 0.37 (sec) , antiderivative size = 1348, normalized size of antiderivative = 4.87 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx =\text {Too large to display} \]
(-(B^2*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]^2) + A^2*Log[f + g*x] - 2*A*B*Log[a/b + x]*Log[f + g*x ] + B^2*Log[a/b + x]^2*Log[f + g*x] + 2*A*B*Log[c/d + x]*Log[f + g*x] - 2* B^2*Log[a/b + x]*Log[c/d + x]*Log[f + g*x] + B^2*Log[c/d + x]^2*Log[f + g* x] + 2*A*B*Log[(e*(a + b*x))/(c + d*x)]*Log[f + g*x] - 2*B^2*Log[a/b + x]* Log[(e*(a + b*x))/(c + d*x)]*Log[f + g*x] + 2*B^2*Log[c/d + x]*Log[(e*(a + b*x))/(c + d*x)]*Log[f + g*x] + B^2*Log[(e*(a + b*x))/(c + d*x)]^2*Log[f + g*x] + 2*A*B*Log[a/b + x]*Log[(b*(f + g*x))/(b*f - a*g)] - B^2*Log[a/b + x]^2*Log[(b*(f + g*x))/(b*f - a*g)] + 2*B^2*Log[a/b + x]*Log[(e*(a + b*x) )/(c + d*x)]*Log[(b*(f + g*x))/(b*f - a*g)] + 2*B^2*Log[a/b + x]*Log[(g*(c + d*x))/(-(d*f) + c*g)]*Log[(b*(f + g*x))/(b*f - a*g)] - B^2*Log[(g*(c + d*x))/(-(d*f) + c*g)]^2*Log[(b*(f + g*x))/(b*f - a*g)] + 2*B^2*Log[(g*(c + d*x))/(-(d*f) + c*g)]*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x)) ]*Log[(b*(f + g*x))/(b*f - a*g)] - B^2*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]^2*Log[(b*(f + g*x))/(b*f - a*g)] - 2*A*B*Log[c/d + x]*Lo g[(d*(f + g*x))/(d*f - c*g)] + 2*B^2*Log[a/b + x]*Log[c/d + x]*Log[(d*(f + g*x))/(d*f - c*g)] - B^2*Log[c/d + x]^2*Log[(d*(f + g*x))/(d*f - c*g)] - 2*B^2*Log[c/d + x]*Log[(e*(a + b*x))/(c + d*x)]*Log[(d*(f + g*x))/(d*f - c *g)] - 2*B^2*Log[a/b + x]*Log[(g*(c + d*x))/(-(d*f) + c*g)]*Log[(d*(f + g* x))/(d*f - c*g)] + B^2*Log[(g*(c + d*x))/(-(d*f) + c*g)]^2*Log[(d*(f + ...
Time = 0.70 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.25, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2954, 2804, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{f+g x} \, dx\) |
| \(\Big \downarrow \) 2954 |
| \(\displaystyle (b c-a d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{\left (b-\frac {d (a+b x)}{c+d x}\right ) \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle (b c-a d) \int \left (\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d) g \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(c g-d f) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d) g \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (b c-a d) \left (\frac {2 B \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g (b c-a d)}+\frac {\log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g (b c-a d)}-\frac {2 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g (b c-a d)}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g (b c-a d)}-\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g (b c-a d)}+\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{g (b c-a d)}\right )\) |
(b*c - a*d)*(-(((A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[1 - (d*(a + b*x ))/(b*(c + d*x))])/((b*c - a*d)*g)) + ((A + B*Log[(e*(a + b*x))/(c + d*x)] )^2*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d) *g) - (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/( b*(c + d*x))])/((b*c - a*d)*g) + (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)]) *PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d) *g) + (2*B^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/((b*c - a*d)*g) - (2 *B^2*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d)*g))
3.3.44.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / ; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d) Subst[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B , n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m ] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(817\) vs. \(2(277)=554\).
Time = 4.26 (sec) , antiderivative size = 818, normalized size of antiderivative = 2.95
| method | result | size |
| parts | \(\frac {A^{2} \ln \left (g x +f \right )}{g}+B^{2} \left (a d -c b \right ) e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \operatorname {Li}_{2}\left (\frac {d \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )-2 \,\operatorname {Li}_{3}\left (\frac {d \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{e g \left (a d -c b \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1+\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{-a e g +b e f}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \operatorname {Li}_{2}\left (-\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{-a e g +b e f}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{-a e g +b e f}\right )}{e g \left (a d -c b \right )}\right )-\frac {2 B A \left (a d -c b \right ) e \left (-\frac {d^{2} \left (c g -d f \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}\right )}{e g \left (a d -c b \right )}+\frac {d^{3} \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \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 (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{e g \left (a d -c b \right )}\right )}{d^{2}}\) | \(818\) |
| derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-d^{2} A^{2} \left (-\frac {\left (c g -d f \right ) \ln \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}{e g \left (a d -c b \right ) \left (-c g +d f \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e g \left (a d -c b \right )}\right )-d^{2} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \operatorname {Li}_{2}\left (\frac {d \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )-2 \,\operatorname {Li}_{3}\left (\frac {d \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{e g \left (a d -c b \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1+\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{-a e g +b e f}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \operatorname {Li}_{2}\left (-\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{-a e g +b e f}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{-a e g +b e f}\right )}{e g \left (a d -c b \right )}\right )-2 d^{2} A B \left (\frac {\left (c g -d f \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}\right )}{e g \left (a d -c b \right )}-\frac {d \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \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 (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{e g \left (a d -c b \right )}\right )\right )}{d^{2}}\) | \(970\) |
| default | \(-\frac {e \left (a d -c b \right ) \left (-d^{2} A^{2} \left (-\frac {\left (c g -d f \right ) \ln \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}{e g \left (a d -c b \right ) \left (-c g +d f \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e g \left (a d -c b \right )}\right )-d^{2} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \operatorname {Li}_{2}\left (\frac {d \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )-2 \,\operatorname {Li}_{3}\left (\frac {d \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{e g \left (a d -c b \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1+\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{-a e g +b e f}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \operatorname {Li}_{2}\left (-\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{-a e g +b e f}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{-a e g +b e f}\right )}{e g \left (a d -c b \right )}\right )-2 d^{2} A B \left (\frac {\left (c g -d f \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}\right )}{e g \left (a d -c b \right )}-\frac {d \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \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 (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{e g \left (a d -c b \right )}\right )\right )}{d^{2}}\) | \(970\) |
| risch | \(\text {Expression too large to display}\) | \(2149\) |
A^2*ln(g*x+f)/g+B^2*(a*d-b*c)*e*(-1/e/g/(a*d-b*c)*(ln(b*e/d+(a*d-b*c)*e/d/ (d*x+c))^2*ln(1-1/b/e*d*(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))*polylog(2,1/b/e*d*(b*e/d+(a*d-b*c)*e/d/(d*x+c)))-2*polylog( 3,1/b/e*d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))))+1/e/g/(a*d-b*c)*(ln(b*e/d+(a*d-b *c)*e/d/(d*x+c))^2*ln(1+(c*g-d*f)/(-a*e*g+b*e*f)*(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))*polylog(2,-(c*g-d*f)/(-a*e*g+b*e*f )*(b*e/d+(a*d-b*c)*e/d/(d*x+c)))-2*polylog(3,-(c*g-d*f)/(-a*e*g+b*e*f)*(b* e/d+(a*d-b*c)*e/d/(d*x+c)))))-2*B*A/d^2*(a*d-b*c)*e*(-d^2*(c*g-d*f)/e/g/(a *d-b*c)*(dilog(((c*g-d*f)*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-a*e*g+b*e*f)/(-a*e *g+b*e*f))/(c*g-d*f)+ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(((c*g-d*f)*(b*e/d+ (a*d-b*c)*e/d/(d*x+c))-a*e*g+b*e*f)/(-a*e*g+b*e*f))/(c*g-d*f))+d^3/e/g/(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))
\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{g x + f} \,d x } \]
integral((B^2*log((b*e*x + a*e)/(d*x + c))^2 + 2*A*B*log((b*e*x + a*e)/(d* x + c)) + A^2)/(g*x + f), x)
Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{g x + f} \,d x } \]
A^2*log(g*x + f)/g - integrate(-(B^2*log(b*x + a)^2 + B^2*log(e)^2 + 2*A*B *log(e) + 2*(B^2*log(e) + A*B)*log(b*x + a) - 2*(B^2*log(b*x + a) + B^2*lo g(e) + A*B)*log(d*x + c))/(g*x + f), x)
\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{g x + f} \,d x } \]
Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{f+g\,x} \,d x \]