\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{f+g x} \, dx\) [244]

Optimal result

Integrand size = 29, antiderivative size = 277 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{g}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{g}-\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g} \]

[Out]

-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*pol
ylog(3,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/g

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2554, 2404, 2354, 2421, 6724} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\frac {2 B \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}+\frac {\log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g}-\frac {2 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g}-\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{g} \]

[In]

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x),x]

[Out]

-((Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/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))])/g - (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)
])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/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))])/g + (2*B^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/g - (2*B^2*PolyLo
g[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/g

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[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]

Rule 2404

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]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2554

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.), x_Symbol] :> Dist[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] && IGt
Q[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = (b c-a d) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{(b-d x) (b f-a g-(d f-c g) x)} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = (b c-a d) \text {Subst}\left (\int \left (\frac {d (A+B \log (e x))^2}{(b c-a d) g (b-d x)}+\frac {(-d f+c g) (A+B \log (e x))^2}{(b c-a d) g (b f-a g-(d f-c g) x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {d \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g}+\frac {((-b c+a d) (d f-c g)) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{b f-a g+(-d f+c g) x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g} \\ & = -\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) \text {Subst}\left (\int \frac {(A+B \log (e x)) \log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g}-\frac {(2 B) \text {Subst}\left (\int \frac {(A+B \log (e x)) \log \left (1+\frac {(-d f+c g) x}{b f-a g}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g} \\ & = -\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 ) \text {Li}_2\left (\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 ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {\left (2 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g}-\frac {\left (2 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {(-d f+c g) x}{b f-a g}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{g} \\ & = -\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 ) \text {Li}_2\left (\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 ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {2 B^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{g}-\frac {2 B^2 \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g} \\ \end{align*}

Mathematica [B] (verified)

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=\frac {-B^2 \log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )+A^2 \log (f+g x)-2 A B \log \left (\frac {a}{b}+x\right ) \log (f+g x)+B^2 \log ^2\left (\frac {a}{b}+x\right ) \log (f+g x)+2 A B \log \left (\frac {c}{d}+x\right ) \log (f+g x)-2 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log (f+g x)+B^2 \log ^2\left (\frac {c}{d}+x\right ) \log (f+g x)+2 A B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (f+g x)-2 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (f+g x)+2 B^2 \log \left (\frac {c}{d}+x\right ) \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (f+g x)+B^2 \log ^2\left (\frac {e (a+b x)}{c+d x}\right ) \log (f+g x)+2 A B \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-B^2 \log ^2\left (\frac {a}{b}+x\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+2 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+2 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-B^2 \log ^2\left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+2 B^2 \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-B^2 \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-2 A B \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )+2 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-B^2 \log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-2 B^2 \log \left (\frac {c}{d}+x\right ) \log \left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-2 B^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )+B^2 \log ^2\left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-2 B^2 \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )+B^2 \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {(-b c+a d) (f+g x)}{(d f-c g) (a+b x)}\right )+2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )+B \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {g (a+b x)}{-b f+a g}\right )-2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )+B \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {g (c+d x)}{-d f+c g}\right )-2 B^2 \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )+2 B^2 \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )+2 B^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )-2 B^2 \operatorname {PolyLog}\left (3,\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{g} \]

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(f + g*x),x]

[Out]

(-(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]*Log[(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*L
og[(g*(c + d*x))/(-(d*f) + c*g)]^2*Log[(d*(f + g*x))/(d*f - c*g)] - 2*B^2*Log[(g*(c + d*x))/(-(d*f) + c*g)]*Lo
g[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]*Log[(d*(f + g*x))/(d*f - c*g)] + B^2*Log[((b*f - a*g)*(c +
d*x))/((d*f - c*g)*(a + b*x))]^2*Log[((-(b*c) + a*d)*(f + g*x))/((d*f - c*g)*(a + b*x))] + 2*B*(A + B*Log[(e*(
a + b*x))/(c + d*x)] + B*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])*PolyLog[2, (g*(a + b*x))/(-(b*f
) + a*g)] - 2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)] + B*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])*
PolyLog[2, (g*(c + d*x))/(-(d*f) + c*g)] - 2*B^2*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]*PolyLog[
2, (b*(c + d*x))/(d*(a + b*x))] + 2*B^2*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]*PolyLog[2, ((b*f
- a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))] + 2*B^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))] - 2*B^2*PolyLog[3,
((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/g

Maple [B] (verified)

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	\(\text {Expression too large to display}\)	\(2149\)

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(g*x+f),x,method=_RETURNVERBOSE)

[Out]

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*polylo
g(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))

Fricas [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{g x + f} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(g*x+f),x, algorithm="fricas")

[Out]

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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\text {Timed out} \]

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(g*x+f),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{g x + f} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(g*x+f),x, algorithm="maxima")

[Out]

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*log(e) + A*B)*log(d*x + c))/(g*x + f), x)

Giac [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{g x + f} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(g*x+f),x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)*e/(d*x + c)) + A)^2/(g*x + f), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{f+g x} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{f+g\,x} \,d x \]

[In]

int((A + B*log((e*(a + b*x))/(c + d*x)))^2/(f + g*x),x)

[Out]

int((A + B*log((e*(a + b*x))/(c + d*x)))^2/(f + g*x), x)