3.3.14 \(\int \frac {(A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2}{a g+b g x} \, dx\) [214]

Optimal. Leaf size=132 \[ -\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{b g}-\frac {4 B \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g}+\frac {8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \]

[Out]

-ln((a*d-b*c)/d/(b*x+a))*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2/b/g-4*B*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))*polylog(2,b
*(d*x+c)/d/(b*x+a))/b/g+8*B^2*polylog(3,b*(d*x+c)/d/(b*x+a))/b/g

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2552, 2354, 2421, 6724} \begin {gather*} -\frac {4 B \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b g}+\frac {8 B^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{b g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/(b*g)) - (4*B*(A + B*Log[(e*(
c + d*x)^2)/(a + b*x)^2])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b*g) + (8*B^2*PolyLog[3, (b*(c + d*x))/(d*
(a + b*x))])/(b*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 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 2552

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)^(m + 1)*(g/d)^m, Subst[Int[(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] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

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*} \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{a g+b g x} \, dx &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B) \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{e (c+d x)^2} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B) \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(c+d x)^2} \, dx}{b e g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(2 B) \int \frac {2 (b c-a d) e \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b e g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 B (b c-a d)) \int \frac {\left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 B (b c-a d)) \int \left (\frac {b \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(b c-a d) (a+b x)}+\frac {d \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{(b c-a d) (c+d x)}\right ) \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 B) \int \frac {\left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{a+b x} \, dx}{g}-\frac {(4 B d) \int \frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 B) \int \left (\frac {A \log (a g+b g x)}{-a-b x}+\frac {B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log (a g+b g x)}{-a-b x}\right ) \, dx}{g}-\frac {(4 B d) \int \left (\frac {A \log (a g+b g x)}{c+d x}+\frac {B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log (a g+b g x)}{c+d x}\right ) \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {(4 A B) \int \frac {\log (a g+b g x)}{-a-b x} \, dx}{g}-\frac {\left (4 B^2\right ) \int \frac {\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log (a g+b g x)}{-a-b x} \, dx}{g}-\frac {(4 A B d) \int \frac {\log (a g+b g x)}{c+d x} \, dx}{b g}-\frac {\left (4 B^2 d\right ) \int \frac {\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=-\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}+\frac {2 B^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}+(4 A B) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right )}{a g+b g x} \, dx+\frac {(4 A B) \text {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,a g+b g x\right )}{b g^2}+\frac {\left (4 B^2\right ) \int \frac {\log ^2(a g+b g x)}{a+b x} \, dx}{g}-\frac {\left (4 B^2 d\right ) \int \frac {\log \left (\frac {1}{(a+b x)^2}\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}-\frac {\left (4 B^2 d\right ) \int \frac {\log \left ((c+d x)^2\right ) \log (a g+b g x)}{c+d x} \, dx}{b g}-\frac {\left (4 B^2 d\right ) \int \frac {\log ^2(a g+b g x)}{c+d x} \, dx}{b g}-\frac {\left (4 B^2 d \left (-\log \left (\frac {1}{(a+b x)^2}\right )-\log \left ((c+d x)^2\right )+\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )\right ) \int \frac {\log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=-\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 B^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}+\left (8 B^2\right ) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right ) \log (a g+b g x)}{a g+b g x} \, dx+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {g \log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g^2}+\frac {(4 A B) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {(4 A B) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}-\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{\left (-\frac {b c-a d}{d}+\frac {b x}{d}\right )^2}\right ) \log \left (-\frac {b c g-a d g}{d}+\frac {b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (x^2\right ) \log \left (\frac {-b c g+a d g}{d}+\frac {b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b g}+\left (4 B^2 \left (-\log \left (\frac {1}{(a+b x)^2}\right )-\log \left ((c+d x)^2\right )+\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )\right ) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right )}{a g+b g x} \, dx\\ &=\frac {2 A B \log ^2(g (a+b x))}{b g}-\frac {4 B^2 \log \left (\frac {1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 B^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {B^2 \text {Subst}\left (\int \frac {\log ^2\left (x^2\right )}{\frac {-b c g+a d g}{d}+\frac {b g x}{d}} \, dx,x,c+d x\right )}{d}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {b c g-a d g}{d}+\frac {b g x}{d}\right )}{-\frac {b c g-a d g}{d}+\frac {b g x}{d}} \, dx,x,c+d x\right )}{d}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {b g \left (\frac {b c g-a d g}{b g}+\frac {d x}{b g}\right )}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {1}{\left (-\frac {b c-a d}{d}+\frac {b x}{d}\right )^2}\right )}{-\frac {b c-a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{d g}+\frac {\left (4 B^2 \left (-\log \left (\frac {1}{(a+b x)^2}\right )-\log \left ((c+d x)^2\right )+\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=\frac {2 A B \log ^2(g (a+b x))}{b g}-\frac {4 B^2 \log \left (\frac {1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 B^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (g (a+b x))\right )}{b g}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{x^2}\right ) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (x^2\right ) \log \left (1+\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {b c g-a d g}{b g}+\frac {d x}{b g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 \log ^2\left (\frac {1}{(a+b x)^2}\right ) \log (c+d x)}{b g}-\frac {4 B^2 \log \left (\frac {1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac {4 B^2 \log ^2(g (a+b x)) \log (c+d x)}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 B^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \log \left ((c+d x)^2\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {8 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {\left (4 B^2 d\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {b c g-a d g}{b g}+\frac {d x}{b g}} \, dx,x,a g+b g x\right )}{b^2 g^2}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b g}+\frac {\left (B^2 d\right ) \text {Subst}\left (\int \frac {\log ^2\left (\frac {1}{x^2}\right )}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b^2 g}\\ &=\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 \log ^2\left (\frac {1}{(a+b x)^2}\right ) \log (c+d x)}{b g}-\frac {4 B^2 \log \left (\frac {1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac {4 B^2 \log ^2(g (a+b x)) \log (c+d x)}{b g}+\frac {B^2 \log ^2\left (\frac {1}{(a+b x)^2}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 B^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \log \left ((c+d x)^2\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {8 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {\left (4 B^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{x^2}\right ) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 \log ^2\left (\frac {1}{(a+b x)^2}\right ) \log (c+d x)}{b g}-\frac {4 B^2 \log \left (\frac {1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac {4 B^2 \log ^2(g (a+b x)) \log (c+d x)}{b g}+\frac {B^2 \log ^2\left (\frac {1}{(a+b x)^2}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 B^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {4 B^2 \log \left (\frac {1}{(a+b x)^2}\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \log \left ((c+d x)^2\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {8 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b g}-\frac {\left (8 B^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=\frac {2 A B \log ^2(g (a+b x))}{b g}+\frac {4 B^2 \log ^3(g (a+b x))}{3 b g}-\frac {B^2 \log ^2\left (\frac {1}{(a+b x)^2}\right ) \log (c+d x)}{b g}-\frac {4 B^2 \log \left (\frac {1}{(a+b x)^2}\right ) \log (g (a+b x)) \log (c+d x)}{b g}-\frac {4 B^2 \log ^2(g (a+b x)) \log (c+d x)}{b g}+\frac {B^2 \log ^2\left (\frac {1}{(a+b x)^2}\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {B^2 \log (g (a+b x)) \log ^2\left ((c+d x)^2\right )}{b g}-\frac {4 A B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \log (a g+b g x)}{b g}-\frac {4 B^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b g}+\frac {2 B^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \log ^2(a g+b g x)}{b g}-\frac {4 A B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {4 B^2 \log \left (\frac {1}{(a+b x)^2}\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \left (\log \left (\frac {1}{(a+b x)^2}\right )+\log \left ((c+d x)^2\right )-\log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {4 B^2 \log \left ((c+d x)^2\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}-\frac {8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 257, normalized size = 1.95 \begin {gather*} \frac {-2 A B \log ^2\left (\frac {a}{b}+x\right )+A^2 \log (a+b x)+4 A B \log \left (\frac {a}{b}+x\right ) \log (a+b x)-4 A B \log \left (\frac {c}{d}+x\right ) \log (a+b x)+4 A B \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+2 A B \log (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )-B^2 \log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+4 A B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-4 B^2 \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )+8 B^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*B*Log[a/b + x]^2 + A^2*Log[a + b*x] + 4*A*B*Log[a/b + x]*Log[a + b*x] - 4*A*B*Log[c/d + x]*Log[a + b*x]
+ 4*A*B*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*A*B*Log[a + b*x]*Log[(e*(c + d*x)^2)/(a + b*x)^2] -
B^2*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(e*(c + d*x)^2)/(a + b*x)^2]^2 + 4*A*B*PolyLog[2, (b*(c + d*x))/(b*
c - a*d)] - 4*B^2*Log[(e*(c + d*x)^2)/(a + b*x)^2]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] + 8*B^2*PolyLog[3,
(b*(c + d*x))/(d*(a + b*x))])/(b*g)

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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )\right )^{2}}{b g x +a g}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*B^2*log(b*x + a)*log(d*x + c)^2/(b*g) + A^2*log(b*g*x + a*g)/(b*g) - integrate(-(2*A*B*b*c + B^2*b*c + 4*(B^
2*b*d*x + B^2*b*c)*log(b*x + a)^2 + (2*A*B*b*d + B^2*b*d)*x - 4*(A*B*b*c + B^2*b*c + (A*B*b*d + B^2*b*d)*x)*lo
g(b*x + a) + 4*(A*B*b*c + B^2*b*c + (A*B*b*d + B^2*b*d)*x - 2*(2*B^2*b*d*x + (b*c + a*d)*B^2)*log(b*x + a))*lo
g(d*x + c))/(b^2*d*g*x^2 + a*b*c*g + (b^2*c*g + a*b*d*g)*x), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {A^{2}}{a + b x}\, dx + \int \frac {B^{2} \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}^{2}}{a + b x}\, dx + \int \frac {2 A B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}{a + b x}\, dx}{g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\right )}^2}{a\,g+b\,g\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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