3.1.59 \(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{a g+b g x} \, dx\) [59]

Optimal. Leaf size=286 \[ \frac {2 B (b c-a d) i \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {d i (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac {(b c-a d) i \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g}+\frac {2 B^2 (b c-a d) i \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 g}+\frac {2 B (b c-a d) i \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g}+\frac {2 B^2 (b c-a d) i \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g} \]

[Out]

2*B*(-a*d+b*c)*i*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^2/g+d*i*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x
+c)))^2/b^2/g-(-a*d+b*c)*i*(A+B*ln(e*(b*x+a)/(d*x+c)))^2*ln(1-b*(d*x+c)/d/(b*x+a))/b^2/g+2*B^2*(-a*d+b*c)*i*po
lylog(2,d*(b*x+a)/b/(d*x+c))/b^2/g+2*B*(-a*d+b*c)*i*(A+B*ln(e*(b*x+a)/(d*x+c)))*polylog(2,b*(d*x+c)/d/(b*x+a))
/b^2/g+2*B^2*(-a*d+b*c)*i*polylog(3,b*(d*x+c)/d/(b*x+a))/b^2/g

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2562, 2389, 2379, 2421, 6724, 2355, 2354, 2438} \begin {gather*} \frac {2 B i (b c-a d) \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g}+\frac {2 B^2 i (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 g}+\frac {2 B^2 i (b c-a d) \text {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g}+\frac {2 B i (b c-a d) \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 )}{b^2 g}+\frac {d i (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g}-\frac {i (b c-a d) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*(b*c - a*d)*i*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g) + (d*i*(a + b*
x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^2*g) - ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log
[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^2*g) + (2*B^2*(b*c - a*d)*i*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^
2*g) + (2*B*(b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^2*g
) + (2*B^2*(b*c - a*d)*i*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^2*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 2355

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

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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 2438

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]

Rule 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(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] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

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 {(59 c+59 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a g+b g x} \, dx &=\int \left (\frac {59 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g (a+b x)}\right ) \, dx\\ &=\frac {(59 d) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{b g}+\frac {(59 (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{b g}\\ &=\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac {(118 B d) \int \frac {(b c-a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b g}-\frac {(118 B (b c-a d)) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{b^2 g}\\ &=\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac {(118 B d (b c-a d)) \int \frac {x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b g}-\frac {(118 B (b c-a d)) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{b^2 e g}\\ &=\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac {(118 B d (b c-a d)) \int \left (-\frac {a \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)}+\frac {c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{b g}-\frac {(118 B (b c-a d)) \int \frac {(b c-a d) e \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 e g}\\ &=\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac {(118 a B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b g}-\frac {(118 B c d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b g}-\frac {\left (118 B (b c-a d)^2\right ) \int \frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}\\ &=\frac {118 a B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac {118 B c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}+\frac {\left (118 B^2 c\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b g}-\frac {\left (118 a B^2 d\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^2 g}-\frac {\left (118 B (b c-a d)^2\right ) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{b^2 g}\\ &=\frac {118 a B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac {118 B c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}-\frac {\left (118 A B (b c-a d)^2\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^2 g}-\frac {\left (118 B^2 (b c-a d)^2\right ) \int \frac {\log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}+\frac {\left (118 B^2 c\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b e g}-\frac {\left (118 a B^2 d\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 e g}\\ &=-\frac {59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^2 g}+\frac {118 a B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac {118 B c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}+\frac {\left (59 B^2 (b c-a d)\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b g}-\frac {\left (118 A B (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^3 g}+\frac {\left (118 B^2 c\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b e g}-\frac {\left (118 a B^2 d\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 e g}\\ &=-\frac {59 B^2 (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^2 g}-\frac {59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^2 g}+\frac {118 a B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}-\frac {118 B c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}+\frac {\left (118 B^2 c\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{g}-\frac {\left (118 a B^2 d\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b g}-\frac {\left (118 B^2 c d\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b g}+\frac {\left (118 a B^2 d^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 g}-\frac {(118 A B (b c-a d)) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}+\frac {(118 A B d (b c-a d)) \text {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b^3 g}+\frac {\left (118 B^2 (b c-a d)^2\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}\\ &=-\frac {59 A B (b c-a d) \log ^2(a+b x)}{b^2 g}-\frac {59 B^2 (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^2 g}-\frac {59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^2 g}+\frac {118 a B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac {118 B^2 c \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac {118 B c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}+\frac {118 a B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {118 A B (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {118 B^2 (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}-\frac {\left (118 B^2 c\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b g}-\frac {\left (118 a B^2 d\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac {\left (118 a B^2 d\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g}-\frac {\left (118 B^2 c d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b g}-\frac {(118 A B (b c-a d)) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac {\left (118 B^2 (b c-a d)^2\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b^2 g}\\ &=-\frac {59 a B^2 d \log ^2(a+b x)}{b^2 g}-\frac {59 A B (b c-a d) \log ^2(a+b x)}{b^2 g}-\frac {59 B^2 (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^2 g}-\frac {59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^2 g}+\frac {118 a B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac {118 B^2 c \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac {118 B c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}-\frac {59 B^2 c \log ^2(c+d x)}{b g}+\frac {118 a B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {118 A B (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {118 A B (b c-a d) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac {118 B^2 (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}+\frac {118 B^2 (b c-a d) \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}-\frac {\left (118 B^2 c\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b g}-\frac {\left (118 a B^2 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}\\ &=-\frac {59 a B^2 d \log ^2(a+b x)}{b^2 g}-\frac {59 A B (b c-a d) \log ^2(a+b x)}{b^2 g}-\frac {59 B^2 (b c-a d) \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^2 g}-\frac {59 B^2 (b c-a d) \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{b^2 g}+\frac {118 a B d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac {59 d x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g}+\frac {59 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g}+\frac {118 B^2 c \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b g}-\frac {118 B c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b g}-\frac {59 B^2 c \log ^2(c+d x)}{b g}+\frac {118 a B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {118 A B (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {118 a B^2 d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac {118 A B (b c-a d) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac {118 B^2 c \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {118 B^2 (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}+\frac {118 B^2 (b c-a d) \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{b^2 g}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(987\) vs. \(2(286)=572\).
time = 0.73, size = 987, normalized size = 3.45 \begin {gather*} \frac {i \left (3 A^2 b d x+3 A^2 (b c-a d) \log (a+b x)-3 A B \left (a d \log ^2\left (\frac {a}{b}+x\right )-2 a d \log \left (\frac {a}{b}+x\right ) (1+\log (a+b x))+2 \left (-b c+a d+\log \left (\frac {c}{d}+x\right ) \left (b c+a d \log (a+b x)-a d \log \left (\frac {d (a+b x)}{-b c+a d}\right )\right )+(-b d x+a d \log (a+b x)) \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 a d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )+3 A b B c \left (\log ^2\left (\frac {a}{b}+x\right )-2 \log (a+b x) \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 \left (\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )-B^2 \left (a d \log ^3\left (\frac {a}{b}+x\right )-3 d \left (2 b x-2 (a+b x) \log \left (\frac {a}{b}+x\right )+(a+b x) \log ^2\left (\frac {a}{b}+x\right )\right )-3 b \left (2 d x-2 (c+d x) \log \left (\frac {c}{d}+x\right )+(c+d x) \log ^2\left (\frac {c}{d}+x\right )\right )-3 d (b x-a \log (a+b x)) \left (-\log \left (\frac {a}{b}+x\right )+\log \left (\frac {c}{d}+x\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+6 \left (a d+2 b d x-b d x \log \left (\frac {c}{d}+x\right )-b c \log (c+d x)+\log \left (\frac {a}{b}+x\right ) \left (-d (a+b x)+d (a+b x) \log \left (\frac {c}{d}+x\right )+(b c-a d) \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+(b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )-3 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \left (-2 b c+2 a d-2 d (a+b x) \log \left (\frac {a}{b}+x\right )+a d \log ^2\left (\frac {a}{b}+x\right )+2 \log \left (\frac {c}{d}+x\right ) \left (b (c+d x)-a d \log \left (\frac {d (a+b x)}{-b c+a d}\right )\right )-2 a d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )-3 a d \left (\log ^2\left (\frac {a}{b}+x\right ) \left (\log \left (\frac {c}{d}+x\right )-\log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \log \left (\frac {a}{b}+x\right ) \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )+2 \text {Li}_3\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+3 a d \left (\log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+2 \log \left (\frac {c}{d}+x\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )-3 b B^2 c \left (\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )-2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )-2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )\right )\right )}{3 b^2 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(i*(3*A^2*b*d*x + 3*A^2*(b*c - a*d)*Log[a + b*x] - 3*A*B*(a*d*Log[a/b + x]^2 - 2*a*d*Log[a/b + x]*(1 + Log[a +
b*x]) + 2*(-(b*c) + a*d + Log[c/d + x]*(b*c + a*d*Log[a + b*x] - a*d*Log[(d*(a + b*x))/(-(b*c) + a*d)]) + (-(
b*d*x) + a*d*Log[a + b*x])*Log[(e*(a + b*x))/(c + d*x)]) - 2*a*d*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 3*A*
b*B*c*(Log[a/b + x]^2 - 2*Log[a + b*x]*(Log[a/b + x] - Log[c/d + x] - Log[(e*(a + b*x))/(c + d*x)]) - 2*(Log[c
/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) - B^2*(a*d*Log[a/b + x]^3
- 3*d*(2*b*x - 2*(a + b*x)*Log[a/b + x] + (a + b*x)*Log[a/b + x]^2) - 3*b*(2*d*x - 2*(c + d*x)*Log[c/d + x] +
(c + d*x)*Log[c/d + x]^2) - 3*d*(b*x - a*Log[a + b*x])*(-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c +
d*x)])^2 + 6*(a*d + 2*b*d*x - b*d*x*Log[c/d + x] - b*c*Log[c + d*x] + Log[a/b + x]*(-(d*(a + b*x)) + d*(a + b*
x)*Log[c/d + x] + (b*c - a*d)*Log[(b*(c + d*x))/(b*c - a*d)]) + (b*c - a*d)*PolyLog[2, (d*(a + b*x))/(-(b*c) +
a*d)]) - 3*(Log[a/b + x] - Log[c/d + x] - Log[(e*(a + b*x))/(c + d*x)])*(-2*b*c + 2*a*d - 2*d*(a + b*x)*Log[a
/b + x] + a*d*Log[a/b + x]^2 + 2*Log[c/d + x]*(b*(c + d*x) - a*d*Log[(d*(a + b*x))/(-(b*c) + a*d)]) - 2*a*d*Po
lyLog[2, (b*(c + d*x))/(b*c - a*d)]) - 3*a*d*(Log[a/b + x]^2*(Log[c/d + x] - Log[(b*(c + d*x))/(b*c - a*d)]) -
2*Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 2*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)]) + 3*a*d
*(Log[c/d + x]^2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*Log[c/d + x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 2*
PolyLog[3, (b*(c + d*x))/(b*c - a*d)])) - 3*b*B^2*c*(Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(e*(a + b*x))/(c +
d*x)]^2 - 2*Log[(e*(a + b*x))/(c + d*x)]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] - 2*PolyLog[3, (b*(c + d*x))/
(d*(a + b*x))])))/(3*b^2*g)

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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{b g x +a g}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

I*A^2*d*(x/(b*g) - a*log(b*x + a)/(b^2*g)) + I*A^2*c*log(b*g*x + a*g)/(b*g) - (-I*B^2*b*d*x + (-I*b*c + I*a*d)
*B^2*log(b*x + a))*log(d*x + c)^2/(b^2*g) - integrate((-2*I*A*B*b^2*c^2 - I*B^2*b^2*c^2 + (-2*I*A*B*b^2*d^2 -
I*B^2*b^2*d^2)*x^2 + (-I*B^2*b^2*d^2*x^2 - 2*I*B^2*b^2*c*d*x - I*B^2*b^2*c^2)*log(b*x + a)^2 - 2*(2*I*A*B*b^2*
c*d + I*B^2*b^2*c*d)*x - 2*(I*A*B*b^2*c^2 + I*B^2*b^2*c^2 + (I*A*B*b^2*d^2 + I*B^2*b^2*d^2)*x^2 + 2*(I*A*B*b^2
*c*d + I*B^2*b^2*c*d)*x)*log(b*x + a) - 2*(-I*A*B*b^2*c^2 - I*B^2*b^2*c^2 + (-I*A*B*b^2*d^2 - 2*I*B^2*b^2*d^2)
*x^2 + (-2*I*A*B*b^2*c*d + (-2*I*b^2*c*d - I*a*b*d^2)*B^2)*x + (-I*B^2*b^2*d^2*x^2 + (-3*I*b^2*c*d + I*a*b*d^2
)*B^2*x + (-I*b^2*c^2 - I*a*b*c*d + I*a^2*d^2)*B^2)*log(b*x + a))*log(d*x + c))/(b^3*d*g*x^2 + a*b^2*c*g + (b^
3*c*g + a*b^2*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((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^2}{a\,g+b\,g\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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