3.2.90 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x) (c i+d i x)} \, dx\) [190]

Optimal. Leaf size=50 \[ \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d) g i n} \]

[Out]

1/3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^3/B/(-a*d+b*c)/g/i/n

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Rubi [A]
time = 0.11, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2561, 2339, 30} \begin {gather*} \frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g i n (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3/(3*B*(b*c - a*d)*g*i*n)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(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] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(190 c+190 d x) (a g+b g x)} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g (c+d x)}\right ) \, dx\\ &=\frac {b \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{190 (b c-a d) g}-\frac {d \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{190 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {(B n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{95 (b c-a d) g}+\frac {(B n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{95 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {(B n) \int \frac {(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{95 (b c-a d) g}+\frac {(B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{95 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {(B n) \int \frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{95 g}+\frac {(B n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{95 g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {(B n) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{95 g}+\frac {(B n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{95 g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {(A B n) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{95 g}-\frac {\left (B^2 n\right ) \int \frac {\log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{95 g}+\frac {(b B n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{95 (b c-a d) g}-\frac {(B d n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{c+d x} \, dx}{95 (b c-a d) g}\\ &=-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{190 (b c-a d) g}-\frac {(A B n) \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 )}{95 b g}+\frac {(b B n) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{95 (b c-a d) g}-\frac {(B d n) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x}\right ) \, dx}{95 (b c-a d) g}\\ &=-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}+\frac {\left (B^2 n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{95 g}-\frac {(A B n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{95 (b c-a d) g}+\frac {(A b B n) \int \frac {\log (c+d x)}{a+b x} \, dx}{95 (b c-a d) g}+\frac {\left (b B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{95 (b c-a d) g}-\frac {(A B d n) \int \frac {\log (c+d x)}{c+d x} \, dx}{95 (b c-a d) g}+\frac {(A B d n) \text {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{95 b (b c-a d) g}-\frac {\left (B^2 d n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{95 (b c-a d) g}\\ &=-\frac {A B n \log ^2(a+b x)}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}+\frac {A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{95 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{190 (b c-a d) g}+\frac {A B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}-\frac {(A B n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}-\frac {(A B n) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{95 (b c-a d) g}+\frac {\left (b B^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{95 (b c-a d) g}+\frac {\left (b B^2 n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{95 (b c-a d) g}-\frac {(A B d n) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{95 (b c-a d) g}-\frac {\left (B^2 n^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}{95 g}+\frac {\left (b B^2 n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{190 (b c-a d) g}-\frac {\left (B^2 d n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{190 (b c-a d) g}+\frac {\left (b B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{95 (b c-a d) g}\\ &=-\frac {A B n \log ^2(a+b x)}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}+\frac {A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{95 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {A B n \log ^2(c+d x)}{190 (b c-a d) g}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{190 (b c-a d) g}+\frac {A B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}-\frac {(A B n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}+\frac {\left (B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{95 (b c-a d) g}+\frac {\left (B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{95 (b c-a d) g}-\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{190 (b c-a d) g}-\frac {\left (B^2 d n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{95 (b c-a d) g}-\frac {\left (B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{95 (b c-a d) g}\\ &=-\frac {A B n \log ^2(a+b x)}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}+\frac {A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{95 (b c-a d) g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {A B n \log ^2(c+d x)}{190 (b c-a d) g}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{190 (b c-a d) g}+\frac {A B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{95 (b c-a d) g}-\frac {B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}-\frac {\left (B^2 d\right ) \text {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{190 b (b c-a d) g}-\frac {\left (B^2 d n\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{95 b (b c-a d) g}-\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{190 (b c-a d) g}-\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}+\frac {\left (B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{95 b (b c-a d) g}-\frac {\left (B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}\\ &=-\frac {A B n \log ^2(a+b x)}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}+\frac {A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{95 (b c-a d) g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {A B n \log ^2(c+d x)}{190 (b c-a d) g}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n^2 \log ^3(c+d x)}{570 (b c-a d) g}+\frac {A B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{190 (b c-a d) g}+\frac {B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{95 (b c-a d) g}-\frac {B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}-\frac {\left (B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}+\frac {\left (B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{95 (b c-a d) g}+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}-\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}\\ &=-\frac {A B n \log ^2(a+b x)}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}+\frac {A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{95 (b c-a d) g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {A B n \log ^2(c+d x)}{190 (b c-a d) g}+\frac {B^2 n^2 \log (a+b x) \log ^2(c+d x)}{190 (b c-a d) g}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n^2 \log ^3(c+d x)}{570 (b c-a d) g}+\frac {A B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{190 (b c-a d) g}+\frac {B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{95 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{190 (b c-a d) g}-\frac {B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}-\frac {B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}-\frac {\left (b B^2\right ) \text {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{190 d (b c-a d) g}+\frac {\left (B^2 n^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 )}{95 (b c-a d) g}-\frac {\left (b B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{190 d (b c-a d) g}\\ &=-\frac {A B n \log ^2(a+b x)}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}+\frac {A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{95 (b c-a d) g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {A B n \log ^2(c+d x)}{190 (b c-a d) g}+\frac {B^2 n^2 \log (a+b x) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n^2 \log ^3(c+d x)}{570 (b c-a d) g}+\frac {A B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{190 (b c-a d) g}+\frac {B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{95 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{190 (b c-a d) g}-\frac {B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}-\frac {\left (B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}\\ &=-\frac {A B n \log ^2(a+b x)}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}+\frac {A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{95 (b c-a d) g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {A B n \log ^2(c+d x)}{190 (b c-a d) g}+\frac {B^2 n^2 \log (a+b x) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n^2 \log ^3(c+d x)}{570 (b c-a d) g}+\frac {A B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{190 (b c-a d) g}+\frac {B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{95 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{190 (b c-a d) g}-\frac {B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}+2 \frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{95 (b c-a d) g}\\ &=-\frac {A B n \log ^2(a+b x)}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}-\frac {B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{190 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{190 (b c-a d) g}+\frac {A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{95 (b c-a d) g}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{190 (b c-a d) g}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{190 (b c-a d) g}-\frac {A B n \log ^2(c+d x)}{190 (b c-a d) g}+\frac {B^2 n^2 \log (a+b x) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{190 (b c-a d) g}-\frac {B^2 n^2 \log ^3(c+d x)}{570 (b c-a d) g}+\frac {A B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{190 (b c-a d) g}+\frac {B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{95 (b c-a d) g}+\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{190 (b c-a d) g}-\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{190 (b c-a d) g}-\frac {B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}-\frac {B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{95 (b c-a d) g}+\frac {B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{95 (b c-a d) g}\\ \end {align*}

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Mathematica [A]
time = 0.14, size = 90, normalized size = 1.80 \begin {gather*} \frac {3 A^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A B \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B^2 \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 b c g i n-3 a d g i n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right ) \left (d i x +c i \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (46) = 92\).
time = 0.30, size = 403, normalized size = 8.06 \begin {gather*} -B^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (-i \, b c + i \, a d\right )} g} - \frac {\log \left (d x + c\right )}{{\left (-i \, b c + i \, a d\right )} g}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )^{2} - 2 \, A B {\left (\frac {\log \left (b x + a\right )}{{\left (-i \, b c + i \, a d\right )} g} - \frac {\log \left (d x + c\right )}{{\left (-i \, b c + i \, a d\right )} g}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {1}{3} \, {\left (\frac {{\left (-i \, \log \left (b x + a\right )^{3} + 3 i \, \log \left (b x + a\right )^{2} \log \left (d x + c\right ) - 3 i \, \log \left (b x + a\right ) \log \left (d x + c\right )^{2} + i \, \log \left (d x + c\right )^{3}\right )} n^{2}}{b c g - a d g} + \frac {3 \, {\left (i \, \log \left (b x + a\right )^{2} - 2 i \, \log \left (b x + a\right ) \log \left (d x + c\right ) + i \, \log \left (d x + c\right )^{2}\right )} n \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{b c g - a d g}\right )} B^{2} + \frac {{\left (i \, \log \left (b x + a\right )^{2} - 2 i \, \log \left (b x + a\right ) \log \left (d x + c\right ) + i \, \log \left (d x + c\right )^{2}\right )} A B n}{b c g - a d g} - A^{2} {\left (\frac {\log \left (b x + a\right )}{{\left (-i \, b c + i \, a d\right )} g} - \frac {\log \left (d x + c\right )}{{\left (-i \, b c + i \, a d\right )} g}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B^2*(log(b*x + a)/((-I*b*c + I*a*d)*g) - log(d*x + c)/((-I*b*c + I*a*d)*g))*log((b*x/(d*x + c) + a/(d*x + c))
^n*e)^2 - 2*A*B*(log(b*x + a)/((-I*b*c + I*a*d)*g) - log(d*x + c)/((-I*b*c + I*a*d)*g))*log((b*x/(d*x + c) + a
/(d*x + c))^n*e) + 1/3*((-I*log(b*x + a)^3 + 3*I*log(b*x + a)^2*log(d*x + c) - 3*I*log(b*x + a)*log(d*x + c)^2
+ I*log(d*x + c)^3)*n^2/(b*c*g - a*d*g) + 3*(I*log(b*x + a)^2 - 2*I*log(b*x + a)*log(d*x + c) + I*log(d*x + c
)^2)*n*log((b*x/(d*x + c) + a/(d*x + c))^n*e)/(b*c*g - a*d*g))*B^2 + (I*log(b*x + a)^2 - 2*I*log(b*x + a)*log(
d*x + c) + I*log(d*x + c)^2)*A*B*n/(b*c*g - a*d*g) - A^2*(log(b*x + a)/((-I*b*c + I*a*d)*g) - log(d*x + c)/((-
I*b*c + I*a*d)*g))

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (46) = 92\).
time = 0.36, size = 100, normalized size = 2.00 \begin {gather*} -\frac {i \, B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{3} - 3 \, {\left (-i \, A B - i \, B^{2}\right )} n \log \left (\frac {b x + a}{d x + c}\right )^{2} - 3 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{3 \, {\left (b c - a d\right )} g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(I*B^2*n^2*log((b*x + a)/(d*x + c))^3 - 3*(-I*A*B - I*B^2)*n*log((b*x + a)/(d*x + c))^2 - 3*(-I*A^2 - 2*I
*A*B - I*B^2)*log((b*x + a)/(d*x + c)))/((b*c - a*d)*g)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (46) = 92\).
time = 4.29, size = 157, normalized size = 3.14 \begin {gather*} -\frac {{\left (i \, B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 3 i \, A B n \log \left (\frac {b x + a}{d x + c}\right )^{2} + 3 i \, B^{2} n \log \left (\frac {b x + a}{d x + c}\right )^{2} + 3 i \, A^{2} \log \left (\frac {b x + a}{d x + c}\right ) + 6 i \, A B \log \left (\frac {b x + a}{d x + c}\right ) + 3 i \, B^{2} \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{3 \, g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(I*B^2*n^2*log((b*x + a)/(d*x + c))^3 + 3*I*A*B*n*log((b*x + a)/(d*x + c))^2 + 3*I*B^2*n*log((b*x + a)/(d
*x + c))^2 + 3*I*A^2*log((b*x + a)/(d*x + c)) + 6*I*A*B*log((b*x + a)/(d*x + c)) + 3*I*B^2*log((b*x + a)/(d*x
+ c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)/g

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Mupad [B]
time = 5.68, size = 122, normalized size = 2.44 \begin {gather*} -\frac {\frac {B^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{3}+A\,B\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{g\,i\,n\,\left (a\,d-b\,c\right )}+\frac {A^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{g\,i\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A^2*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*2i)/(g*i*(a*d - b*c)) - ((B^2*log(e*((a + b*x)/(c + d*x))^
n)^3)/3 + A*B*log(e*((a + b*x)/(c + d*x))^n)^2)/(g*i*n*(a*d - b*c))

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