3.3.4 \(\int \frac {(a g+b g x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(c i+d i x)^3} \, dx\) [204]

Optimal. Leaf size=151 \[ \frac {B^2 g n^2 (a+b x)^2}{4 (b c-a d) i^3 (c+d x)^2}-\frac {B g n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d) i^3 (c+d x)^2}+\frac {g (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d) i^3 (c+d x)^2} \]

[Out]

1/4*B^2*g*n^2*(b*x+a)^2/(-a*d+b*c)/i^3/(d*x+c)^2-1/2*B*g*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c
)/i^3/(d*x+c)^2+1/2*g*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)/i^3/(d*x+c)^2

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2561, 2342, 2341} \begin {gather*} \frac {g (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 i^3 (c+d x)^2 (b c-a d)}-\frac {B g n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)}+\frac {B^2 g n^2 (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

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 {(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(204 c+204 d x)^3} \, dx &=\int \left (\frac {(-b c+a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d (c+d x)^3}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d (c+d x)^2}\right ) \, dx\\ &=\frac {(b g) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2} \, dx}{8489664 d}-\frac {((b c-a d) g) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3} \, dx}{8489664 d}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac {(b B g 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 )}{(a+b x) (c+d x)^2} \, dx}{4244832 d^2}-\frac {(B (b c-a d) g 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 )}{(a+b x) (c+d x)^3} \, dx}{8489664 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac {(b B (b c-a d) g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^2} \, dx}{4244832 d^2}-\frac {\left (B (b c-a d)^2 g n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^3} \, dx}{8489664 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac {(b B (b c-a d) g n) \int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{4244832 d^2}-\frac {\left (B (b c-a d)^2 g n\right ) \int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{8489664 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac {(b B g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{8489664 d}-\frac {(b B g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{4244832 d}-\frac {\left (b^3 B g n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8489664 d^2 (b c-a d)}+\frac {\left (b^3 B g n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{4244832 d^2 (b c-a d)}+\frac {\left (b^2 B g n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8489664 d (b c-a d)}-\frac {\left (b^2 B g n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{4244832 d (b c-a d)}+\frac {(B (b c-a d) g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{8489664 d}\\ &=-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac {b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}-\frac {b^2 B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}+\frac {\left (b B^2 g n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8489664 d^2}-\frac {\left (b B^2 g n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{4244832 d^2}+\frac {\left (b^2 B^2 g n^2\right ) \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)}{a+b x} \, dx}{8489664 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{8489664 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \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)}{a+b x} \, dx}{4244832 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{4244832 d^2 (b c-a d)}+\frac {\left (B^2 (b c-a d) g n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{16979328 d^2}\\ &=-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac {b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}-\frac {b^2 B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{8489664 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{8489664 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{4244832 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{4244832 d^2 (b c-a d)}+\frac {\left (b B^2 (b c-a d) g n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8489664 d^2}-\frac {\left (b B^2 (b c-a d) g n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{4244832 d^2}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{16979328 d^2}\\ &=-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac {b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}-\frac {b^2 B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}+\frac {\left (b^3 B^2 g n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8489664 d^2 (b c-a d)}-\frac {\left (b^3 B^2 g n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8489664 d^2 (b c-a d)}-\frac {\left (b^3 B^2 g n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{4244832 d^2 (b c-a d)}+\frac {\left (b^3 B^2 g n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4244832 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8489664 d (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8489664 d (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{4244832 d (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{4244832 d (b c-a d)}+\frac {\left (b B^2 (b c-a d) g n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{8489664 d^2}-\frac {\left (b B^2 (b c-a d) g n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{4244832 d^2}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{16979328 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{33958656 d^2 (c+d x)^2}-\frac {b B^2 g n^2}{16979328 d^2 (c+d x)}-\frac {b^2 B^2 g n^2 \log (a+b x)}{16979328 d^2 (b c-a d)}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac {b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac {b^2 B^2 g n^2 \log (c+d x)}{16979328 d^2 (b c-a d)}+\frac {b^2 B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac {b^2 B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}+\frac {b^2 B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8489664 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8489664 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{4244832 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4244832 d^2 (b c-a d)}+\frac {\left (b^3 B^2 g n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8489664 d^2 (b c-a d)}-\frac {\left (b^3 B^2 g n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{4244832 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8489664 d (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4244832 d (b c-a d)}\\ &=\frac {B^2 (b c-a d) g n^2}{33958656 d^2 (c+d x)^2}-\frac {b B^2 g n^2}{16979328 d^2 (c+d x)}-\frac {b^2 B^2 g n^2 \log (a+b x)}{16979328 d^2 (b c-a d)}-\frac {b^2 B^2 g n^2 \log ^2(a+b x)}{16979328 d^2 (b c-a d)}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac {b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac {b^2 B^2 g n^2 \log (c+d x)}{16979328 d^2 (b c-a d)}+\frac {b^2 B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac {b^2 B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac {b^2 B^2 g n^2 \log ^2(c+d x)}{16979328 d^2 (b c-a d)}+\frac {b^2 B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{8489664 d^2 (b c-a d)}+\frac {\left (b^2 B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8489664 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{4244832 d^2 (b c-a d)}-\frac {\left (b^2 B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4244832 d^2 (b c-a d)}\\ &=\frac {B^2 (b c-a d) g n^2}{33958656 d^2 (c+d x)^2}-\frac {b B^2 g n^2}{16979328 d^2 (c+d x)}-\frac {b^2 B^2 g n^2 \log (a+b x)}{16979328 d^2 (b c-a d)}-\frac {b^2 B^2 g n^2 \log ^2(a+b x)}{16979328 d^2 (b c-a d)}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16979328 d^2 (c+d x)^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (c+d x)}+\frac {b^2 B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8489664 d^2 (b c-a d)}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16979328 d^2 (c+d x)^2}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8489664 d^2 (c+d x)}+\frac {b^2 B^2 g n^2 \log (c+d x)}{16979328 d^2 (b c-a d)}+\frac {b^2 B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac {b^2 B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8489664 d^2 (b c-a d)}-\frac {b^2 B^2 g n^2 \log ^2(c+d x)}{16979328 d^2 (b c-a d)}+\frac {b^2 B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}+\frac {b^2 B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}+\frac {b^2 B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8489664 d^2 (b c-a d)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.67, size = 803, normalized size = 5.32 \begin {gather*} \frac {g \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-4 b (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+4 b B n (c+d x) \left (2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 b (c+d x) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 b (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-2 B n (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-b B n (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+b B n (c+d x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )-B n \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 b (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 b^2 (c+d x)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-4 b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-4 b B n (c+d x) (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-B n \left ((b c-a d)^2+2 b (b c-a d) (c+d x)+2 b^2 (c+d x)^2 \log (a+b x)-2 b^2 (c+d x)^2 \log (c+d x)\right )-2 b^2 B n (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+2 b^2 B n (c+d x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 d^2 (b c-a d) i^3 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (d i x +c i \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(d*i*x+c*i)^3,x)

[Out]

int((b*g*x+a*g)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(d*i*x+c*i)^3,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. 1891 vs. \(2 (138) = 276\).
time = 0.52, size = 1891, normalized size = 12.52 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b^2*log(b*x + a)/(2*I*b^2*c^2*d - 4*I*a*b*c*d^2 + 2*I*a^2*d^3) - b^2*log(d*x + c)/(2*I*b^2*c^2*d - 4*I*a*b
*c*d^2 + 2*I*a^2*d^3) - (2*b*d*x + 3*b*c - a*d)/(-4*I*b*c^3*d + 4*I*a*c^2*d^2 - 4*(I*b*c*d^3 - I*a*d^4)*x^2 -
8*(I*b*c^2*d^2 - I*a*c*d^3)*x))*A*B*a*g*n - 2*A*B*b*g*n*((b^2*c - 2*a*b*d)*log(b*x + a)/(2*I*b^2*c^2*d^2 - 4*I
*a*b*c*d^3 + 2*I*a^2*d^4) - (b^2*c - 2*a*b*d)*log(d*x + c)/(2*I*b^2*c^2*d^2 - 4*I*a*b*c*d^3 + 2*I*a^2*d^4) - (
b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/(-4*I*b*c^3*d^2 + 4*I*a*c^2*d^3 - 4*(I*b*c*d^4 - I*a*d^5)*x^2 - 8*(I*
b*c^2*d^3 - I*a*c*d^4)*x)) + (2*d*x + c)*B^2*b*g*log((b*x/(d*x + c) + a/(d*x + c))^n*e)^2/(2*I*d^4*x^2 + 4*I*c
*d^3*x + 2*I*c^2*d^2) - 1/4*(8*(b^2*log(b*x + a)/(2*I*b^2*c^2*d - 4*I*a*b*c*d^2 + 2*I*a^2*d^3) - b^2*log(d*x +
c)/(2*I*b^2*c^2*d - 4*I*a*b*c*d^2 + 2*I*a^2*d^3) - (2*b*d*x + 3*b*c - a*d)/(-4*I*b*c^3*d + 4*I*a*c^2*d^2 - 4*
(I*b*c*d^3 - I*a*d^4)*x^2 - 8*(I*b*c^2*d^2 - I*a*c*d^3)*x))*n*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + (7*I*b^
2*c^2 - 8*I*a*b*c*d + I*a^2*d^2 - 2*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I*b^2*c^2)*log(b*x + a)^2 - 2*(-I*b^2*d^
2*x^2 - 2*I*b^2*c*d*x - I*b^2*c^2)*log(d*x + c)^2 - 6*(-I*b^2*c*d + I*a*b*d^2)*x - 6*(-I*b^2*d^2*x^2 - 2*I*b^2
*c*d*x - I*b^2*c^2)*log(b*x + a) - 2*(3*I*b^2*d^2*x^2 + 6*I*b^2*c*d*x + 3*I*b^2*c^2 + 2*(I*b^2*d^2*x^2 + 2*I*b
^2*c*d*x + I*b^2*c^2)*log(b*x + a))*log(d*x + c))*n^2/(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^2*d^3
- 2*a*b*c*d^4 + a^2*d^5)*x^2 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x))*B^2*a*g - 1/4*(8*n*((b^2*c - 2*
a*b*d)*log(b*x + a)/(2*I*b^2*c^2*d^2 - 4*I*a*b*c*d^3 + 2*I*a^2*d^4) - (b^2*c - 2*a*b*d)*log(d*x + c)/(2*I*b^2*
c^2*d^2 - 4*I*a*b*c*d^3 + 2*I*a^2*d^4) - (b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/(-4*I*b*c^3*d^2 + 4*I*a*c^2
*d^3 - 4*(I*b*c*d^4 - I*a*d^5)*x^2 - 8*(I*b*c^2*d^3 - I*a*c*d^4)*x))*log((b*x/(d*x + c) + a/(d*x + c))^n*e) -
(-I*b^2*c^3 + 8*I*a*b*c^2*d - 7*I*a^2*c*d^2 - 2*(I*b^2*c^3 - 2*I*a*b*c^2*d + (I*b^2*c*d^2 - 2*I*a*b*d^3)*x^2 +
2*(I*b^2*c^2*d - 2*I*a*b*c*d^2)*x)*log(b*x + a)^2 - 2*(I*b^2*c^3 - 2*I*a*b*c^2*d + (I*b^2*c*d^2 - 2*I*a*b*d^3
)*x^2 + 2*(I*b^2*c^2*d - 2*I*a*b*c*d^2)*x)*log(d*x + c)^2 - 2*(I*b^2*c^2*d - 5*I*a*b*c*d^2 + 4*I*a^2*d^3)*x -
2*(I*b^2*c^3 - 4*I*a*b*c^2*d + (I*b^2*c*d^2 - 4*I*a*b*d^3)*x^2 + 2*(I*b^2*c^2*d - 4*I*a*b*c*d^2)*x)*log(b*x +
a) - 2*(-I*b^2*c^3 + 4*I*a*b*c^2*d + (-I*b^2*c*d^2 + 4*I*a*b*d^3)*x^2 + 2*(-I*b^2*c^2*d + 4*I*a*b*c*d^2)*x + 2
*(-I*b^2*c^3 + 2*I*a*b*c^2*d + (-I*b^2*c*d^2 + 2*I*a*b*d^3)*x^2 + 2*(-I*b^2*c^2*d + 2*I*a*b*c*d^2)*x)*log(b*x
+ a))*log(d*x + c))*n^2/(b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4 + (b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*x^2
+ 2*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*x))*B^2*b*g + 2*(2*d*x + c)*A*B*b*g*log((b*x/(d*x + c) + a/(d*x
+ c))^n*e)/(2*I*d^4*x^2 + 4*I*c*d^3*x + 2*I*c^2*d^2) + B^2*a*g*log((b*x/(d*x + c) + a/(d*x + c))^n*e)^2/(2*I*
d^3*x^2 + 4*I*c*d^2*x + 2*I*c^2*d) + (2*d*x + c)*A^2*b*g/(2*I*d^4*x^2 + 4*I*c*d^3*x + 2*I*c^2*d^2) + 2*A*B*a*g
*log((b*x/(d*x + c) + a/(d*x + c))^n*e)/(2*I*d^3*x^2 + 4*I*c*d^2*x + 2*I*c^2*d) + A^2*a*g/(2*I*d^3*x^2 + 4*I*c
*d^2*x + 2*I*c^2*d)

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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. 483 vs. \(2 (138) = 276\).
time = 0.37, size = 483, normalized size = 3.20 \begin {gather*} -\frac {{\left (i \, B^{2} b^{2} c^{2} - i \, B^{2} a^{2} d^{2}\right )} g n^{2} - 2 \, {\left ({\left (i \, A B + i \, B^{2}\right )} b^{2} c^{2} + {\left (-i \, A B - i \, B^{2}\right )} a^{2} d^{2}\right )} g n - 2 \, {\left (i \, B^{2} b^{2} d^{2} g n^{2} x^{2} + 2 i \, B^{2} a b d^{2} g n^{2} x + i \, B^{2} a^{2} d^{2} g n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 2 \, {\left ({\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} b^{2} c^{2} + {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a^{2} d^{2}\right )} g - 2 \, {\left ({\left (-i \, B^{2} b^{2} c d + i \, B^{2} a b d^{2}\right )} g n^{2} + 2 \, {\left ({\left (i \, A B + i \, B^{2}\right )} b^{2} c d + {\left (-i \, A B - i \, B^{2}\right )} a b d^{2}\right )} g n + 2 \, {\left ({\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} b^{2} c d + {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b d^{2}\right )} g\right )} x - 2 \, {\left (-i \, B^{2} a^{2} d^{2} g n^{2} + 2 \, {\left (i \, A B + i \, B^{2}\right )} a^{2} d^{2} g n + {\left (-i \, B^{2} b^{2} d^{2} g n^{2} + 2 \, {\left (i \, A B + i \, B^{2}\right )} b^{2} d^{2} g n\right )} x^{2} + 2 \, {\left (-i \, B^{2} a b d^{2} g n^{2} + 2 \, {\left (i \, A B + i \, B^{2}\right )} a b d^{2} g n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left (b c^{3} d^{2} - a c^{2} d^{3} + {\left (b c d^{4} - a d^{5}\right )} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((I*B^2*b^2*c^2 - I*B^2*a^2*d^2)*g*n^2 - 2*((I*A*B + I*B^2)*b^2*c^2 + (-I*A*B - I*B^2)*a^2*d^2)*g*n - 2*(
I*B^2*b^2*d^2*g*n^2*x^2 + 2*I*B^2*a*b*d^2*g*n^2*x + I*B^2*a^2*d^2*g*n^2)*log((b*x + a)/(d*x + c))^2 - 2*((-I*A
^2 - 2*I*A*B - I*B^2)*b^2*c^2 + (I*A^2 + 2*I*A*B + I*B^2)*a^2*d^2)*g - 2*((-I*B^2*b^2*c*d + I*B^2*a*b*d^2)*g*n
^2 + 2*((I*A*B + I*B^2)*b^2*c*d + (-I*A*B - I*B^2)*a*b*d^2)*g*n + 2*((-I*A^2 - 2*I*A*B - I*B^2)*b^2*c*d + (I*A
^2 + 2*I*A*B + I*B^2)*a*b*d^2)*g)*x - 2*(-I*B^2*a^2*d^2*g*n^2 + 2*(I*A*B + I*B^2)*a^2*d^2*g*n + (-I*B^2*b^2*d^
2*g*n^2 + 2*(I*A*B + I*B^2)*b^2*d^2*g*n)*x^2 + 2*(-I*B^2*a*b*d^2*g*n^2 + 2*(I*A*B + I*B^2)*a*b*d^2*g*n)*x)*log
((b*x + a)/(d*x + c)))/(b*c^3*d^2 - a*c^2*d^3 + (b*c*d^4 - a*d^5)*x^2 + 2*(b*c^2*d^3 - a*c*d^4)*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {g \left (\int \frac {A^{2} a}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {A^{2} b x}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {B^{2} a \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 A B a \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {B^{2} b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 A B b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx\right )}{i^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(d*i*x+c*i)**3,x)

[Out]

g*(Integral(A**2*a/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(A**2*b*x/(c**3 + 3*c**2*d*x
+ 3*c*d**2*x**2 + d**3*x**3), x) + Integral(B**2*a*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c**3 + 3*c**2*d
*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(2*A*B*a*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c**3 + 3*c**2*
d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(B**2*b*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c**3 + 3*
c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(2*A*B*b*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c**3 +
3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x))/i**3

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Giac [A]
time = 5.54, size = 178, normalized size = 1.18 \begin {gather*} -\frac {1}{4} \, {\left (-\frac {2 i \, {\left (b x + a\right )}^{2} B^{2} g n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{2}} + \frac {2 \, {\left (i \, B^{2} g n^{2} - 2 i \, A B g n - 2 i \, B^{2} g n\right )} {\left (b x + a\right )}^{2} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {{\left (-i \, B^{2} g n^{2} + 2 i \, A B g n + 2 i \, B^{2} g n - 2 i \, A^{2} g - 4 i \, A B g - 2 i \, B^{2} g\right )} {\left (b x + a\right )}^{2}}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-2*I*(b*x + a)^2*B^2*g*n^2*log((b*x + a)/(d*x + c))^2/(d*x + c)^2 + 2*(I*B^2*g*n^2 - 2*I*A*B*g*n - 2*I*B
^2*g*n)*(b*x + a)^2*log((b*x + a)/(d*x + c))/(d*x + c)^2 + (-I*B^2*g*n^2 + 2*I*A*B*g*n + 2*I*B^2*g*n - 2*I*A^2
*g - 4*I*A*B*g - 2*I*B^2*g)*(b*x + a)^2/(d*x + c)^2)*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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Mupad [B]
time = 7.50, size = 565, normalized size = 3.74 \begin {gather*} -{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {\frac {B^2\,a\,g}{2\,d}+\frac {B^2\,b\,g\,x}{d}+\frac {B^2\,b\,c\,g}{2\,d^2}}{c^2\,i^3+2\,c\,d\,i^3\,x+d^2\,i^3\,x^2}+\frac {B^2\,b^2\,g}{2\,d^2\,i^3\,\left (a\,d-b\,c\right )}\right )-\frac {x\,\left (2\,b\,d\,g\,A^2-2\,b\,d\,g\,A\,B\,n+b\,d\,g\,B^2\,n^2\right )+A^2\,a\,d\,g+A^2\,b\,c\,g+\frac {B^2\,a\,d\,g\,n^2}{2}+\frac {B^2\,b\,c\,g\,n^2}{2}-A\,B\,a\,d\,g\,n-A\,B\,b\,c\,g\,n}{2\,c^2\,d^2\,i^3+4\,c\,d^3\,i^3\,x+2\,d^4\,i^3\,x^2}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {A\,B\,a\,d\,g+A\,B\,b\,c\,g-B^2\,a\,d\,g\,n+B^2\,b\,c\,g\,n+2\,A\,B\,b\,d\,g\,x}{c^2\,d^2\,i^3+2\,c\,d^3\,i^3\,x+d^4\,i^3\,x^2}-\frac {B^2\,b^2\,g\,\left (\frac {c\,d^2\,i^3\,n\,\left (a\,d-b\,c\right )}{2\,b}+\frac {d^3\,i^3\,n\,x\,\left (a\,d-b\,c\right )}{b}-\frac {d^2\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )}{2\,b^2}\right )}{d^2\,i^3\,\left (a\,d-b\,c\right )\,\left (c^2\,d^2\,i^3+2\,c\,d^3\,i^3\,x+d^4\,i^3\,x^2\right )}\right )-\frac {B\,b^2\,g\,n\,\mathrm {atan}\left (\frac {B\,b^2\,g\,n\,\left (2\,A-B\,n\right )\,\left (\frac {a\,d^3\,i^3+b\,c\,d^2\,i^3}{d^2\,i^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (B^2\,b^2\,g\,n^2-2\,A\,B\,b^2\,g\,n\right )}\right )\,\left (2\,A-B\,n\right )\,1{}\mathrm {i}}{d^2\,i^3\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- log(e*((a + b*x)/(c + d*x))^n)^2*(((B^2*a*g)/(2*d) + (B^2*b*g*x)/d + (B^2*b*c*g)/(2*d^2))/(c^2*i^3 + d^2*i^3
*x^2 + 2*c*d*i^3*x) + (B^2*b^2*g)/(2*d^2*i^3*(a*d - b*c))) - (x*(2*A^2*b*d*g + B^2*b*d*g*n^2 - 2*A*B*b*d*g*n)
+ A^2*a*d*g + A^2*b*c*g + (B^2*a*d*g*n^2)/2 + (B^2*b*c*g*n^2)/2 - A*B*a*d*g*n - A*B*b*c*g*n)/(2*c^2*d^2*i^3 +
2*d^4*i^3*x^2 + 4*c*d^3*i^3*x) - log(e*((a + b*x)/(c + d*x))^n)*((A*B*a*d*g + A*B*b*c*g - B^2*a*d*g*n + B^2*b*
c*g*n + 2*A*B*b*d*g*x)/(c^2*d^2*i^3 + d^4*i^3*x^2 + 2*c*d^3*i^3*x) - (B^2*b^2*g*((c*d^2*i^3*n*(a*d - b*c))/(2*
b) + (d^3*i^3*n*x*(a*d - b*c))/b - (d^2*i^3*n*(a*d - b*c)*(a*d - 2*b*c))/(2*b^2)))/(d^2*i^3*(a*d - b*c)*(c^2*d
^2*i^3 + d^4*i^3*x^2 + 2*c*d^3*i^3*x))) - (B*b^2*g*n*atan((B*b^2*g*n*(2*A - B*n)*((a*d^3*i^3 + b*c*d^2*i^3)/(d
^2*i^3) + 2*b*d*x)*1i)/((a*d - b*c)*(B^2*b^2*g*n^2 - 2*A*B*b^2*g*n)))*(2*A - B*n)*1i)/(d^2*i^3*(a*d - b*c))

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