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

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

[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 [C]  time = 2.00, antiderivative size = 686, normalized size of antiderivative = 4.54, number of steps used = 54, number of rules used = 11, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac {b^2 B^2 g n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac {b^2 B^2 g n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac {b^2 B g n \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i^3 (b c-a d)}-\frac {b^2 B g n \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i^3 (b c-a d)}-\frac {b g \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^2 i^3 (c+d x)}+\frac {b B g n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i^3 (c+d x)}+\frac {g (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d^2 i^3 (c+d x)^2}-\frac {B g n (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^2 i^3 (c+d x)^2}-\frac {b^2 B^2 g n^2 \log ^2(a+b x)}{2 d^2 i^3 (b c-a d)}-\frac {b^2 B^2 g n^2 \log ^2(c+d x)}{2 d^2 i^3 (b c-a d)}-\frac {b^2 B^2 g n^2 \log (a+b x)}{2 d^2 i^3 (b c-a d)}+\frac {b^2 B^2 g n^2 \log (c+d x)}{2 d^2 i^3 (b c-a d)}+\frac {b^2 B^2 g n^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac {b^2 B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d^2 i^3 (b c-a d)}+\frac {B^2 g n^2 (b c-a d)}{4 d^2 i^3 (c+d x)^2}-\frac {b B^2 g n^2}{2 d^2 i^3 (c+d x)} \]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2391

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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2524

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

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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]  time = 0.99, size = 803, normalized size = 5.32 \[ \frac {g \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-4 b (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+4 b B n (c+d x) \left (2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 b (c+d x) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 b (c+d x) \log (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B n (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-b B n (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+b B n (c+d x) \left (\left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )-B n \left (2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2+4 b (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)+4 b^2 (c+d x)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-4 b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-4 b B n (c+d x) (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-B n \left ((b c-a d)^2+2 b (c+d x) (b c-a d)+2 b^2 (c+d x)^2 \log (a+b x)-2 b^2 (c+d x)^2 \log (c+d x)\right )-2 b^2 B n (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+2 b^2 B n (c+d x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 d^2 (b c-a d) i^3 (c+d x)^2} \]

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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fricas [B]  time = 0.87, size = 600, normalized size = 3.97 \[ -\frac {{\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} g n^{2} - 2 \, {\left (A B b^{2} c^{2} - A B a^{2} d^{2}\right )} g n + 2 \, {\left (2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} g x + {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} g\right )} \log \relax (e)^{2} - 2 \, {\left (B^{2} b^{2} d^{2} g n^{2} x^{2} + 2 \, B^{2} a b d^{2} g n^{2} x + B^{2} a^{2} d^{2} g n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, {\left (A^{2} b^{2} c^{2} - A^{2} a^{2} d^{2}\right )} g + 2 \, {\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} g n^{2} - 2 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} g n + 2 \, {\left (A^{2} b^{2} c d - A^{2} a b d^{2}\right )} g\right )} x - 2 \, {\left ({\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} g n - 2 \, {\left (A B b^{2} c^{2} - A B a^{2} d^{2}\right )} g + 2 \, {\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} g n - 2 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} g\right )} x + 2 \, {\left (B^{2} b^{2} d^{2} g n x^{2} + 2 \, B^{2} a b d^{2} g n x + B^{2} a^{2} d^{2} g n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 2 \, {\left (B^{2} a^{2} d^{2} g n^{2} - 2 \, A B a^{2} d^{2} g n + {\left (B^{2} b^{2} d^{2} g n^{2} - 2 \, A B b^{2} d^{2} g n\right )} x^{2} + 2 \, {\left (B^{2} a b d^{2} g n^{2} - 2 \, A B a b d^{2} g n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b c d^{4} - a d^{5}\right )} i^{3} x^{2} + 2 \, {\left (b c^{2} d^{3} - a c d^{4}\right )} i^{3} x + {\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} i^{3}\right )}} \]

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

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giac [A]  time = 19.66, size = 186, normalized size = 1.23 \[ \frac {1}{4} \, {\left (\frac {2 \, {\left (b x + a\right )}^{2} B^{2} g i n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{2}} - \frac {2 \, {\left (B^{2} g i n^{2} - 2 \, A B g i n - 2 \, B^{2} g i n\right )} {\left (b x + a\right )}^{2} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {{\left (B^{2} g i n^{2} - 2 \, A B g i n - 2 \, B^{2} g i n + 2 \, A^{2} g i + 4 \, A B g i + 2 \, B^{2} g i\right )} {\left (b x + a\right )}^{2}}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

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*(b*x + a)^2*B^2*g*i*n^2*log((b*x + a)/(d*x + c))^2/(d*x + c)^2 - 2*(B^2*g*i*n^2 - 2*A*B*g*i*n - 2*B^2*g
*i*n)*(b*x + a)^2*log((b*x + a)/(d*x + c))/(d*x + c)^2 + (B^2*g*i*n^2 - 2*A*B*g*i*n - 2*B^2*g*i*n + 2*A^2*g*i
+ 4*A*B*g*i + 2*B^2*g*i)*(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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maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (d i x +c i \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.35, size = 1995, normalized size = 13.21 \[ \text {result too large to display} \]

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]

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

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

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

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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