3.62 \(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=287 \[ -\frac {b i (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)^2}-\frac {2 b B i (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)^2}+\frac {d i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^4 (a+b x)^2 (b c-a d)^2}+\frac {B d i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 (a+b x)^2 (b c-a d)^2}-\frac {2 b B^2 i (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)^2}+\frac {B^2 d i (c+d x)^2}{4 g^4 (a+b x)^2 (b c-a d)^2} \]

[Out]

1/4*B^2*d*i*(d*x+c)^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-2/27*b*B^2*i*(d*x+c)^3/(-a*d+b*c)^2/g^4/(b*x+a)^3+1/2*B*d*i*(
d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^2/g^4/(b*x+a)^2-2/9*b*B*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c))
)/(-a*d+b*c)^2/g^4/(b*x+a)^3+1/2*d*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-1/3*b*
i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^2/g^4/(b*x+a)^3

________________________________________________________________________________________

Rubi [C]  time = 2.29, antiderivative size = 741, normalized size of antiderivative = 2.58, number of steps used = 66, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B^2 d^3 i \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{3 b^2 g^4 (b c-a d)^2}+\frac {B^2 d^3 i \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 b^2 g^4 (b c-a d)^2}+\frac {B d^3 i \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^4 (b c-a d)^2}-\frac {B d^3 i \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^4 (b c-a d)^2}+\frac {B d^2 i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^4 (a+b x) (b c-a d)}-\frac {d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b^2 g^4 (a+b x)^2}-\frac {B d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 b^2 g^4 (a+b x)^2}-\frac {i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {2 B i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 b^2 g^4 (a+b x)^3}+\frac {5 B^2 d^2 i}{18 b^2 g^4 (a+b x) (b c-a d)}-\frac {B^2 d^3 i \log ^2(a+b x)}{6 b^2 g^4 (b c-a d)^2}-\frac {B^2 d^3 i \log ^2(c+d x)}{6 b^2 g^4 (b c-a d)^2}+\frac {5 B^2 d^3 i \log (a+b x)}{18 b^2 g^4 (b c-a d)^2}-\frac {5 B^2 d^3 i \log (c+d x)}{18 b^2 g^4 (b c-a d)^2}+\frac {B^2 d^3 i \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b^2 g^4 (b c-a d)^2}+\frac {B^2 d^3 i \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^2 g^4 (b c-a d)^2}-\frac {2 B^2 i (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac {B^2 d i}{36 b^2 g^4 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2*(b*c - a*d)*i)/(27*b^2*g^4*(a + b*x)^3) + (B^2*d*i)/(36*b^2*g^4*(a + b*x)^2) + (5*B^2*d^2*i)/(18*b^2*(
b*c - a*d)*g^4*(a + b*x)) + (5*B^2*d^3*i*Log[a + b*x])/(18*b^2*(b*c - a*d)^2*g^4) - (B^2*d^3*i*Log[a + b*x]^2)
/(6*b^2*(b*c - a*d)^2*g^4) - (2*B*(b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*b^2*g^4*(a + b*x)^3)
- (B*d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^2*g^4*(a + b*x)^2) + (B*d^2*i*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(3*b^2*(b*c - a*d)*g^4*(a + b*x)) + (B*d^3*i*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b
^2*(b*c - a*d)^2*g^4) - ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*b^2*g^4*(a + b*x)^3) - (d*i*
(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^2*g^4*(a + b*x)^2) - (5*B^2*d^3*i*Log[c + d*x])/(18*b^2*(b*c - a*
d)^2*g^4) + (B^2*d^3*i*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*b^2*(b*c - a*d)^2*g^4) - (B*d^3*i*(A
+ B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(3*b^2*(b*c - a*d)^2*g^4) - (B^2*d^3*i*Log[c + d*x]^2)/(6*b^2
*(b*c - a*d)^2*g^4) + (B^2*d^3*i*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*b^2*(b*c - a*d)^2*g^4) + (B^2
*d^3*i*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*b^2*(b*c - a*d)^2*g^4) + (B^2*d^3*i*PolyLog[2, (b*(c + d*x
))/(b*c - a*d)])/(3*b^2*(b*c - a*d)^2*g^4)

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 {(62 c+62 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx &=\int \left (\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^4 (a+b x)^4}+\frac {62 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b g^4 (a+b x)^3}\right ) \, dx\\ &=\frac {(62 d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{b g^4}+\frac {(62 (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{b g^4}\\ &=-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}+\frac {(62 B d) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac {(124 B (b c-a d)) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}+\frac {(62 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac {\left (124 B (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}+\frac {(62 B d (b c-a d)) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^4}+\frac {\left (124 B (b c-a d)^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^4}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^2 g^4}\\ &=-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac {(124 B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{3 b g^4}+\frac {(62 B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^4}-\frac {\left (124 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{3 b (b c-a d)^2 g^4}+\frac {\left (62 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d)^2 g^4}+\frac {\left (124 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3 b^2 (b c-a d)^2 g^4}-\frac {\left (62 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d)^2 g^4}+\frac {\left (124 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{3 b (b c-a d) g^4}-\frac {\left (62 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b (b c-a d) g^4}+\frac {(124 B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{3 b g^4}\\ &=-\frac {124 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac {31 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac {62 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac {62 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac {62 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac {\left (62 B^2 d\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^2 g^4}+\frac {\left (31 B^2 d\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac {\left (124 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{3 b^2 (b c-a d)^2 g^4}-\frac {\left (124 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{3 b^2 (b c-a d)^2 g^4}-\frac {\left (62 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^2 (b c-a d)^2 g^4}+\frac {\left (62 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^2 (b c-a d) g^4}-\frac {\left (62 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 (b c-a d) g^4}+\frac {\left (124 B^2 (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^2 g^4}\\ &=-\frac {124 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac {31 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac {62 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac {62 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac {62 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b^2 g^4}-\frac {\left (62 B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^4}-\frac {\left (62 B^2 d (b c-a d)\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b^2 g^4}+\frac {\left (31 B^2 d (b c-a d)\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac {\left (124 B^2 (b c-a d)^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b^2 g^4}+\frac {\left (124 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d)^2 e g^4}-\frac {\left (124 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 b^2 (b c-a d)^2 e g^4}-\frac {\left (62 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 e g^4}+\frac {\left (62 B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 e g^4}\\ &=-\frac {124 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac {31 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac {62 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac {62 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac {62 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3 b^2 g^4}-\frac {\left (62 B^2 d^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^4}-\frac {\left (62 B^2 d (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 b^2 g^4}+\frac {\left (31 B^2 d (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^4}+\frac {\left (124 B^2 (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{9 b^2 g^4}+\frac {\left (124 B^2 d^3\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{3 b^2 (b c-a d)^2 e g^4}-\frac {\left (124 B^2 d^3\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{3 b^2 (b c-a d)^2 e g^4}-\frac {\left (62 B^2 d^3\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 (b c-a d)^2 e g^4}+\frac {\left (62 B^2 d^3\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^2 (b c-a d)^2 e g^4}\\ &=-\frac {124 B^2 (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac {31 B^2 d}{18 b^2 g^4 (a+b x)^2}+\frac {155 B^2 d^2}{9 b^2 (b c-a d) g^4 (a+b x)}+\frac {155 B^2 d^3 \log (a+b x)}{9 b^2 (b c-a d)^2 g^4}-\frac {124 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac {31 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac {62 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac {62 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac {155 B^2 d^3 \log (c+d x)}{9 b^2 (b c-a d)^2 g^4}-\frac {62 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b (b c-a d)^2 g^4}-\frac {\left (124 B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b (b c-a d)^2 g^4}-\frac {\left (62 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b (b c-a d)^2 g^4}+\frac {\left (62 B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b (b c-a d)^2 g^4}-\frac {\left (124 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^2 (b c-a d)^2 g^4}+\frac {\left (62 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d)^2 g^4}-\frac {\left (62 B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d)^2 g^4}\\ &=-\frac {124 B^2 (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac {31 B^2 d}{18 b^2 g^4 (a+b x)^2}+\frac {155 B^2 d^2}{9 b^2 (b c-a d) g^4 (a+b x)}+\frac {155 B^2 d^3 \log (a+b x)}{9 b^2 (b c-a d)^2 g^4}-\frac {124 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac {31 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac {62 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac {62 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac {155 B^2 d^3 \log (c+d x)}{9 b^2 (b c-a d)^2 g^4}+\frac {62 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac {62 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac {62 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^2 (b c-a d)^2 g^4}-\frac {\left (62 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d)^2 g^4}-\frac {\left (62 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b (b c-a d)^2 g^4}-\frac {\left (62 B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^2 (b c-a d)^2 g^4}-\frac {\left (62 B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d)^2 g^4}\\ &=-\frac {124 B^2 (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac {31 B^2 d}{18 b^2 g^4 (a+b x)^2}+\frac {155 B^2 d^2}{9 b^2 (b c-a d) g^4 (a+b x)}+\frac {155 B^2 d^3 \log (a+b x)}{9 b^2 (b c-a d)^2 g^4}-\frac {31 B^2 d^3 \log ^2(a+b x)}{3 b^2 (b c-a d)^2 g^4}-\frac {124 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac {31 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac {62 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac {62 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac {155 B^2 d^3 \log (c+d x)}{9 b^2 (b c-a d)^2 g^4}+\frac {62 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac {62 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac {31 B^2 d^3 \log ^2(c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac {62 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b^2 (b c-a d)^2 g^4}+\frac {\left (124 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^2 (b c-a d)^2 g^4}-\frac {\left (62 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d)^2 g^4}-\frac {\left (62 B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d)^2 g^4}\\ &=-\frac {124 B^2 (b c-a d)}{27 b^2 g^4 (a+b x)^3}+\frac {31 B^2 d}{18 b^2 g^4 (a+b x)^2}+\frac {155 B^2 d^2}{9 b^2 (b c-a d) g^4 (a+b x)}+\frac {155 B^2 d^3 \log (a+b x)}{9 b^2 (b c-a d)^2 g^4}-\frac {31 B^2 d^3 \log ^2(a+b x)}{3 b^2 (b c-a d)^2 g^4}-\frac {124 B (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 b^2 g^4 (a+b x)^3}-\frac {31 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^2}+\frac {62 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac {62 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 (b c-a d)^2 g^4}-\frac {62 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^2 g^4 (a+b x)^3}-\frac {31 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}-\frac {155 B^2 d^3 \log (c+d x)}{9 b^2 (b c-a d)^2 g^4}+\frac {62 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac {62 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}-\frac {31 B^2 d^3 \log ^2(c+d x)}{3 b^2 (b c-a d)^2 g^4}+\frac {62 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}+\frac {62 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}+\frac {62 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^2 (b c-a d)^2 g^4}\\ \end {align*}

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Mathematica [C]  time = 1.04, size = 1035, normalized size = 3.61 \[ -\frac {i \left (36 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^3+54 d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^2+27 B d (a+b x) \left (2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2+4 d (a d-b c) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 (a+b 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 d^2 (a+b 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 )+2 B \left (12 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3-18 d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2+36 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)+36 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-36 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+36 B d^2 (a+b x)^2 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-9 B d (a+b x) \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B \left (2 (b c-a d)^3-3 d (a+b x) (b c-a d)^2+6 d^2 (a+b x)^2 (b c-a d)+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )-18 B d^3 (a+b x)^3 \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 )+18 B d^3 (a+b x)^3 \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 )}{108 b^2 (b c-a d)^2 g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/108*(i*(36*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 54*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[(
e*(a + b*x))/(c + d*x)])^2 + 27*B*d*(a + b*x)*(2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*d*(-(b
*c) + a*d)*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 4*d^2*(a + b*x)^2*Log[a + b*x]*(A + B*Log[(e*(a +
b*x))/(c + d*x)]) + 4*d^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 4*B*d*(a + b*x)*(b*c
- a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + B*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x
) - 2*d^2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*d^2*(a + b*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*d^2*(a + b*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)
])) + 2*B*(12*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 18*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*
(a + b*x))/(c + d*x)]) + 36*d^2*(b*c - a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 36*d^3*(a + b*x
)^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 36*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]
)*Log[c + d*x] + 36*B*d^2*(a + b*x)^2*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - 9*B*
d*(a + b*x)*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2
*Log[c + d*x]) + 2*B*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d)^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a
+ b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]) - 18*B*d^3*(a + b*x)^3*(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)]) + 18*B*d^3*(a + b*x)^3*((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^2
*(b*c - a*d)^2*g^4*(a + b*x)^3)

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fricas [B]  time = 0.84, size = 601, normalized size = 2.09 \[ \frac {6 \, {\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} c d^{2} - {\left (6 \, A B + 5 \, B^{2}\right )} a b^{2} d^{3}\right )} i x^{2} - 3 \, {\left ({\left (18 \, A^{2} + 6 \, A B - B^{2}\right )} b^{3} c^{2} d - 18 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c d^{2} + {\left (18 \, A^{2} + 30 \, A B + 19 \, B^{2}\right )} a^{2} b d^{3}\right )} i x + 18 \, {\left (B^{2} b^{3} d^{3} i x^{3} + 3 \, B^{2} a b^{2} d^{3} i x^{2} - 3 \, {\left (B^{2} b^{3} c^{2} d - 2 \, B^{2} a b^{2} c d^{2}\right )} i x - {\left (2 \, B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - {\left (4 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 27 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c^{2} d + {\left (18 \, A^{2} + 30 \, A B + 19 \, B^{2}\right )} a^{3} d^{3}\right )} i + 6 \, {\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} d^{3} i x^{3} + 3 \, {\left (2 \, B^{2} b^{3} c d^{2} + 3 \, {\left (2 \, A B + B^{2}\right )} a b^{2} d^{3}\right )} i x^{2} - 3 \, {\left ({\left (6 \, A B + B^{2}\right )} b^{3} c^{2} d - 6 \, {\left (2 \, A B + B^{2}\right )} a b^{2} c d^{2}\right )} i x - {\left (4 \, {\left (3 \, A B + B^{2}\right )} b^{3} c^{3} - 9 \, {\left (2 \, A B + B^{2}\right )} a b^{2} c^{2} d\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{108 \, {\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

1/108*(6*((6*A*B + 5*B^2)*b^3*c*d^2 - (6*A*B + 5*B^2)*a*b^2*d^3)*i*x^2 - 3*((18*A^2 + 6*A*B - B^2)*b^3*c^2*d -
18*(2*A^2 + 2*A*B + B^2)*a*b^2*c*d^2 + (18*A^2 + 30*A*B + 19*B^2)*a^2*b*d^3)*i*x + 18*(B^2*b^3*d^3*i*x^3 + 3*
B^2*a*b^2*d^3*i*x^2 - 3*(B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^2)*i*x - (2*B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d)*i)*log((
b*e*x + a*e)/(d*x + c))^2 - (4*(9*A^2 + 6*A*B + 2*B^2)*b^3*c^3 - 27*(2*A^2 + 2*A*B + B^2)*a*b^2*c^2*d + (18*A^
2 + 30*A*B + 19*B^2)*a^3*d^3)*i + 6*((6*A*B + 5*B^2)*b^3*d^3*i*x^3 + 3*(2*B^2*b^3*c*d^2 + 3*(2*A*B + B^2)*a*b^
2*d^3)*i*x^2 - 3*((6*A*B + B^2)*b^3*c^2*d - 6*(2*A*B + B^2)*a*b^2*c*d^2)*i*x - (4*(3*A*B + B^2)*b^3*c^3 - 9*(2
*A*B + B^2)*a*b^2*c^2*d)*i)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(
a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*
b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4)

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giac [A]  time = 1.32, size = 437, normalized size = 1.52 \[ -\frac {{\left (36 \, B^{2} b i e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} - \frac {54 \, {\left (b x e + a e\right )} B^{2} d i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} + 72 \, A B b i e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + 24 \, B^{2} b i e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {108 \, {\left (b x e + a e\right )} A B d i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {54 \, {\left (b x e + a e\right )} B^{2} d i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + 36 \, A^{2} b i e^{4} + 24 \, A B b i e^{4} + 8 \, B^{2} b i e^{4} - \frac {54 \, {\left (b x e + a e\right )} A^{2} d i e^{3}}{d x + c} - \frac {54 \, {\left (b x e + a e\right )} A B d i e^{3}}{d x + c} - \frac {27 \, {\left (b x e + a e\right )} B^{2} d i e^{3}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{108 \, {\left (\frac {{\left (b x e + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x e + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

-1/108*(36*B^2*b*i*e^4*log((b*x*e + a*e)/(d*x + c))^2 - 54*(b*x*e + a*e)*B^2*d*i*e^3*log((b*x*e + a*e)/(d*x +
c))^2/(d*x + c) + 72*A*B*b*i*e^4*log((b*x*e + a*e)/(d*x + c)) + 24*B^2*b*i*e^4*log((b*x*e + a*e)/(d*x + c)) -
108*(b*x*e + a*e)*A*B*d*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 54*(b*x*e + a*e)*B^2*d*i*e^3*log((b*x*e
+ a*e)/(d*x + c))/(d*x + c) + 36*A^2*b*i*e^4 + 24*A*B*b*i*e^4 + 8*B^2*b*i*e^4 - 54*(b*x*e + a*e)*A^2*d*i*e^3/
(d*x + c) - 54*(b*x*e + a*e)*A*B*d*i*e^3/(d*x + c) - 27*(b*x*e + a*e)*B^2*d*i*e^3/(d*x + c))*(b*c/((b*c*e - a*
d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/((b*x*e + a*e)^3*b*c*g^4/(d*x + c)^3 - (b*x*e + a*e)^3*
a*d*g^4/(d*x + c)^3)

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maple [B]  time = 0.05, size = 1765, normalized size = 6.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)*(B*ln((b*x+a)/(d*x+c)*e)+A)^2/(b*g*x+a*g)^4,x)

[Out]

1/2*d^2*e^2*i/(a*d-b*c)^3/g^4*A^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*a-1/2*d*e^2*i/(a*d-b*c)^3/g^4*A^2/
(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*b*c-1/3*d*e^3*i/(a*d-b*c)^3/g^4*A^2*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d
*e+b/d*e)^3*a+1/3*e^3*i/(a*d-b*c)^3/g^4*A^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*c+d^2*e^2*i/(a*d-b*c
)^3/g^4*A*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-d*e^2*i/(a*d-b*c)^3/g^
4*A*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b*c+1/2*d^2*e^2*i/(a*d-b*c)^3/
g^4*A*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*a-1/2*d*e^2*i/(a*d-b*c)^3/g^4*A*B/(1/(d*x+c)*a*e-1/(d*x+c)*b
*c/d*e+b/d*e)^2*b*c-2/3*d*e^3*i/(a*d-b*c)^3/g^4*A*B*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-
b*c)/(d*x+c)/d*e)*a+2/3*e^3*i/(a*d-b*c)^3/g^4*A*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-
b*c)/(d*x+c)/d*e)*c-2/9*d*e^3*i/(a*d-b*c)^3/g^4*A*B*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*a+2/9*e^3*i/(a
*d-b*c)^3/g^4*A*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*c+1/2*d^2*e^2*i/(a*d-b*c)^3/g^4*B^2/(1/(d*x+c)
*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*a-1/2*d*e^2*i/(a*d-b*c)^3/g^4*B^2/(1/(d*x+c)
*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*b*c+1/2*d^2*e^2*i/(a*d-b*c)^3/g^4*B^2/(1/(d*
x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-1/2*d*e^2*i/(a*d-b*c)^3/g^4*B^2/(1/(d*x+
c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b*c+1/4*d^2*e^2*i/(a*d-b*c)^3/g^4*B^2/(1/(d*
x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*a-1/4*d*e^2*i/(a*d-b*c)^3/g^4*B^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^
2*b*c-1/3*d*e^3*i/(a*d-b*c)^3/g^4*B^2*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d
*e)^2*a+1/3*e^3*i/(a*d-b*c)^3/g^4*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)
/d*e)^2*c-2/9*d*e^3*i/(a*d-b*c)^3/g^4*B^2*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+
c)/d*e)*a+2/9*e^3*i/(a*d-b*c)^3/g^4*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+
c)/d*e)*c-2/27*d*e^3*i/(a*d-b*c)^3/g^4*B^2*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*a+2/27*e^3*i/(a*d-b*c)^
3/g^4*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*c

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maxima [B]  time = 3.46, size = 3282, normalized size = 11.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-1/6*(3*b*x + a)*B^2*d*i*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4
*x + a^3*b^2*g^4) - 1/54*(6*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/
((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*
b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x
+ a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^
2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (4*b^3*c^3 - 27*a*b^2*c^2*d +
108*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^
3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 -
3*(5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^
3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^
2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))/(a^3*b^4*c^3*g^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^
5*b^2*c*d^2*g^4 - a^6*b*d^3*g^4 + (b^7*c^3*g^4 - 3*a*b^6*c^2*d*g^4 + 3*a^2*b^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^
3 + 3*(a*b^6*c^3*g^4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 - a^4*b^3*d^3*g^4)*x^2 + 3*(a^2*b^5*c^3*g^4 -
3*a^3*b^4*c^2*d*g^4 + 3*a^4*b^3*c*d^2*g^4 - a^5*b^2*d^3*g^4)*x))*B^2*c*i - 1/108*(6*((5*a*b^2*c^2 - 22*a^2*b*
c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*
a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^
3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(
b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/
((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (19*a*
b^3*c^3 - 189*a^2*b^2*c^2*d + 189*a^3*b*c*d^2 - 19*a^4*d^3 - 6*(27*b^4*c^2*d - 32*a*b^3*c*d^2 + 5*a^2*b^2*d^3)
*x^2 + 18*(3*a^3*b*c*d^2 - a^4*d^3 + (3*b^4*c*d^2 - a*b^3*d^3)*x^3 + 3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(
3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log(b*x + a)^2 + 18*(3*a^3*b*c*d^2 - a^4*d^3 + (3*b^4*c*d^2 - a*b^3*d^3)*x^3 +
3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log(d*x + c)^2 + 3*(9*b^4*c^3 - 125*
a*b^3*c^2*d + 135*a^2*b^2*c*d^2 - 19*a^3*b*d^3)*x - 6*(27*a^3*b*c*d^2 - 5*a^4*d^3 + (27*b^4*c*d^2 - 5*a*b^3*d^
3)*x^3 + 3*(27*a*b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2 + 3*(27*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x)*log(b*x + a) + 6*(27*a
^3*b*c*d^2 - 5*a^4*d^3 + (27*b^4*c*d^2 - 5*a*b^3*d^3)*x^3 + 3*(27*a*b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2 + 3*(27*a^2
*b^2*c*d^2 - 5*a^3*b*d^3)*x - 6*(3*a^3*b*c*d^2 - a^4*d^3 + (3*b^4*c*d^2 - a*b^3*d^3)*x^3 + 3*(3*a*b^3*c*d^2 -
a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log(b*x + a))*log(d*x + c))/(a^3*b^5*c^3*g^4 - 3*a^4*b^4
*c^2*d*g^4 + 3*a^5*b^3*c*d^2*g^4 - a^6*b^2*d^3*g^4 + (b^8*c^3*g^4 - 3*a*b^7*c^2*d*g^4 + 3*a^2*b^6*c*d^2*g^4 -
a^3*b^5*d^3*g^4)*x^3 + 3*(a*b^7*c^3*g^4 - 3*a^2*b^6*c^2*d*g^4 + 3*a^3*b^5*c*d^2*g^4 - a^4*b^4*d^3*g^4)*x^2 + 3
*(a^2*b^6*c^3*g^4 - 3*a^3*b^5*c^2*d*g^4 + 3*a^4*b^4*c*d^2*g^4 - a^5*b^3*d^3*g^4)*x))*B^2*d*i - 1/18*A*B*d*i*(6
*(3*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g
^4) + (5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d +
5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g
^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4
) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*
b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) - 1/9*A*B*c*i*(
(6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2
*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4
*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^4
*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a
^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g
^4)) - 1/3*B^2*c*i*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a
^3*b*g^4) - 1/6*(3*b*x + a)*A^2*d*i/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/3*A^2*
c*i/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)

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mupad [B]  time = 7.70, size = 955, normalized size = 3.33 \[ -{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,c\,i}{3\,b^2\,g^4}+\frac {B^2\,a\,d\,i}{6\,b^3\,g^4}+\frac {B^2\,d\,i\,x}{2\,b^2\,g^4}}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B^2\,d^3\,i}{6\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\frac {18\,i\,A^2\,a^2\,d^2+18\,i\,A^2\,a\,b\,c\,d-36\,i\,A^2\,b^2\,c^2+30\,i\,A\,B\,a^2\,d^2+30\,i\,A\,B\,a\,b\,c\,d-24\,i\,A\,B\,b^2\,c^2+19\,i\,B^2\,a^2\,d^2+19\,i\,B^2\,a\,b\,c\,d-8\,i\,B^2\,b^2\,c^2}{6\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (5\,i\,B^2\,b^2\,d^2+6\,A\,i\,B\,b^2\,d^2\right )}{a\,d-b\,c}+\frac {x\,\left (-18\,c\,i\,A^2\,b^2\,d+18\,a\,i\,A^2\,b\,d^2-6\,c\,i\,A\,B\,b^2\,d+30\,a\,i\,A\,B\,b\,d^2+c\,i\,B^2\,b^2\,d+19\,a\,i\,B^2\,b\,d^2\right )}{2\,\left (a\,d-b\,c\right )}}{18\,a^3\,b^2\,g^4+54\,a^2\,b^3\,g^4\,x+54\,a\,b^4\,g^4\,x^2+18\,b^5\,g^4\,x^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (\frac {A\,B\,i}{b^2\,g^4}+\frac {B^2\,d^3\,i\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )+\frac {A\,B\,a\,i}{3\,b^3\,g^4}+\frac {B\,i\,\left (2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{3\,b^3\,d\,g^4}+\frac {B^2\,d^3\,i\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3}{3\,b\,d^4}\right )}{3\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,d^3\,i\,x^2\,\left (\frac {b^2\,c-a\,b\,d}{3\,d^2}-\frac {2\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b^2\,g^4\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{\frac {3\,a^2\,x}{d}+\frac {a^3}{b\,d}+\frac {b^2\,x^3}{d}+\frac {3\,a\,b\,x^2}{d}}-\frac {B\,d^3\,i\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x-\frac {18\,b^4\,c^2\,g^4-18\,a^2\,b^2\,d^2\,g^4}{18\,b^2\,g^4\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (6\,A+5\,B\right )\,1{}\mathrm {i}}{9\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(a*g + b*g*x)^4,x)

[Out]

- log((e*(a + b*x))/(c + d*x))^2*(((B^2*c*i)/(3*b^2*g^4) + (B^2*a*d*i)/(6*b^3*g^4) + (B^2*d*i*x)/(2*b^2*g^4))/
(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b*x^2) - (B^2*d^3*i)/(6*b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - ((18*A^2*
a^2*d^2*i - 36*A^2*b^2*c^2*i + 19*B^2*a^2*d^2*i - 8*B^2*b^2*c^2*i + 30*A*B*a^2*d^2*i - 24*A*B*b^2*c^2*i + 18*A
^2*a*b*c*d*i + 19*B^2*a*b*c*d*i + 30*A*B*a*b*c*d*i)/(6*(a*d - b*c)) + (x^2*(5*B^2*b^2*d^2*i + 6*A*B*b^2*d^2*i)
)/(a*d - b*c) + (x*(18*A^2*a*b*d^2*i + 19*B^2*a*b*d^2*i - 18*A^2*b^2*c*d*i + B^2*b^2*c*d*i + 30*A*B*a*b*d^2*i
- 6*A*B*b^2*c*d*i))/(2*(a*d - b*c)))/(18*a^3*b^2*g^4 + 18*b^5*g^4*x^3 + 54*a^2*b^3*g^4*x + 54*a*b^4*g^4*x^2) -
(log((e*(a + b*x))/(c + d*x))*(x*((A*B*i)/(b^2*g^4) + (B^2*d^3*i*(b*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d
^3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(3*d^3) + (2*a*(a*d - b*c))/(3*d^2)))/(3*
b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + (A*B*a*i)/(3*b^3*g^4) + (B*i*(2*A*b*c - B*a*d + B*b*c))/(3*b^3*d*g
^4) + (B^2*d^3*i*(a*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^3*d^3 - b
^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2)/(3*b*d^4)))/(3*b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (B^2*d^3*i*x
^2*((b^2*c - a*b*d)/(3*d^2) - (2*b*(a*d - b*c))/(3*d^2)))/(3*b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))))/((3*a^
2*x)/d + a^3/(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/d) - (B*d^3*i*atan(((2*b*d*x - (18*b^4*c^2*g^4 - 18*a^2*b^2*d^2
*g^4)/(18*b^2*g^4*(a*d - b*c)))*1i)/(a*d - b*c))*(6*A + 5*B)*1i)/(9*b^2*g^4*(a*d - b*c)^2)

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sympy [B]  time = 29.46, size = 1387, normalized size = 4.83 \[ - \frac {B d^{3} i \left (6 A + 5 B\right ) \log {\left (x + \frac {6 A B a d^{4} i + 6 A B b c d^{3} i + 5 B^{2} a d^{4} i + 5 B^{2} b c d^{3} i - \frac {B a^{3} d^{6} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{5} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{4} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} + \frac {B b^{3} c^{3} d^{3} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}}}{12 A B b d^{4} i + 10 B^{2} b d^{4} i} \right )}}{18 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {B d^{3} i \left (6 A + 5 B\right ) \log {\left (x + \frac {6 A B a d^{4} i + 6 A B b c d^{3} i + 5 B^{2} a d^{4} i + 5 B^{2} b c d^{3} i + \frac {B a^{3} d^{6} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{5} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{4} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}} - \frac {B b^{3} c^{3} d^{3} i \left (6 A + 5 B\right )}{\left (a d - b c\right )^{2}}}{12 A B b d^{4} i + 10 B^{2} b d^{4} i} \right )}}{18 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {\left (3 B^{2} a c^{2} d i + 6 B^{2} a c d^{2} i x + 3 B^{2} a d^{3} i x^{2} - 2 B^{2} b c^{3} i - 3 B^{2} b c^{2} d i x + B^{2} b d^{3} i x^{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{6 a^{5} d^{2} g^{4} - 12 a^{4} b c d g^{4} + 18 a^{4} b d^{2} g^{4} x + 6 a^{3} b^{2} c^{2} g^{4} - 36 a^{3} b^{2} c d g^{4} x + 18 a^{3} b^{2} d^{2} g^{4} x^{2} + 18 a^{2} b^{3} c^{2} g^{4} x - 36 a^{2} b^{3} c d g^{4} x^{2} + 6 a^{2} b^{3} d^{2} g^{4} x^{3} + 18 a b^{4} c^{2} g^{4} x^{2} - 12 a b^{4} c d g^{4} x^{3} + 6 b^{5} c^{2} g^{4} x^{3}} + \frac {\left (- 6 A B a^{2} d^{2} i - 6 A B a b c d i - 18 A B a b d^{2} i x + 12 A B b^{2} c^{2} i + 18 A B b^{2} c d i x - 5 B^{2} a^{2} d^{2} i - 5 B^{2} a b c d i - 15 B^{2} a b d^{2} i x + 4 B^{2} b^{2} c^{2} i + 3 B^{2} b^{2} c d i x - 6 B^{2} b^{2} d^{2} i x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{18 a^{4} b^{2} d g^{4} - 18 a^{3} b^{3} c g^{4} + 54 a^{3} b^{3} d g^{4} x - 54 a^{2} b^{4} c g^{4} x + 54 a^{2} b^{4} d g^{4} x^{2} - 54 a b^{5} c g^{4} x^{2} + 18 a b^{5} d g^{4} x^{3} - 18 b^{6} c g^{4} x^{3}} + \frac {- 18 A^{2} a^{2} d^{2} i - 18 A^{2} a b c d i + 36 A^{2} b^{2} c^{2} i - 30 A B a^{2} d^{2} i - 30 A B a b c d i + 24 A B b^{2} c^{2} i - 19 B^{2} a^{2} d^{2} i - 19 B^{2} a b c d i + 8 B^{2} b^{2} c^{2} i + x^{2} \left (- 36 A B b^{2} d^{2} i - 30 B^{2} b^{2} d^{2} i\right ) + x \left (- 54 A^{2} a b d^{2} i + 54 A^{2} b^{2} c d i - 90 A B a b d^{2} i + 18 A B b^{2} c d i - 57 B^{2} a b d^{2} i - 3 B^{2} b^{2} c d i\right )}{108 a^{4} b^{2} d g^{4} - 108 a^{3} b^{3} c g^{4} + x^{3} \left (108 a b^{5} d g^{4} - 108 b^{6} c g^{4}\right ) + x^{2} \left (324 a^{2} b^{4} d g^{4} - 324 a b^{5} c g^{4}\right ) + x \left (324 a^{3} b^{3} d g^{4} - 324 a^{2} b^{4} c g^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**4,x)

[Out]

-B*d**3*i*(6*A + 5*B)*log(x + (6*A*B*a*d**4*i + 6*A*B*b*c*d**3*i + 5*B**2*a*d**4*i + 5*B**2*b*c*d**3*i - B*a**
3*d**6*i*(6*A + 5*B)/(a*d - b*c)**2 + 3*B*a**2*b*c*d**5*i*(6*A + 5*B)/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**4*i*
(6*A + 5*B)/(a*d - b*c)**2 + B*b**3*c**3*d**3*i*(6*A + 5*B)/(a*d - b*c)**2)/(12*A*B*b*d**4*i + 10*B**2*b*d**4*
i))/(18*b**2*g**4*(a*d - b*c)**2) + B*d**3*i*(6*A + 5*B)*log(x + (6*A*B*a*d**4*i + 6*A*B*b*c*d**3*i + 5*B**2*a
*d**4*i + 5*B**2*b*c*d**3*i + B*a**3*d**6*i*(6*A + 5*B)/(a*d - b*c)**2 - 3*B*a**2*b*c*d**5*i*(6*A + 5*B)/(a*d
- b*c)**2 + 3*B*a*b**2*c**2*d**4*i*(6*A + 5*B)/(a*d - b*c)**2 - B*b**3*c**3*d**3*i*(6*A + 5*B)/(a*d - b*c)**2)
/(12*A*B*b*d**4*i + 10*B**2*b*d**4*i))/(18*b**2*g**4*(a*d - b*c)**2) + (3*B**2*a*c**2*d*i + 6*B**2*a*c*d**2*i*
x + 3*B**2*a*d**3*i*x**2 - 2*B**2*b*c**3*i - 3*B**2*b*c**2*d*i*x + B**2*b*d**3*i*x**3)*log(e*(a + b*x)/(c + d*
x))**2/(6*a**5*d**2*g**4 - 12*a**4*b*c*d*g**4 + 18*a**4*b*d**2*g**4*x + 6*a**3*b**2*c**2*g**4 - 36*a**3*b**2*c
*d*g**4*x + 18*a**3*b**2*d**2*g**4*x**2 + 18*a**2*b**3*c**2*g**4*x - 36*a**2*b**3*c*d*g**4*x**2 + 6*a**2*b**3*
d**2*g**4*x**3 + 18*a*b**4*c**2*g**4*x**2 - 12*a*b**4*c*d*g**4*x**3 + 6*b**5*c**2*g**4*x**3) + (-6*A*B*a**2*d*
*2*i - 6*A*B*a*b*c*d*i - 18*A*B*a*b*d**2*i*x + 12*A*B*b**2*c**2*i + 18*A*B*b**2*c*d*i*x - 5*B**2*a**2*d**2*i -
5*B**2*a*b*c*d*i - 15*B**2*a*b*d**2*i*x + 4*B**2*b**2*c**2*i + 3*B**2*b**2*c*d*i*x - 6*B**2*b**2*d**2*i*x**2)
*log(e*(a + b*x)/(c + d*x))/(18*a**4*b**2*d*g**4 - 18*a**3*b**3*c*g**4 + 54*a**3*b**3*d*g**4*x - 54*a**2*b**4*
c*g**4*x + 54*a**2*b**4*d*g**4*x**2 - 54*a*b**5*c*g**4*x**2 + 18*a*b**5*d*g**4*x**3 - 18*b**6*c*g**4*x**3) + (
-18*A**2*a**2*d**2*i - 18*A**2*a*b*c*d*i + 36*A**2*b**2*c**2*i - 30*A*B*a**2*d**2*i - 30*A*B*a*b*c*d*i + 24*A*
B*b**2*c**2*i - 19*B**2*a**2*d**2*i - 19*B**2*a*b*c*d*i + 8*B**2*b**2*c**2*i + x**2*(-36*A*B*b**2*d**2*i - 30*
B**2*b**2*d**2*i) + x*(-54*A**2*a*b*d**2*i + 54*A**2*b**2*c*d*i - 90*A*B*a*b*d**2*i + 18*A*B*b**2*c*d*i - 57*B
**2*a*b*d**2*i - 3*B**2*b**2*c*d*i))/(108*a**4*b**2*d*g**4 - 108*a**3*b**3*c*g**4 + x**3*(108*a*b**5*d*g**4 -
108*b**6*c*g**4) + x**2*(324*a**2*b**4*d*g**4 - 324*a*b**5*c*g**4) + x*(324*a**3*b**3*d*g**4 - 324*a**2*b**4*c
*g**4))

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