3.81 \(\int \frac{(c i+d i x)^3 (A+B \log (\frac{e (a+b x)}{c+d x}))^2}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=147 \[ -\frac{i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)}-\frac{B i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)}-\frac{B^2 i^3 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)} \]

[Out]

-(B^2*i^3*(c + d*x)^4)/(32*(b*c - a*d)*g^5*(a + b*x)^4) - (B*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x
)]))/(8*(b*c - a*d)*g^5*(a + b*x)^4) - (i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(b*c - a*d)
*g^5*(a + b*x)^4)

________________________________________________________________________________________

Rubi [C]  time = 4.5405, antiderivative size = 970, normalized size of antiderivative = 6.6, number of steps used = 130, number of rules used = 11, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.262, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B^2 i^3 \log ^2(a+b x) d^4}{4 b^4 (b c-a d) g^5}+\frac{B^2 i^3 \log ^2(c+d x) d^4}{4 b^4 (b c-a d) g^5}-\frac{B^2 i^3 \log (a+b x) d^4}{8 b^4 (b c-a d) g^5}-\frac{B i^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^4}{2 b^4 (b c-a d) g^5}+\frac{B^2 i^3 \log (c+d x) d^4}{8 b^4 (b c-a d) g^5}-\frac{B^2 i^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{2 b^4 (b c-a d) g^5}+\frac{B i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x) d^4}{2 b^4 (b c-a d) g^5}-\frac{B^2 i^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^4}{2 b^4 (b c-a d) g^5}-\frac{B^2 i^3 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^4}{2 b^4 (b c-a d) g^5}-\frac{B^2 i^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^4}{2 b^4 (b c-a d) g^5}-\frac{i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 d^3}{b^4 g^5 (a+b x)}-\frac{B i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^3}{2 b^4 g^5 (a+b x)}-\frac{B^2 i^3 d^3}{8 b^4 g^5 (a+b x)}-\frac{3 (b c-a d) i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 d^2}{2 b^4 g^5 (a+b x)^2}-\frac{3 B (b c-a d) i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^2}{4 b^4 g^5 (a+b x)^2}-\frac{3 B^2 (b c-a d) i^3 d^2}{16 b^4 g^5 (a+b x)^2}-\frac{(b c-a d)^2 i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 d}{b^4 g^5 (a+b x)^3}-\frac{B (b c-a d)^2 i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d}{2 b^4 g^5 (a+b x)^3}-\frac{B^2 (b c-a d)^2 i^3 d}{8 b^4 g^5 (a+b x)^3}-\frac{(b c-a d)^3 i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{B (b c-a d)^3 i^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^4 g^5 (a+b x)^4}-\frac{B^2 (b c-a d)^3 i^3}{32 b^4 g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x)^5,x]

[Out]

-(B^2*(b*c - a*d)^3*i^3)/(32*b^4*g^5*(a + b*x)^4) - (B^2*d*(b*c - a*d)^2*i^3)/(8*b^4*g^5*(a + b*x)^3) - (3*B^2
*d^2*(b*c - a*d)*i^3)/(16*b^4*g^5*(a + b*x)^2) - (B^2*d^3*i^3)/(8*b^4*g^5*(a + b*x)) - (B^2*d^4*i^3*Log[a + b*
x])/(8*b^4*(b*c - a*d)*g^5) + (B^2*d^4*i^3*Log[a + b*x]^2)/(4*b^4*(b*c - a*d)*g^5) - (B*(b*c - a*d)^3*i^3*(A +
B*Log[(e*(a + b*x))/(c + d*x)]))/(8*b^4*g^5*(a + b*x)^4) - (B*d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(2*b^4*g^5*(a + b*x)^3) - (3*B*d^2*(b*c - a*d)*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^4*g^5
*(a + b*x)^2) - (B*d^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^4*g^5*(a + b*x)) - (B*d^4*i^3*Log[a + b*
x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^4*(b*c - a*d)*g^5) - ((b*c - a*d)^3*i^3*(A + B*Log[(e*(a + b*x))
/(c + d*x)])^2)/(4*b^4*g^5*(a + b*x)^4) - (d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^4*g^
5*(a + b*x)^3) - (3*d^2*(b*c - a*d)*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^4*g^5*(a + b*x)^2) - (d^3
*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^4*g^5*(a + b*x)) + (B^2*d^4*i^3*Log[c + d*x])/(8*b^4*(b*c - a*
d)*g^5) - (B^2*d^4*i^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*b^4*(b*c - a*d)*g^5) + (B*d^4*i^3*(A
+ B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(2*b^4*(b*c - a*d)*g^5) + (B^2*d^4*i^3*Log[c + d*x]^2)/(4*b^4
*(b*c - a*d)*g^5) - (B^2*d^4*i^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(2*b^4*(b*c - a*d)*g^5) - (B^2*d
^4*i^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(2*b^4*(b*c - a*d)*g^5) - (B^2*d^4*i^3*PolyLog[2, (b*(c + d*x
))/(b*c - a*d)])/(2*b^4*(b*c - a*d)*g^5)

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]

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 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 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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(81 c+81 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx &=\int \left (\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^5}+\frac{1594323 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}+\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}+\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}\right ) \, dx\\ &=\frac{\left (531441 d^3\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b^3 g^5}+\frac{\left (1594323 d^2 (b c-a d)\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{b^3 g^5}+\frac{\left (1594323 d (b c-a d)^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{b^3 g^5}+\frac{\left (531441 (b c-a d)^3\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^5} \, dx}{b^3 g^5}\\ &=-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}+\frac{\left (1062882 B d^3\right ) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (1594323 B d^2 (b c-a d)\right ) \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^4 g^5}+\frac{\left (1062882 B d (b c-a d)^2\right ) \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}{b^4 g^5}+\frac{\left (531441 B (b c-a d)^3\right ) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b^4 g^5}\\ &=-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}+\frac{\left (1062882 B d^3 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (1594323 B d^2 (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (1062882 B d (b c-a d)^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (531441 B (b c-a d)^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b^4 g^5}\\ &=-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}+\frac{\left (1062882 B d^3 (b c-a d)\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)^2}-\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)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^5}+\frac{\left (1594323 B d^2 (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)^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^4 g^5}+\frac{\left (1062882 B d (b c-a d)^3\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}{b^4 g^5}+\frac{\left (531441 B (b c-a d)^4\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)^5}-\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)^4}+\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)^3}-\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)^2}+\frac{b d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b^4 g^5}\\ &=-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}-\frac{\left (531441 B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{2 b^3 g^5}+2 \frac{\left (1062882 B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^3 g^5}-\frac{\left (1594323 B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^3 g^5}+\frac{\left (531441 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{2 b^3 (b c-a d) g^5}-2 \frac{\left (1062882 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 (b c-a d) g^5}+\frac{\left (1594323 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^3 (b c-a d) g^5}-\frac{\left (531441 B d^5\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{2 b^4 (b c-a d) g^5}+2 \frac{\left (1062882 B d^5\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^4 (b c-a d) g^5}-\frac{\left (1594323 B d^5\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^4 (b c-a d) g^5}+\frac{\left (531441 B d^2 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{2 b^3 g^5}-\frac{\left (1062882 B d^2 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^5}+\frac{\left (1594323 B d^2 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^5}-\frac{\left (531441 B d (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} \, dx}{2 b^3 g^5}+\frac{\left (1062882 B d (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} \, dx}{b^3 g^5}+\frac{\left (531441 B (b c-a d)^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{2 b^3 g^5}\\ &=-\frac{531441 B (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^4 g^5 (a+b x)^4}-\frac{531441 B d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)^3}-\frac{1594323 B d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^5 (a+b x)^2}+\frac{3720087 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)}+\frac{3720087 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 (b c-a d) g^5}-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}-\frac{3720087 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}-\frac{\left (531441 B^2 d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2 b^4 g^5}+2 \left (-\frac{1062882 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{\left (1062882 B^2 d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}\right )-\frac{\left (1594323 B^2 d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}-\frac{\left (531441 B^2 d^4\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}{2 b^4 (b c-a d) g^5}+\frac{\left (531441 B^2 d^4\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}{2 b^4 (b c-a d) g^5}-2 \left (\frac{1062882 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\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^4 (b c-a d) g^5}\right )+2 \left (\frac{1062882 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\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^4 (b c-a d) g^5}\right )-\frac{\left (1594323 B^2 d^4\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^4 (b c-a d) g^5}+\frac{\left (1594323 B^2 d^4\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^4 (b c-a d) g^5}+\frac{\left (531441 B^2 d^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{4 b^4 g^5}-\frac{\left (531441 B^2 d^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (1594323 B^2 d^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^5}-\frac{\left (177147 B^2 d (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{2 b^4 g^5}+\frac{\left (354294 B^2 d (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (531441 B^2 (b c-a d)^3\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{8 b^4 g^5}\\ &=-\frac{531441 B (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^4 g^5 (a+b x)^4}-\frac{531441 B d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)^3}-\frac{1594323 B d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^5 (a+b x)^2}+\frac{3720087 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)}+\frac{3720087 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 (b c-a d) g^5}-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}-\frac{3720087 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}-\frac{\left (531441 B^2 d^3 (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{2 b^4 g^5}+2 \left (-\frac{1062882 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{\left (1062882 B^2 d^3 (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}\right )-\frac{\left (1594323 B^2 d^3 (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (531441 B^2 d^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{4 b^4 g^5}-\frac{\left (531441 B^2 d^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (1594323 B^2 d^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^5}-\frac{\left (177147 B^2 d (b c-a d)^3\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{2 b^4 g^5}+\frac{\left (354294 B^2 d (b c-a d)^3\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^5}+\frac{\left (531441 B^2 (b c-a d)^4\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{8 b^4 g^5}-\frac{\left (531441 B^2 d^4\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}{2 b^4 (b c-a d) e g^5}+\frac{\left (531441 B^2 d^4\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}{2 b^4 (b c-a d) e g^5}-2 \left (\frac{1062882 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\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^4 (b c-a d) e g^5}\right )+2 \left (\frac{1062882 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\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^4 (b c-a d) e g^5}\right )-\frac{\left (1594323 B^2 d^4\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^4 (b c-a d) e g^5}+\frac{\left (1594323 B^2 d^4\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^4 (b c-a d) e g^5}\\ &=-\frac{531441 B (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^4 g^5 (a+b x)^4}-\frac{531441 B d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)^3}-\frac{1594323 B d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^5 (a+b x)^2}+\frac{3720087 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)}+\frac{3720087 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 (b c-a d) g^5}-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}-\frac{3720087 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}-\frac{\left (531441 B^2 d^3 (b c-a d)\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}{2 b^4 g^5}+2 \left (-\frac{1062882 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{\left (1062882 B^2 d^3 (b c-a d)\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^4 g^5}\right )-\frac{\left (1594323 B^2 d^3 (b c-a d)\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^4 g^5}+\frac{\left (531441 B^2 d^2 (b c-a d)^2\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}{4 b^4 g^5}-\frac{\left (531441 B^2 d^2 (b c-a d)^2\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^4 g^5}+\frac{\left (1594323 B^2 d^2 (b c-a d)^2\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}{2 b^4 g^5}-\frac{\left (177147 B^2 d (b c-a d)^3\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}{2 b^4 g^5}+\frac{\left (354294 B^2 d (b c-a d)^3\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}{b^4 g^5}+\frac{\left (531441 B^2 (b c-a d)^4\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{8 b^4 g^5}-\frac{\left (531441 B^2 d^4\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}{2 b^4 (b c-a d) e g^5}+\frac{\left (531441 B^2 d^4\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}{2 b^4 (b c-a d) e g^5}-2 \left (\frac{1062882 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\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^4 (b c-a d) e g^5}\right )+2 \left (\frac{1062882 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\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^4 (b c-a d) e g^5}\right )-\frac{\left (1594323 B^2 d^4\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^4 (b c-a d) e g^5}+\frac{\left (1594323 B^2 d^4\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^4 (b c-a d) e g^5}\\ &=-\frac{531441 B^2 (b c-a d)^3}{32 b^4 g^5 (a+b x)^4}-\frac{531441 B^2 d (b c-a d)^2}{8 b^4 g^5 (a+b x)^3}-\frac{1594323 B^2 d^2 (b c-a d)}{16 b^4 g^5 (a+b x)^2}+\frac{16474671 B^2 d^3}{8 b^4 g^5 (a+b x)}+\frac{16474671 B^2 d^4 \log (a+b x)}{8 b^4 (b c-a d) g^5}-\frac{531441 B (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^4 g^5 (a+b x)^4}-\frac{531441 B d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)^3}-\frac{1594323 B d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^5 (a+b x)^2}+\frac{3720087 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)}+\frac{3720087 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 (b c-a d) g^5}-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}-\frac{16474671 B^2 d^4 \log (c+d x)}{8 b^4 (b c-a d) g^5}-\frac{3720087 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}+2 \left (-\frac{1062882 B^2 d^3}{b^4 g^5 (a+b x)}-\frac{1062882 B^2 d^4 \log (a+b x)}{b^4 (b c-a d) g^5}-\frac{1062882 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{1062882 B^2 d^4 \log (c+d x)}{b^4 (b c-a d) g^5}\right )-\frac{\left (531441 B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{2 b^3 (b c-a d) g^5}+\frac{\left (531441 B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2 b^3 (b c-a d) g^5}-\frac{\left (1594323 B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d) g^5}+\frac{\left (1594323 B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d) g^5}+\frac{\left (531441 B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{2 b^4 (b c-a d) g^5}-\frac{\left (531441 B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{2 b^4 (b c-a d) g^5}-2 \left (\frac{1062882 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d) g^5}+\frac{\left (1062882 B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 (b c-a d) g^5}\right )+2 \left (\frac{1062882 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d) g^5}+\frac{\left (1062882 B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^4 (b c-a d) g^5}\right )+\frac{\left (1594323 B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 (b c-a d) g^5}-\frac{\left (1594323 B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^4 (b c-a d) g^5}\\ &=-\frac{531441 B^2 (b c-a d)^3}{32 b^4 g^5 (a+b x)^4}-\frac{531441 B^2 d (b c-a d)^2}{8 b^4 g^5 (a+b x)^3}-\frac{1594323 B^2 d^2 (b c-a d)}{16 b^4 g^5 (a+b x)^2}+\frac{16474671 B^2 d^3}{8 b^4 g^5 (a+b x)}+\frac{16474671 B^2 d^4 \log (a+b x)}{8 b^4 (b c-a d) g^5}-\frac{531441 B (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^4 g^5 (a+b x)^4}-\frac{531441 B d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)^3}-\frac{1594323 B d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^5 (a+b x)^2}+\frac{3720087 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)}+\frac{3720087 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 (b c-a d) g^5}-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}-\frac{16474671 B^2 d^4 \log (c+d x)}{8 b^4 (b c-a d) g^5}+\frac{3720087 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}-\frac{3720087 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}+2 \left (-\frac{1062882 B^2 d^3}{b^4 g^5 (a+b x)}-\frac{1062882 B^2 d^4 \log (a+b x)}{b^4 (b c-a d) g^5}-\frac{1062882 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{1062882 B^2 d^4 \log (c+d x)}{b^4 (b c-a d) g^5}\right )+\frac{3720087 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 (b c-a d) g^5}-\frac{\left (531441 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2 b^4 (b c-a d) g^5}-\frac{\left (531441 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2 b^4 (b c-a d) g^5}-\frac{\left (1594323 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 (b c-a d) g^5}-\frac{\left (1594323 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^4 (b c-a d) g^5}-\frac{\left (531441 B^2 d^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 b^3 (b c-a d) g^5}-2 \left (\frac{1062882 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 (b c-a d) g^5}+\frac{1062882 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 (b c-a d) g^5}\right )-\frac{\left (1594323 B^2 d^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 (b c-a d) g^5}-\frac{\left (531441 B^2 d^5\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b^4 (b c-a d) g^5}+2 \left (-\frac{1062882 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^4 (b c-a d) g^5}+\frac{1062882 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 (b c-a d) g^5}+\frac{\left (1062882 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^4 (b c-a d) g^5}+\frac{\left (1062882 B^2 d^5\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^4 (b c-a d) g^5}\right )-\frac{\left (1594323 B^2 d^5\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^4 (b c-a d) g^5}\\ &=-\frac{531441 B^2 (b c-a d)^3}{32 b^4 g^5 (a+b x)^4}-\frac{531441 B^2 d (b c-a d)^2}{8 b^4 g^5 (a+b x)^3}-\frac{1594323 B^2 d^2 (b c-a d)}{16 b^4 g^5 (a+b x)^2}+\frac{16474671 B^2 d^3}{8 b^4 g^5 (a+b x)}+\frac{16474671 B^2 d^4 \log (a+b x)}{8 b^4 (b c-a d) g^5}-\frac{3720087 B^2 d^4 \log ^2(a+b x)}{4 b^4 (b c-a d) g^5}-\frac{531441 B (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^4 g^5 (a+b x)^4}-\frac{531441 B d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)^3}-\frac{1594323 B d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^5 (a+b x)^2}+\frac{3720087 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)}+\frac{3720087 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 (b c-a d) g^5}-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}-\frac{16474671 B^2 d^4 \log (c+d x)}{8 b^4 (b c-a d) g^5}+\frac{3720087 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}-\frac{3720087 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}-\frac{3720087 B^2 d^4 \log ^2(c+d x)}{4 b^4 (b c-a d) g^5}+2 \left (-\frac{1062882 B^2 d^3}{b^4 g^5 (a+b x)}-\frac{1062882 B^2 d^4 \log (a+b x)}{b^4 (b c-a d) g^5}-\frac{1062882 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{1062882 B^2 d^4 \log (c+d x)}{b^4 (b c-a d) g^5}\right )+\frac{3720087 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 (b c-a d) g^5}-\frac{\left (531441 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2 b^4 (b c-a d) g^5}-\frac{\left (531441 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2 b^4 (b c-a d) g^5}-2 \left (-\frac{531441 B^2 d^4 \log ^2(a+b x)}{b^4 (b c-a d) g^5}+\frac{1062882 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 (b c-a d) g^5}+\frac{1062882 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 (b c-a d) g^5}-\frac{\left (1062882 B^2 d^4\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^4 (b c-a d) g^5}\right )+2 \left (-\frac{1062882 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^4 (b c-a d) g^5}+\frac{1062882 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 (b c-a d) g^5}+\frac{531441 B^2 d^4 \log ^2(c+d x)}{b^4 (b c-a d) g^5}+\frac{\left (1062882 B^2 d^4\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^4 (b c-a d) g^5}\right )-\frac{\left (1594323 B^2 d^4\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^4 (b c-a d) g^5}-\frac{\left (1594323 B^2 d^4\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^4 (b c-a d) g^5}\\ &=-\frac{531441 B^2 (b c-a d)^3}{32 b^4 g^5 (a+b x)^4}-\frac{531441 B^2 d (b c-a d)^2}{8 b^4 g^5 (a+b x)^3}-\frac{1594323 B^2 d^2 (b c-a d)}{16 b^4 g^5 (a+b x)^2}+\frac{16474671 B^2 d^3}{8 b^4 g^5 (a+b x)}+\frac{16474671 B^2 d^4 \log (a+b x)}{8 b^4 (b c-a d) g^5}-\frac{3720087 B^2 d^4 \log ^2(a+b x)}{4 b^4 (b c-a d) g^5}-\frac{531441 B (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^4 g^5 (a+b x)^4}-\frac{531441 B d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)^3}-\frac{1594323 B d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^5 (a+b x)^2}+\frac{3720087 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^5 (a+b x)}+\frac{3720087 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 (b c-a d) g^5}-\frac{531441 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g^5 (a+b x)^4}-\frac{531441 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)^3}-\frac{1594323 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^4 g^5 (a+b x)^2}-\frac{531441 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^4 g^5 (a+b x)}-\frac{16474671 B^2 d^4 \log (c+d x)}{8 b^4 (b c-a d) g^5}+\frac{3720087 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}-\frac{3720087 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^4 (b c-a d) g^5}-\frac{3720087 B^2 d^4 \log ^2(c+d x)}{4 b^4 (b c-a d) g^5}+2 \left (-\frac{1062882 B^2 d^3}{b^4 g^5 (a+b x)}-\frac{1062882 B^2 d^4 \log (a+b x)}{b^4 (b c-a d) g^5}-\frac{1062882 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^5 (a+b x)}+\frac{1062882 B^2 d^4 \log (c+d x)}{b^4 (b c-a d) g^5}\right )+\frac{3720087 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 (b c-a d) g^5}+\frac{3720087 B^2 d^4 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b^4 (b c-a d) g^5}-2 \left (-\frac{531441 B^2 d^4 \log ^2(a+b x)}{b^4 (b c-a d) g^5}+\frac{1062882 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 (b c-a d) g^5}+\frac{1062882 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 (b c-a d) g^5}+\frac{1062882 B^2 d^4 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 (b c-a d) g^5}\right )+\frac{3720087 B^2 d^4 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^4 (b c-a d) g^5}+2 \left (-\frac{1062882 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^4 (b c-a d) g^5}+\frac{1062882 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^4 (b c-a d) g^5}+\frac{531441 B^2 d^4 \log ^2(c+d x)}{b^4 (b c-a d) g^5}-\frac{1062882 B^2 d^4 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 (b c-a d) g^5}\right )\\ \end{align*}

Mathematica [C]  time = 1.54465, size = 2470, normalized size = 16.8 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x)^5,x]

[Out]

-(i^3*(8*A^2*b^4*c^4 + 4*A*b^4*B*c^4 + b^4*B^2*c^4 - 8*a^4*A^2*d^4 - 4*a^4*A*B*d^4 - a^4*B^2*d^4 + 32*A^2*b^4*
c^3*d*x + 16*A*b^4*B*c^3*d*x + 4*b^4*B^2*c^3*d*x - 32*a^3*A^2*b*d^4*x - 16*a^3*A*b*B*d^4*x - 4*a^3*b*B^2*d^4*x
+ 48*A^2*b^4*c^2*d^2*x^2 + 24*A*b^4*B*c^2*d^2*x^2 + 6*b^4*B^2*c^2*d^2*x^2 - 48*a^2*A^2*b^2*d^4*x^2 - 24*a^2*A
*b^2*B*d^4*x^2 - 6*a^2*b^2*B^2*d^4*x^2 + 32*A^2*b^4*c*d^3*x^3 + 16*A*b^4*B*c*d^3*x^3 + 4*b^4*B^2*c*d^3*x^3 - 3
2*a*A^2*b^3*d^4*x^3 - 16*a*A*b^3*B*d^4*x^3 - 4*a*b^3*B^2*d^4*x^3 + 16*a^4*A*B*d^4*Log[a + b*x] + 4*a^4*B^2*d^4
*Log[a + b*x] + 64*a^3*A*b*B*d^4*x*Log[a + b*x] + 16*a^3*b*B^2*d^4*x*Log[a + b*x] + 96*a^2*A*b^2*B*d^4*x^2*Log
[a + b*x] + 24*a^2*b^2*B^2*d^4*x^2*Log[a + b*x] + 64*a*A*b^3*B*d^4*x^3*Log[a + b*x] + 16*a*b^3*B^2*d^4*x^3*Log
[a + b*x] + 16*A*b^4*B*d^4*x^4*Log[a + b*x] + 4*b^4*B^2*d^4*x^4*Log[a + b*x] - 8*a^4*B^2*d^4*Log[a + b*x]^2 -
32*a^3*b*B^2*d^4*x*Log[a + b*x]^2 - 48*a^2*b^2*B^2*d^4*x^2*Log[a + b*x]^2 - 32*a*b^3*B^2*d^4*x^3*Log[a + b*x]^
2 - 8*b^4*B^2*d^4*x^4*Log[a + b*x]^2 + 16*A*b^4*B*c^4*Log[(e*(a + b*x))/(c + d*x)] + 4*b^4*B^2*c^4*Log[(e*(a +
b*x))/(c + d*x)] - 16*a^4*A*B*d^4*Log[(e*(a + b*x))/(c + d*x)] - 4*a^4*B^2*d^4*Log[(e*(a + b*x))/(c + d*x)] +
64*A*b^4*B*c^3*d*x*Log[(e*(a + b*x))/(c + d*x)] + 16*b^4*B^2*c^3*d*x*Log[(e*(a + b*x))/(c + d*x)] - 64*a^3*A*
b*B*d^4*x*Log[(e*(a + b*x))/(c + d*x)] - 16*a^3*b*B^2*d^4*x*Log[(e*(a + b*x))/(c + d*x)] + 96*A*b^4*B*c^2*d^2*
x^2*Log[(e*(a + b*x))/(c + d*x)] + 24*b^4*B^2*c^2*d^2*x^2*Log[(e*(a + b*x))/(c + d*x)] - 96*a^2*A*b^2*B*d^4*x^
2*Log[(e*(a + b*x))/(c + d*x)] - 24*a^2*b^2*B^2*d^4*x^2*Log[(e*(a + b*x))/(c + d*x)] + 64*A*b^4*B*c*d^3*x^3*Lo
g[(e*(a + b*x))/(c + d*x)] + 16*b^4*B^2*c*d^3*x^3*Log[(e*(a + b*x))/(c + d*x)] - 64*a*A*b^3*B*d^4*x^3*Log[(e*(
a + b*x))/(c + d*x)] - 16*a*b^3*B^2*d^4*x^3*Log[(e*(a + b*x))/(c + d*x)] + 16*a^4*B^2*d^4*Log[a + b*x]*Log[(e*
(a + b*x))/(c + d*x)] + 64*a^3*b*B^2*d^4*x*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] + 96*a^2*b^2*B^2*d^4*x^2*
Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] + 64*a*b^3*B^2*d^4*x^3*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] + 1
6*b^4*B^2*d^4*x^4*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] + 8*b^4*B^2*c^4*Log[(e*(a + b*x))/(c + d*x)]^2 - 8
*a^4*B^2*d^4*Log[(e*(a + b*x))/(c + d*x)]^2 + 32*b^4*B^2*c^3*d*x*Log[(e*(a + b*x))/(c + d*x)]^2 - 32*a^3*b*B^2
*d^4*x*Log[(e*(a + b*x))/(c + d*x)]^2 + 48*b^4*B^2*c^2*d^2*x^2*Log[(e*(a + b*x))/(c + d*x)]^2 - 48*a^2*b^2*B^2
*d^4*x^2*Log[(e*(a + b*x))/(c + d*x)]^2 + 32*b^4*B^2*c*d^3*x^3*Log[(e*(a + b*x))/(c + d*x)]^2 - 32*a*b^3*B^2*d
^4*x^3*Log[(e*(a + b*x))/(c + d*x)]^2 - 16*a^4*A*B*d^4*Log[c + d*x] - 4*a^4*B^2*d^4*Log[c + d*x] - 64*a^3*A*b*
B*d^4*x*Log[c + d*x] - 16*a^3*b*B^2*d^4*x*Log[c + d*x] - 96*a^2*A*b^2*B*d^4*x^2*Log[c + d*x] - 24*a^2*b^2*B^2*
d^4*x^2*Log[c + d*x] - 64*a*A*b^3*B*d^4*x^3*Log[c + d*x] - 16*a*b^3*B^2*d^4*x^3*Log[c + d*x] - 16*A*b^4*B*d^4*
x^4*Log[c + d*x] - 4*b^4*B^2*d^4*x^4*Log[c + d*x] + 16*a^4*B^2*d^4*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d
*x] + 64*a^3*b*B^2*d^4*x*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 96*a^2*b^2*B^2*d^4*x^2*Log[(d*(a + b
*x))/(-(b*c) + a*d)]*Log[c + d*x] + 64*a*b^3*B^2*d^4*x^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 16*b
^4*B^2*d^4*x^4*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] - 16*a^4*B^2*d^4*Log[(e*(a + b*x))/(c + d*x)]*Lo
g[c + d*x] - 64*a^3*b*B^2*d^4*x*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x] - 96*a^2*b^2*B^2*d^4*x^2*Log[(e*(a +
b*x))/(c + d*x)]*Log[c + d*x] - 64*a*b^3*B^2*d^4*x^3*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x] - 16*b^4*B^2*d
^4*x^4*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x] - 8*a^4*B^2*d^4*Log[c + d*x]^2 - 32*a^3*b*B^2*d^4*x*Log[c + d
*x]^2 - 48*a^2*b^2*B^2*d^4*x^2*Log[c + d*x]^2 - 32*a*b^3*B^2*d^4*x^3*Log[c + d*x]^2 - 8*b^4*B^2*d^4*x^4*Log[c
+ d*x]^2 + 16*a^4*B^2*d^4*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 64*a^3*b*B^2*d^4*x*Log[a + b*x]*Log[(b
*(c + d*x))/(b*c - a*d)] + 96*a^2*b^2*B^2*d^4*x^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 64*a*b^3*B^2*d
^4*x^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 16*b^4*B^2*d^4*x^4*Log[a + b*x]*Log[(b*(c + d*x))/(b*c -
a*d)] + 16*B^2*d^4*(a + b*x)^4*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 16*B^2*d^4*(a + b*x)^4*PolyLog[2, (b
*(c + d*x))/(b*c - a*d)]))/(32*b^4*(b*c - a*d)*g^5*(a + b*x)^4)

________________________________________________________________________________________

Maple [B]  time = 0.053, size = 890, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x)

[Out]

1/4*e^4*d*i^3/(a*d-b*c)^2/g^5*A^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-1/4*e^4*i^3/(a*d-b*c)^2/g^5*A^2/(b*e
/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*b*c+1/2*e^4*d*i^3/(a*d-b*c)^2/g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*
ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-1/2*e^4*i^3/(a*d-b*c)^2/g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e
/d+(a*d-b*c)*e/d/(d*x+c))*b*c+1/8*e^4*d*i^3/(a*d-b*c)^2/g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-1/8*e^
4*i^3/(a*d-b*c)^2/g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*b*c+1/4*e^4*d*i^3/(a*d-b*c)^2/g^5*B^2/(b*e/d+e
/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a-1/4*e^4*i^3/(a*d-b*c)^2/g^5*B^2/(b*e/d+e/(d*
x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*b*c+1/8*e^4*d*i^3/(a*d-b*c)^2/g^5*B^2/(b*e/d+e/(d*
x+c)*a-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-1/8*e^4*i^3/(a*d-b*c)^2/g^5*B^2/(b*e/d+e/(d*x+c)*a
-e/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b*c+1/32*e^4*d*i^3/(a*d-b*c)^2/g^5*B^2/(b*e/d+e/(d*x+c)*a-
e/d/(d*x+c)*b*c)^4*a-1/32*e^4*i^3/(a*d-b*c)^2/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*b*c

________________________________________________________________________________________

Maxima [B]  time = 6.38507, size = 15779, normalized size = 107.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

-1/4*(4*b*x + a)*B^2*c^2*d*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b
^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/4*(6*b^2*x^2 + 4*a*b*x + a^2)*B^2*c*d^2*i^3*log(b*e*x/(d*x + c
) + a*e/(d*x + c))^2/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) - 1/4
*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)*B^2*d^3*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^8*g^5*x^4 +
4*a*b^7*g^5*x^3 + 6*a^2*b^6*g^5*x^2 + 4*a^3*b^5*g^5*x + a^4*b^4*g^5) + 1/288*(12*((12*b^3*d^3*x^3 - 3*b^3*c^3
+ 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d
^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^
2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^
5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 -
3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^
3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*
d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - (9*b^4*c^4 - 64*a*b^3*c^3*d +
216*a^2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(13*b^4*c^2*d^2 - 17
6*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x +
a^4*d^4)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*lo
g(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*x - 300*(b^4*d^4*x^4 + 4
*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(25*b^4*d^4*x^4 + 100*a*b^3*d^
4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*x + 25*a^4*d^4 - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4
*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a))*log(d*x + c))/(a^4*b^5*c^4*g^5 - 4*a^5*b^4*c^3*d*g^5 + 6*a^6*b^3
*c^2*d^2*g^5 - 4*a^7*b^2*c*d^3*g^5 + a^8*b*d^4*g^5 + (b^9*c^4*g^5 - 4*a*b^8*c^3*d*g^5 + 6*a^2*b^7*c^2*d^2*g^5
- 4*a^3*b^6*c*d^3*g^5 + a^4*b^5*d^4*g^5)*x^4 + 4*(a*b^8*c^4*g^5 - 4*a^2*b^7*c^3*d*g^5 + 6*a^3*b^6*c^2*d^2*g^5
- 4*a^4*b^5*c*d^3*g^5 + a^5*b^4*d^4*g^5)*x^3 + 6*(a^2*b^7*c^4*g^5 - 4*a^3*b^6*c^3*d*g^5 + 6*a^4*b^5*c^2*d^2*g^
5 - 4*a^5*b^4*c*d^3*g^5 + a^6*b^3*d^4*g^5)*x^2 + 4*(a^3*b^6*c^4*g^5 - 4*a^4*b^5*c^3*d*g^5 + 6*a^5*b^4*c^2*d^2*
g^5 - 4*a^6*b^3*c*d^3*g^5 + a^7*b^2*d^4*g^5)*x))*B^2*c^3*i^3 - 1/288*(12*((7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75
*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3
)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*
b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6
*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d +
3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) +
12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*
d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^
3 + a^4*b^2*d^4)*g^5))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (37*a*b^4*c^4 - 304*a^2*b^3*c^3*d + 1512*a^3*b^2
*c^2*d^2 - 1360*a^4*b*c*d^3 + 115*a^5*d^4 + 12*(88*b^5*c^2*d^2 - 101*a*b^4*c*d^3 + 13*a^2*b^3*d^4)*x^3 - 6*(40
*b^5*c^3*d - 609*a*b^4*c^2*d^2 + 648*a^2*b^3*c*d^3 - 79*a^3*b^2*d^4)*x^2 - 72*(4*a^4*b*c*d^3 - a^5*d^4 + (4*b^
5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^3 + 6*(4*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2 + 4*(4*
a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(b*x + a)^2 - 72*(4*a^4*b*c*d^3 - a^5*d^4 + (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4
*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^3 + 6*(4*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2 + 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x
)*log(d*x + c)^2 + 4*(16*b^5*c^4 - 163*a*b^4*c^3*d + 1068*a^2*b^3*c^2*d^2 - 1036*a^3*b^2*c*d^3 + 115*a^4*b*d^4
)*x + 12*(88*a^4*b*c*d^3 - 13*a^5*d^4 + (88*b^5*c*d^3 - 13*a*b^4*d^4)*x^4 + 4*(88*a*b^4*c*d^3 - 13*a^2*b^3*d^4
)*x^3 + 6*(88*a^2*b^3*c*d^3 - 13*a^3*b^2*d^4)*x^2 + 4*(88*a^3*b^2*c*d^3 - 13*a^4*b*d^4)*x)*log(b*x + a) - 12*(
88*a^4*b*c*d^3 - 13*a^5*d^4 + (88*b^5*c*d^3 - 13*a*b^4*d^4)*x^4 + 4*(88*a*b^4*c*d^3 - 13*a^2*b^3*d^4)*x^3 + 6*
(88*a^2*b^3*c*d^3 - 13*a^3*b^2*d^4)*x^2 + 4*(88*a^3*b^2*c*d^3 - 13*a^4*b*d^4)*x - 12*(4*a^4*b*c*d^3 - a^5*d^4
+ (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^3 + 6*(4*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2
+ 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(b*x + a))*log(d*x + c))/(a^4*b^6*c^4*g^5 - 4*a^5*b^5*c^3*d*g^5 + 6*a^
6*b^4*c^2*d^2*g^5 - 4*a^7*b^3*c*d^3*g^5 + a^8*b^2*d^4*g^5 + (b^10*c^4*g^5 - 4*a*b^9*c^3*d*g^5 + 6*a^2*b^8*c^2*
d^2*g^5 - 4*a^3*b^7*c*d^3*g^5 + a^4*b^6*d^4*g^5)*x^4 + 4*(a*b^9*c^4*g^5 - 4*a^2*b^8*c^3*d*g^5 + 6*a^3*b^7*c^2*
d^2*g^5 - 4*a^4*b^6*c*d^3*g^5 + a^5*b^5*d^4*g^5)*x^3 + 6*(a^2*b^8*c^4*g^5 - 4*a^3*b^7*c^3*d*g^5 + 6*a^4*b^6*c^
2*d^2*g^5 - 4*a^5*b^5*c*d^3*g^5 + a^6*b^4*d^4*g^5)*x^2 + 4*(a^3*b^7*c^4*g^5 - 4*a^4*b^6*c^3*d*g^5 + 6*a^5*b^5*
c^2*d^2*g^5 - 4*a^6*b^4*c*d^3*g^5 + a^7*b^3*d^4*g^5)*x))*B^2*c^2*d*i^3 - 1/288*(12*((13*a^2*b^3*c^3 - 75*a^3*b
^2*c^2*d + 33*a^4*b*c*d^2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 6*(6*b^5*c^3 - 46
*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 + 4*(10*a*b^4*c^3 - 63*a^2*b^3*c^2*d + 33*a^3*b^2*c*d^2 -
7*a^4*b*d^3)*x)/((b^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5*x^4 + 4*(a*b^9*c^3 - 3*a^2*b^
8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^
5*d^3)*g^5*x^2 + 4*(a^3*b^7*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*g^5*x + (a^4*b^6*c^3 - 3*a^
5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(b*x + a)/((
b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5) + 12*(6*b^2*c^2*d^2 - 4*a*b*
c*d^3 + a^2*d^4)*log(d*x + c)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g
^5))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (115*a^2*b^4*c^4 - 1360*a^3*b^3*c^3*d + 1512*a^4*b^2*c^2*d^2 - 304
*a^5*b*c*d^3 + 37*a^6*d^4 - 12*(108*b^6*c^3*d - 148*a*b^5*c^2*d^2 + 47*a^2*b^4*c*d^3 - 7*a^3*b^3*d^4)*x^3 + 6*
(36*b^6*c^4 - 712*a*b^5*c^3*d + 903*a^2*b^4*c^2*d^2 - 264*a^3*b^3*c*d^3 + 37*a^4*b^2*d^4)*x^2 + 72*(6*a^4*b^2*
c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4 + (6*b^6*c^2*d^2 - 4*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 + 4*(6*a*b^5*c^2*d^2 - 4
*a^2*b^4*c*d^3 + a^3*b^3*d^4)*x^3 + 6*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^2 + 4*(6*a^3*b^3*c
^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4)*x)*log(b*x + a)^2 + 72*(6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4 + (6
*b^6*c^2*d^2 - 4*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 + 4*(6*a*b^5*c^2*d^2 - 4*a^2*b^4*c*d^3 + a^3*b^3*d^4)*x^3 + 6*
(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^2 + 4*(6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4)*
x)*log(d*x + c)^2 + 4*(76*a*b^5*c^4 - 1057*a^2*b^4*c^3*d + 1248*a^3*b^3*c^2*d^2 - 304*a^4*b^2*c*d^3 + 37*a^5*b
*d^4)*x - 12*(108*a^4*b^2*c^2*d^2 - 40*a^5*b*c*d^3 + 7*a^6*d^4 + (108*b^6*c^2*d^2 - 40*a*b^5*c*d^3 + 7*a^2*b^4
*d^4)*x^4 + 4*(108*a*b^5*c^2*d^2 - 40*a^2*b^4*c*d^3 + 7*a^3*b^3*d^4)*x^3 + 6*(108*a^2*b^4*c^2*d^2 - 40*a^3*b^3
*c*d^3 + 7*a^4*b^2*d^4)*x^2 + 4*(108*a^3*b^3*c^2*d^2 - 40*a^4*b^2*c*d^3 + 7*a^5*b*d^4)*x)*log(b*x + a) + 12*(1
08*a^4*b^2*c^2*d^2 - 40*a^5*b*c*d^3 + 7*a^6*d^4 + (108*b^6*c^2*d^2 - 40*a*b^5*c*d^3 + 7*a^2*b^4*d^4)*x^4 + 4*(
108*a*b^5*c^2*d^2 - 40*a^2*b^4*c*d^3 + 7*a^3*b^3*d^4)*x^3 + 6*(108*a^2*b^4*c^2*d^2 - 40*a^3*b^3*c*d^3 + 7*a^4*
b^2*d^4)*x^2 + 4*(108*a^3*b^3*c^2*d^2 - 40*a^4*b^2*c*d^3 + 7*a^5*b*d^4)*x - 12*(6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*
d^3 + a^6*d^4 + (6*b^6*c^2*d^2 - 4*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 + 4*(6*a*b^5*c^2*d^2 - 4*a^2*b^4*c*d^3 + a^3
*b^3*d^4)*x^3 + 6*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^2 + 4*(6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c
*d^3 + a^5*b*d^4)*x)*log(b*x + a))*log(d*x + c))/(a^4*b^7*c^4*g^5 - 4*a^5*b^6*c^3*d*g^5 + 6*a^6*b^5*c^2*d^2*g^
5 - 4*a^7*b^4*c*d^3*g^5 + a^8*b^3*d^4*g^5 + (b^11*c^4*g^5 - 4*a*b^10*c^3*d*g^5 + 6*a^2*b^9*c^2*d^2*g^5 - 4*a^3
*b^8*c*d^3*g^5 + a^4*b^7*d^4*g^5)*x^4 + 4*(a*b^10*c^4*g^5 - 4*a^2*b^9*c^3*d*g^5 + 6*a^3*b^8*c^2*d^2*g^5 - 4*a^
4*b^7*c*d^3*g^5 + a^5*b^6*d^4*g^5)*x^3 + 6*(a^2*b^9*c^4*g^5 - 4*a^3*b^8*c^3*d*g^5 + 6*a^4*b^7*c^2*d^2*g^5 - 4*
a^5*b^6*c*d^3*g^5 + a^6*b^5*d^4*g^5)*x^2 + 4*(a^3*b^8*c^4*g^5 - 4*a^4*b^7*c^3*d*g^5 + 6*a^5*b^6*c^2*d^2*g^5 -
4*a^6*b^5*c*d^3*g^5 + a^7*b^4*d^4*g^5)*x))*B^2*c*d^2*i^3 - 1/288*(12*((25*a^3*b^3*c^3 - 23*a^4*b^2*c^2*d + 13*
a^5*b*c*d^2 - 3*a^6*d^3 + 12*(4*b^6*c^3 - 6*a*b^5*c^2*d + 4*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^3 + 6*(18*a*b^5*c^3
- 22*a^2*b^4*c^2*d + 13*a^3*b^3*c*d^2 - 3*a^4*b^2*d^3)*x^2 + 4*(22*a^2*b^4*c^3 - 23*a^3*b^3*c^2*d + 13*a^4*b^
2*c*d^2 - 3*a^5*b*d^3)*x)/((b^11*c^3 - 3*a*b^10*c^2*d + 3*a^2*b^9*c*d^2 - a^3*b^8*d^3)*g^5*x^4 + 4*(a*b^10*c^3
- 3*a^2*b^9*c^2*d + 3*a^3*b^8*c*d^2 - a^4*b^7*d^3)*g^5*x^3 + 6*(a^2*b^9*c^3 - 3*a^3*b^8*c^2*d + 3*a^4*b^7*c*d
^2 - a^5*b^6*d^3)*g^5*x^2 + 4*(a^3*b^8*c^3 - 3*a^4*b^7*c^2*d + 3*a^5*b^6*c*d^2 - a^6*b^5*d^3)*g^5*x + (a^4*b^7
*c^3 - 3*a^5*b^6*c^2*d + 3*a^6*b^5*c*d^2 - a^7*b^4*d^3)*g^5) + 12*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d
^3 - a^3*d^4)*log(b*x + a)/((b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*g^5)
- 12*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*log(d*x + c)/((b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2
*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*g^5))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (415*a^3*b^4*c^4 -
576*a^4*b^3*c^3*d + 216*a^5*b^2*c^2*d^2 - 64*a^6*b*c*d^3 + 9*a^7*d^4 + 12*(48*b^7*c^4 - 84*a*b^6*c^3*d + 52*a^
2*b^5*c^2*d^2 - 19*a^3*b^4*c*d^3 + 3*a^4*b^3*d^4)*x^3 + 6*(252*a*b^6*c^4 - 400*a^2*b^5*c^3*d + 203*a^3*b^4*c^2
*d^2 - 64*a^4*b^3*c*d^3 + 9*a^5*b^2*d^4)*x^2 - 72*(4*a^4*b^3*c^3*d - 6*a^5*b^2*c^2*d^2 + 4*a^6*b*c*d^3 - a^7*d
^4 + (4*b^7*c^3*d - 6*a*b^6*c^2*d^2 + 4*a^2*b^5*c*d^3 - a^3*b^4*d^4)*x^4 + 4*(4*a*b^6*c^3*d - 6*a^2*b^5*c^2*d^
2 + 4*a^3*b^4*c*d^3 - a^4*b^3*d^4)*x^3 + 6*(4*a^2*b^5*c^3*d - 6*a^3*b^4*c^2*d^2 + 4*a^4*b^3*c*d^3 - a^5*b^2*d^
4)*x^2 + 4*(4*a^3*b^4*c^3*d - 6*a^4*b^3*c^2*d^2 + 4*a^5*b^2*c*d^3 - a^6*b*d^4)*x)*log(b*x + a)^2 - 72*(4*a^4*b
^3*c^3*d - 6*a^5*b^2*c^2*d^2 + 4*a^6*b*c*d^3 - a^7*d^4 + (4*b^7*c^3*d - 6*a*b^6*c^2*d^2 + 4*a^2*b^5*c*d^3 - a^
3*b^4*d^4)*x^4 + 4*(4*a*b^6*c^3*d - 6*a^2*b^5*c^2*d^2 + 4*a^3*b^4*c*d^3 - a^4*b^3*d^4)*x^3 + 6*(4*a^2*b^5*c^3*
d - 6*a^3*b^4*c^2*d^2 + 4*a^4*b^3*c*d^3 - a^5*b^2*d^4)*x^2 + 4*(4*a^3*b^4*c^3*d - 6*a^4*b^3*c^2*d^2 + 4*a^5*b^
2*c*d^3 - a^6*b*d^4)*x)*log(d*x + c)^2 + 4*(340*a^2*b^5*c^4 - 501*a^3*b^4*c^3*d + 216*a^4*b^3*c^2*d^2 - 64*a^5
*b^2*c*d^3 + 9*a^6*b*d^4)*x + 12*(48*a^4*b^3*c^3*d - 36*a^5*b^2*c^2*d^2 + 16*a^6*b*c*d^3 - 3*a^7*d^4 + (48*b^7
*c^3*d - 36*a*b^6*c^2*d^2 + 16*a^2*b^5*c*d^3 - 3*a^3*b^4*d^4)*x^4 + 4*(48*a*b^6*c^3*d - 36*a^2*b^5*c^2*d^2 + 1
6*a^3*b^4*c*d^3 - 3*a^4*b^3*d^4)*x^3 + 6*(48*a^2*b^5*c^3*d - 36*a^3*b^4*c^2*d^2 + 16*a^4*b^3*c*d^3 - 3*a^5*b^2
*d^4)*x^2 + 4*(48*a^3*b^4*c^3*d - 36*a^4*b^3*c^2*d^2 + 16*a^5*b^2*c*d^3 - 3*a^6*b*d^4)*x)*log(b*x + a) - 12*(4
8*a^4*b^3*c^3*d - 36*a^5*b^2*c^2*d^2 + 16*a^6*b*c*d^3 - 3*a^7*d^4 + (48*b^7*c^3*d - 36*a*b^6*c^2*d^2 + 16*a^2*
b^5*c*d^3 - 3*a^3*b^4*d^4)*x^4 + 4*(48*a*b^6*c^3*d - 36*a^2*b^5*c^2*d^2 + 16*a^3*b^4*c*d^3 - 3*a^4*b^3*d^4)*x^
3 + 6*(48*a^2*b^5*c^3*d - 36*a^3*b^4*c^2*d^2 + 16*a^4*b^3*c*d^3 - 3*a^5*b^2*d^4)*x^2 + 4*(48*a^3*b^4*c^3*d - 3
6*a^4*b^3*c^2*d^2 + 16*a^5*b^2*c*d^3 - 3*a^6*b*d^4)*x - 12*(4*a^4*b^3*c^3*d - 6*a^5*b^2*c^2*d^2 + 4*a^6*b*c*d^
3 - a^7*d^4 + (4*b^7*c^3*d - 6*a*b^6*c^2*d^2 + 4*a^2*b^5*c*d^3 - a^3*b^4*d^4)*x^4 + 4*(4*a*b^6*c^3*d - 6*a^2*b
^5*c^2*d^2 + 4*a^3*b^4*c*d^3 - a^4*b^3*d^4)*x^3 + 6*(4*a^2*b^5*c^3*d - 6*a^3*b^4*c^2*d^2 + 4*a^4*b^3*c*d^3 - a
^5*b^2*d^4)*x^2 + 4*(4*a^3*b^4*c^3*d - 6*a^4*b^3*c^2*d^2 + 4*a^5*b^2*c*d^3 - a^6*b*d^4)*x)*log(b*x + a))*log(d
*x + c))/(a^4*b^8*c^4*g^5 - 4*a^5*b^7*c^3*d*g^5 + 6*a^6*b^6*c^2*d^2*g^5 - 4*a^7*b^5*c*d^3*g^5 + a^8*b^4*d^4*g^
5 + (b^12*c^4*g^5 - 4*a*b^11*c^3*d*g^5 + 6*a^2*b^10*c^2*d^2*g^5 - 4*a^3*b^9*c*d^3*g^5 + a^4*b^8*d^4*g^5)*x^4 +
4*(a*b^11*c^4*g^5 - 4*a^2*b^10*c^3*d*g^5 + 6*a^3*b^9*c^2*d^2*g^5 - 4*a^4*b^8*c*d^3*g^5 + a^5*b^7*d^4*g^5)*x^3
+ 6*(a^2*b^10*c^4*g^5 - 4*a^3*b^9*c^3*d*g^5 + 6*a^4*b^8*c^2*d^2*g^5 - 4*a^5*b^7*c*d^3*g^5 + a^6*b^6*d^4*g^5)*
x^2 + 4*(a^3*b^9*c^4*g^5 - 4*a^4*b^8*c^3*d*g^5 + 6*a^5*b^7*c^2*d^2*g^5 - 4*a^6*b^6*c*d^3*g^5 + a^7*b^5*d^4*g^5
)*x))*B^2*d^3*i^3 - 1/24*A*B*d^3*i^3*(12*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)*log(b*e*x/(d*x + c) + a*e
/(d*x + c))/(b^8*g^5*x^4 + 4*a*b^7*g^5*x^3 + 6*a^2*b^6*g^5*x^2 + 4*a^3*b^5*g^5*x + a^4*b^4*g^5) + (25*a^3*b^3*
c^3 - 23*a^4*b^2*c^2*d + 13*a^5*b*c*d^2 - 3*a^6*d^3 + 12*(4*b^6*c^3 - 6*a*b^5*c^2*d + 4*a^2*b^4*c*d^2 - a^3*b^
3*d^3)*x^3 + 6*(18*a*b^5*c^3 - 22*a^2*b^4*c^2*d + 13*a^3*b^3*c*d^2 - 3*a^4*b^2*d^3)*x^2 + 4*(22*a^2*b^4*c^3 -
23*a^3*b^3*c^2*d + 13*a^4*b^2*c*d^2 - 3*a^5*b*d^3)*x)/((b^11*c^3 - 3*a*b^10*c^2*d + 3*a^2*b^9*c*d^2 - a^3*b^8*
d^3)*g^5*x^4 + 4*(a*b^10*c^3 - 3*a^2*b^9*c^2*d + 3*a^3*b^8*c*d^2 - a^4*b^7*d^3)*g^5*x^3 + 6*(a^2*b^9*c^3 - 3*a
^3*b^8*c^2*d + 3*a^4*b^7*c*d^2 - a^5*b^6*d^3)*g^5*x^2 + 4*(a^3*b^8*c^3 - 3*a^4*b^7*c^2*d + 3*a^5*b^6*c*d^2 - a
^6*b^5*d^3)*g^5*x + (a^4*b^7*c^3 - 3*a^5*b^6*c^2*d + 3*a^6*b^5*c*d^2 - a^7*b^4*d^3)*g^5) + 12*(4*b^3*c^3*d - 6
*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*log(b*x + a)/((b^8*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b
^5*c*d^3 + a^4*b^4*d^4)*g^5) - 12*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*log(d*x + c)/((b^8
*c^4 - 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*g^5)) - 1/24*A*B*c*d^2*i^3*(12*(6*b^
2*x^2 + 4*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2
+ 4*a^3*b^4*g^5*x + a^4*b^3*g^5) + (13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*d^2 - 7*a^5*d^3 - 12*(6*b^
5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 6*(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)
*x^2 + 4*(10*a*b^4*c^3 - 63*a^2*b^3*c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^10*c^3 - 3*a*b^9*c^2*d + 3*
a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5*x^4 + 4*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3
+ 6*(a^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^3*b^7*c^3 - 3*a^4*b^6*c^2*
d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*g^5*x + (a^4*b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5
) - 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*
a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5) + 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(d*x + c)/((b^7*c^4 - 4*a*b^
6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5)) - 1/24*A*B*c^2*d*i^3*(12*(4*b*x + a)*log(b*
e*x/(d*x + c) + a*e/(d*x + c))/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*
g^5) + (7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4
*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b
*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d +
3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^
5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2
*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a
^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5
*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)) + 1/24*A*B*c^3*i^3*((12*b^3*d^3*x^3 - 3*b^3*
c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*
c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3
*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 -
a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3
- 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) - 12*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^5*x^4
+ 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4
*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d
+ 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 1/4*B^2*c^3*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c
))^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) - 1/4*(4*b*x + a)*A^2*c
^2*d*i^3/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/4*(6*b^2*x^2
+ 4*a*b*x + a^2)*A^2*c*d^2*i^3/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*
g^5) - 1/4*(4*b^3*x^3 + 6*a*b^2*x^2 + 4*a^2*b*x + a^3)*A^2*d^3*i^3/(b^8*g^5*x^4 + 4*a*b^7*g^5*x^3 + 6*a^2*b^6*
g^5*x^2 + 4*a^3*b^5*g^5*x + a^4*b^4*g^5) - 1/4*A^2*c^3*i^3/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2
+ 4*a^3*b^2*g^5*x + a^4*b*g^5)

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Fricas [B]  time = 0.584212, size = 1127, normalized size = 7.67 \begin{align*} -\frac{4 \,{\left ({\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c d^{3} -{\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} a b^{3} d^{4}\right )} i^{3} x^{3} + 6 \,{\left ({\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{2} d^{2} -{\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} a^{2} b^{2} d^{4}\right )} i^{3} x^{2} + 4 \,{\left ({\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{3} d -{\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} a^{3} b d^{4}\right )} i^{3} x +{\left ({\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{4} -{\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} a^{4} d^{4}\right )} i^{3} + 8 \,{\left (B^{2} b^{4} d^{4} i^{3} x^{4} + 4 \, B^{2} b^{4} c d^{3} i^{3} x^{3} + 6 \, B^{2} b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B^{2} b^{4} c^{3} d i^{3} x + B^{2} b^{4} c^{4} i^{3}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 4 \,{\left ({\left (4 \, A B + B^{2}\right )} b^{4} d^{4} i^{3} x^{4} + 4 \,{\left (4 \, A B + B^{2}\right )} b^{4} c d^{3} i^{3} x^{3} + 6 \,{\left (4 \, A B + B^{2}\right )} b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \,{\left (4 \, A B + B^{2}\right )} b^{4} c^{3} d i^{3} x +{\left (4 \, A B + B^{2}\right )} b^{4} c^{4} i^{3}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{32 \,{\left ({\left (b^{9} c - a b^{8} d\right )} g^{5} x^{4} + 4 \,{\left (a b^{8} c - a^{2} b^{7} d\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{7} c - a^{3} b^{6} d\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{6} c - a^{4} b^{5} d\right )} g^{5} x +{\left (a^{4} b^{5} c - a^{5} b^{4} d\right )} g^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/32*(4*((8*A^2 + 4*A*B + B^2)*b^4*c*d^3 - (8*A^2 + 4*A*B + B^2)*a*b^3*d^4)*i^3*x^3 + 6*((8*A^2 + 4*A*B + B^2
)*b^4*c^2*d^2 - (8*A^2 + 4*A*B + B^2)*a^2*b^2*d^4)*i^3*x^2 + 4*((8*A^2 + 4*A*B + B^2)*b^4*c^3*d - (8*A^2 + 4*A
*B + B^2)*a^3*b*d^4)*i^3*x + ((8*A^2 + 4*A*B + B^2)*b^4*c^4 - (8*A^2 + 4*A*B + B^2)*a^4*d^4)*i^3 + 8*(B^2*b^4*
d^4*i^3*x^4 + 4*B^2*b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*c^3*d*i^3*x + B^2*b^4*c^4*i^3)*l
og((b*e*x + a*e)/(d*x + c))^2 + 4*((4*A*B + B^2)*b^4*d^4*i^3*x^4 + 4*(4*A*B + B^2)*b^4*c*d^3*i^3*x^3 + 6*(4*A*
B + B^2)*b^4*c^2*d^2*i^3*x^2 + 4*(4*A*B + B^2)*b^4*c^3*d*i^3*x + (4*A*B + B^2)*b^4*c^4*i^3)*log((b*e*x + a*e)/
(d*x + c)))/((b^9*c - a*b^8*d)*g^5*x^4 + 4*(a*b^8*c - a^2*b^7*d)*g^5*x^3 + 6*(a^2*b^7*c - a^3*b^6*d)*g^5*x^2 +
4*(a^3*b^6*c - a^4*b^5*d)*g^5*x + (a^4*b^5*c - a^5*b^4*d)*g^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}^{3}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

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