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

Optimal. Leaf size=299 \[ -\frac{b i^2 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^2}-\frac{b B i^2 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^2}+\frac{d i^2 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^5 (a+b x)^3 (b c-a d)^2}+\frac{2 B d i^2 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{9 g^5 (a+b x)^3 (b c-a d)^2}-\frac{b B^2 i^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^2}+\frac{2 B^2 d i^2 (c+d x)^3}{27 g^5 (a+b x)^3 (b c-a d)^2} \]

[Out]

(2*B^2*d*i^2*(c + d*x)^3)/(27*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*B^2*i^2*(c + d*x)^4)/(32*(b*c - a*d)^2*g^5*(
a + b*x)^4) + (2*B*d*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)^2*g^5*(a + b*x)^3) -
(b*B*i^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(b*c - a*d)^2*g^5*(a + b*x)^4) + (d*i^2*(c + d*
x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*i^2*(c + d*x)^4*(A + B*Log
[(e*(a + b*x))/(c + d*x)])^2)/(4*(b*c - a*d)^2*g^5*(a + b*x)^4)

________________________________________________________________________________________

Rubi [C]  time = 3.69071, antiderivative size = 920, normalized size of antiderivative = 3.08, number of steps used = 104, 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} \[ -\frac{B^2 i^2 \log ^2(a+b x) d^4}{12 b^3 (b c-a d)^2 g^5}-\frac{B^2 i^2 \log ^2(c+d x) d^4}{12 b^3 (b c-a d)^2 g^5}+\frac{7 B^2 i^2 \log (a+b x) d^4}{72 b^3 (b c-a d)^2 g^5}+\frac{B i^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^4}{6 b^3 (b c-a d)^2 g^5}-\frac{7 B^2 i^2 \log (c+d x) d^4}{72 b^3 (b c-a d)^2 g^5}+\frac{B^2 i^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{6 b^3 (b c-a d)^2 g^5}-\frac{B i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac{B^2 i^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac{B^2 i^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac{B^2 i^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac{B i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^3}{6 b^3 (b c-a d) g^5 (a+b x)}+\frac{7 B^2 i^2 d^3}{72 b^3 (b c-a d) g^5 (a+b x)}-\frac{i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 d^2}{2 b^3 g^5 (a+b x)^2}-\frac{B i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^2}{12 b^3 g^5 (a+b x)^2}+\frac{5 B^2 i^2 d^2}{144 b^3 g^5 (a+b x)^2}-\frac{2 (b c-a d) i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 d}{3 b^3 g^5 (a+b x)^3}-\frac{5 B (b c-a d) i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d}{18 b^3 g^5 (a+b x)^3}-\frac{11 B^2 (b c-a d) i^2 d}{216 b^3 g^5 (a+b x)^3}-\frac{(b c-a d)^2 i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^3 g^5 (a+b x)^4}-\frac{B (b c-a d)^2 i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^3 g^5 (a+b x)^4}-\frac{B^2 (b c-a d)^2 i^2}{32 b^3 g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^2*(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)^2*i^2)/(32*b^3*g^5*(a + b*x)^4) - (11*B^2*d*(b*c - a*d)*i^2)/(216*b^3*g^5*(a + b*x)^3) + (5*
B^2*d^2*i^2)/(144*b^3*g^5*(a + b*x)^2) + (7*B^2*d^3*i^2)/(72*b^3*(b*c - a*d)*g^5*(a + b*x)) + (7*B^2*d^4*i^2*L
og[a + b*x])/(72*b^3*(b*c - a*d)^2*g^5) - (B^2*d^4*i^2*Log[a + b*x]^2)/(12*b^3*(b*c - a*d)^2*g^5) - (B*(b*c -
a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*b^3*g^5*(a + b*x)^4) - (5*B*d*(b*c - a*d)*i^2*(A + B*Log[(
e*(a + b*x))/(c + d*x)]))/(18*b^3*g^5*(a + b*x)^3) - (B*d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^3*
g^5*(a + b*x)^2) + (B*d^3*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3*(b*c - a*d)*g^5*(a + b*x)) + (B*d^4
*i^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3*(b*c - a*d)^2*g^5) - ((b*c - a*d)^2*i^2*(A + B*
Log[(e*(a + b*x))/(c + d*x)])^2)/(4*b^3*g^5*(a + b*x)^4) - (2*d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c +
d*x)])^2)/(3*b^3*g^5*(a + b*x)^3) - (d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^3*g^5*(a + b*x)^2) -
(7*B^2*d^4*i^2*Log[c + d*x])/(72*b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[
c + d*x])/(6*b^3*(b*c - a*d)^2*g^5) - (B*d^4*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(6*b^3*(b*
c - a*d)^2*g^5) - (B^2*d^4*i^2*Log[c + d*x]^2)/(12*b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^2*Log[a + b*x]*Log[(b*(
c + d*x))/(b*c - a*d)])/(6*b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(6*
b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(6*b^3*(b*c - a*d)^2*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{(72 c+72 d x)^2 \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{5184 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^5}+\frac{10368 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^4}+\frac{5184 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}\right ) \, dx\\ &=\frac{\left (5184 d^2\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^2 g^5}+\frac{(10368 d (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^2 g^5}+\frac{\left (5184 (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)^5} \, dx}{b^2 g^5}\\ &=-\frac{1296 (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{3456 d (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{2592 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)^2}+\frac{\left (5184 B 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)^3 (c+d x)} \, dx}{b^3 g^5}+\frac{(6912 B d (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}{b^3 g^5}+\frac{\left (2592 B (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)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac{1296 (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{3456 d (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{2592 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)^2}+\frac{\left (5184 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 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (6912 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 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (2592 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 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac{1296 (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{3456 d (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{2592 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)^2}+\frac{\left (5184 B d^2 (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)^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^3 g^5}+\frac{\left (6912 B d (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}{b^3 g^5}+\frac{\left (2592 B (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)^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}{b^3 g^5}\\ &=-\frac{1296 (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{3456 d (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{2592 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)^2}+\frac{\left (2592 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^5}+\frac{\left (5184 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^5}-\frac{\left (6912 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^5}+\frac{\left (2592 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (5184 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac{\left (6912 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac{\left (2592 B d^5\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (5184 B d^5\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac{\left (6912 B d^5\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (2592 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^2 (b c-a d) g^5}-\frac{\left (5184 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^2 (b c-a d) g^5}+\frac{\left (6912 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^2 (b c-a d) g^5}-\frac{(2592 B d (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^2 g^5}+\frac{(6912 B d (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^2 g^5}+\frac{\left (2592 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)^5} \, dx}{b^2 g^5}\\ &=-\frac{648 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{1440 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{432 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{864 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac{864 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac{1296 (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{3456 d (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{2592 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)^2}-\frac{864 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac{\left (1296 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (2592 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac{\left (3456 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac{\left (2592 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^3 (b c-a d)^2 g^5}+\frac{\left (2592 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^3 (b c-a d)^2 g^5}-\frac{\left (5184 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^3 (b c-a d)^2 g^5}+\frac{\left (5184 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^3 (b c-a d)^2 g^5}+\frac{\left (6912 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^3 (b c-a d)^2 g^5}-\frac{\left (6912 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^3 (b c-a d)^2 g^5}-\frac{\left (2592 B^2 d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}-\frac{\left (5184 B^2 d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}+\frac{\left (6912 B^2 d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}-\frac{\left (864 B^2 d (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (2304 B^2 d (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (648 B^2 (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac{648 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{1440 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{432 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{864 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac{864 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac{1296 (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{3456 d (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{2592 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)^2}-\frac{864 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac{\left (2592 B^2 d^3\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}-\frac{\left (5184 B^2 d^3\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (6912 B^2 d^3\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (1296 B^2 d^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (2592 B^2 d^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac{\left (3456 B^2 d^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac{\left (864 B^2 d (b c-a d)^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (2304 B^2 d (b c-a d)^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac{\left (648 B^2 (b c-a d)^3\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}-\frac{\left (2592 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^3 (b c-a d)^2 e g^5}+\frac{\left (2592 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^3 (b c-a d)^2 e g^5}-\frac{\left (5184 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^3 (b c-a d)^2 e g^5}+\frac{\left (5184 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^3 (b c-a d)^2 e g^5}+\frac{\left (6912 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^3 (b c-a d)^2 e g^5}-\frac{\left (6912 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^3 (b c-a d)^2 e g^5}\\ &=-\frac{648 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{1440 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{432 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{864 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac{864 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac{1296 (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{3456 d (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{2592 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)^2}-\frac{864 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac{\left (2592 B^2 d^3\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^3 g^5}-\frac{\left (5184 B^2 d^3\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^3 g^5}+\frac{\left (6912 B^2 d^3\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^3 g^5}+\frac{\left (1296 B^2 d^2 (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^3 g^5}+\frac{\left (2592 B^2 d^2 (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^3 g^5}-\frac{\left (3456 B^2 d^2 (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^3 g^5}-\frac{\left (864 B^2 d (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}{b^3 g^5}+\frac{\left (2304 B^2 d (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}{b^3 g^5}+\frac{\left (648 B^2 (b c-a d)^3\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}{b^3 g^5}-\frac{\left (2592 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^3 (b c-a d)^2 e g^5}+\frac{\left (2592 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^3 (b c-a d)^2 e g^5}-\frac{\left (5184 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^3 (b c-a d)^2 e g^5}+\frac{\left (5184 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^3 (b c-a d)^2 e g^5}+\frac{\left (6912 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^3 (b c-a d)^2 e g^5}-\frac{\left (6912 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^3 (b c-a d)^2 e g^5}\\ &=-\frac{162 B^2 (b c-a d)^2}{b^3 g^5 (a+b x)^4}-\frac{264 B^2 d (b c-a d)}{b^3 g^5 (a+b x)^3}+\frac{180 B^2 d^2}{b^3 g^5 (a+b x)^2}+\frac{504 B^2 d^3}{b^3 (b c-a d) g^5 (a+b x)}+\frac{504 B^2 d^4 \log (a+b x)}{b^3 (b c-a d)^2 g^5}-\frac{648 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{1440 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{432 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{864 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac{864 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac{1296 (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{3456 d (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{2592 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)^2}-\frac{504 B^2 d^4 \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac{864 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac{\left (2592 B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (2592 B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac{\left (5184 B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (5184 B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (6912 B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac{\left (6912 B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (2592 B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (2592 B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac{\left (5184 B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (5184 B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (6912 B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac{\left (6912 B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}\\ &=-\frac{162 B^2 (b c-a d)^2}{b^3 g^5 (a+b x)^4}-\frac{264 B^2 d (b c-a d)}{b^3 g^5 (a+b x)^3}+\frac{180 B^2 d^2}{b^3 g^5 (a+b x)^2}+\frac{504 B^2 d^3}{b^3 (b c-a d) g^5 (a+b x)}+\frac{504 B^2 d^4 \log (a+b x)}{b^3 (b c-a d)^2 g^5}-\frac{648 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{1440 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{432 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{864 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac{864 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac{1296 (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{3456 d (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{2592 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)^2}-\frac{504 B^2 d^4 \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac{864 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac{864 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac{864 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (2592 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (2592 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (5184 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (5184 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}+\frac{\left (6912 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}+\frac{\left (6912 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (2592 B^2 d^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac{\left (5184 B^2 d^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (6912 B^2 d^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac{\left (2592 B^2 d^5\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac{\left (5184 B^2 d^5\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac{\left (6912 B^2 d^5\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}\\ &=-\frac{162 B^2 (b c-a d)^2}{b^3 g^5 (a+b x)^4}-\frac{264 B^2 d (b c-a d)}{b^3 g^5 (a+b x)^3}+\frac{180 B^2 d^2}{b^3 g^5 (a+b x)^2}+\frac{504 B^2 d^3}{b^3 (b c-a d) g^5 (a+b x)}+\frac{504 B^2 d^4 \log (a+b x)}{b^3 (b c-a d)^2 g^5}-\frac{432 B^2 d^4 \log ^2(a+b x)}{b^3 (b c-a d)^2 g^5}-\frac{648 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{1440 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{432 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{864 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac{864 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac{1296 (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{3456 d (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{2592 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)^2}-\frac{504 B^2 d^4 \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac{864 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac{864 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac{432 B^2 d^4 \log ^2(c+d x)}{b^3 (b c-a d)^2 g^5}+\frac{864 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}-\frac{\left (2592 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^3 (b c-a d)^2 g^5}-\frac{\left (2592 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^3 (b c-a d)^2 g^5}-\frac{\left (5184 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^3 (b c-a d)^2 g^5}-\frac{\left (5184 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^3 (b c-a d)^2 g^5}+\frac{\left (6912 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^3 (b c-a d)^2 g^5}+\frac{\left (6912 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^3 (b c-a d)^2 g^5}\\ &=-\frac{162 B^2 (b c-a d)^2}{b^3 g^5 (a+b x)^4}-\frac{264 B^2 d (b c-a d)}{b^3 g^5 (a+b x)^3}+\frac{180 B^2 d^2}{b^3 g^5 (a+b x)^2}+\frac{504 B^2 d^3}{b^3 (b c-a d) g^5 (a+b x)}+\frac{504 B^2 d^4 \log (a+b x)}{b^3 (b c-a d)^2 g^5}-\frac{432 B^2 d^4 \log ^2(a+b x)}{b^3 (b c-a d)^2 g^5}-\frac{648 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac{1440 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac{432 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac{864 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac{864 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac{1296 (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{3456 d (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{2592 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)^2}-\frac{504 B^2 d^4 \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac{864 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac{864 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac{432 B^2 d^4 \log ^2(c+d x)}{b^3 (b c-a d)^2 g^5}+\frac{864 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}+\frac{864 B^2 d^4 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}+\frac{864 B^2 d^4 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}\\ \end{align*}

Mathematica [C]  time = 3.10902, size = 1788, normalized size = 5.98 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(i^2*(216*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 576*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[
(e*(a + b*x))/(c + d*x)])^2 + 432*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 216*B
*d^2*(a + b*x)^2*(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*L
og[(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)])) + 32*B*d*(a + b*x)*(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)])) + 3*B*(36*(b*c - a*d)^
4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 48*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])
+ 72*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(
A + B*Log[(e*(a + b*x))/(c + d*x)]) - 144*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) +
144*d^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 144*B*d^3*(a + b*x)^3*(b*c - a*d + d*(
a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + 36*B*d^2*(a + b*x)^2*((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]) - 8*B*d*(a + b*x)*(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]) + 3*B*(3*(b*c - a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b*c - a*d)^2*(a + b*x)^2
+ 12*d^3*(-(b*c) + a*d)*(a + b*x)^3 - 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x]) + 72*
B*d^4*(a + b*x)^4*(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)]) - 72*B*d^4*(a + b*x)^4*((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)]))))/(864*b^3*(b*c - a*d)^2*g^5*(a + b*x)^4)

________________________________________________________________________________________

Maple [B]  time = 0.053, size = 1814, 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)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x)

[Out]

1/3*e^3*d^2*i^2/(a*d-b*c)^3/g^5*A^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-1/3*e^3*d*i^2/(a*d-b*c)^3/g^5*A^2/
(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*b*c-1/4*e^4*d*i^2/(a*d-b*c)^3/g^5*A^2*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b
*c)^4*a+1/4*e^4*i^2/(a*d-b*c)^3/g^5*A^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c+2/3*e^3*d^2*i^2/(a*d-b*c)^
3/g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-2/3*e^3*d*i^2/(a*d-b*c)^3/g^
5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b*c+2/9*e^3*d^2*i^2/(a*d-b*c)^3/g^
5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-2/9*e^3*d*i^2/(a*d-b*c)^3/g^5*A*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)
*b*c)^3*b*c-1/2*e^4*d*i^2/(a*d-b*c)^3/g^5*A*B*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^2/(a*d-b*c)^3/g^5*A*B*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))*c-1/8*e^4*d*i^2/(a*d-b*c)^3/g^5*A*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a+1/8*e^4*i^2/(a*d-b*c)^3/g^
5*A*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c+1/3*e^3*d^2*i^2/(a*d-b*c)^3/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(
d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a-1/3*e^3*d*i^2/(a*d-b*c)^3/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*
x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*b*c+2/9*e^3*d^2*i^2/(a*d-b*c)^3/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(
d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-2/9*e^3*d*i^2/(a*d-b*c)^3/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+
c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b*c+2/27*e^3*d^2*i^2/(a*d-b*c)^3/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x
+c)*b*c)^3*a-2/27*e^3*d*i^2/(a*d-b*c)^3/g^5*B^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*b*c-1/4*e^4*d*i^2/(a*d-b
*c)^3/g^5*B^2*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))^2*a+1/4*e^4*i^2/(a*d-b*c
)^3/g^5*B^2*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*c-1/8*e^4*d*i^2/(a*d-b
*c)^3/g^5*B^2*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+1/8*e^4*i^2/(a*d-b*c)^
3/g^5*B^2*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))*c-1/32*e^4*d*i^2/(a*d-b*c)
^3/g^5*B^2*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a+1/32*e^4*i^2/(a*d-b*c)^3/g^5*B^2*b^2/(b*e/d+e/(d*x+c)*a-e
/d/(d*x+c)*b*c)^4*c

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Maxima [B]  time = 4.87487, size = 10842, normalized size = 36.26 \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)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

-1/6*(4*b*x + a)*B^2*c*d*i^2*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) - 1/12*(6*b^2*x^2 + 4*a*b*x + a^2)*B^2*d^2*i^2*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) + 1/288*
(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 - 176*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)*log(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) + 1
2*(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^2*i^2 - 1/432*(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 - 10
36*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*d*i^2 - 1/864*(
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*(10
8*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*(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 - 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*d^2*i^2 - 1/72*A*B*d^2*i^2*(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)) - 1/36*A*B*c*d*i^2*(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^2*i^2*((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^2*i^2*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) - 1/6*(4*b*x + a)*A^2*
c*d*i^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) - 1/12*(6*b^2*x^2
+ 4*a*b*x + a^2)*A^2*d^2*i^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*A^2*c^2*i^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)

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Fricas [B]  time = 0.590224, size = 1733, normalized size = 5.8 \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)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

1/864*(12*((12*A*B + 7*B^2)*b^4*c*d^3 - (12*A*B + 7*B^2)*a*b^3*d^4)*i^2*x^3 - 6*((72*A^2 + 12*A*B - 5*B^2)*b^4
*c^2*d^2 - 16*(9*A^2 + 6*A*B + 2*B^2)*a*b^3*c*d^3 + (72*A^2 + 84*A*B + 37*B^2)*a^2*b^2*d^4)*i^2*x^2 - 4*((144*
A^2 + 60*A*B + 11*B^2)*b^4*c^3*d - 24*(9*A^2 + 6*A*B + 2*B^2)*a*b^3*c^2*d^2 + (72*A^2 + 84*A*B + 37*B^2)*a^3*b
*d^4)*i^2*x - (27*(8*A^2 + 4*A*B + B^2)*b^4*c^4 - 32*(9*A^2 + 6*A*B + 2*B^2)*a*b^3*c^3*d + (72*A^2 + 84*A*B +
37*B^2)*a^4*d^4)*i^2 + 72*(B^2*b^4*d^4*i^2*x^4 + 4*B^2*a*b^3*d^4*i^2*x^3 - 6*(B^2*b^4*c^2*d^2 - 2*B^2*a*b^3*c*
d^3)*i^2*x^2 - 4*(2*B^2*b^4*c^3*d - 3*B^2*a*b^3*c^2*d^2)*i^2*x - (3*B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d)*i^2)*log(
(b*e*x + a*e)/(d*x + c))^2 + 12*((12*A*B + 7*B^2)*b^4*d^4*i^2*x^4 + 4*(3*B^2*b^4*c*d^3 + 4*(3*A*B + B^2)*a*b^3
*d^4)*i^2*x^3 - 6*((12*A*B + B^2)*b^4*c^2*d^2 - 8*(3*A*B + B^2)*a*b^3*c*d^3)*i^2*x^2 - 4*((24*A*B + 5*B^2)*b^4
*c^3*d - 12*(3*A*B + B^2)*a*b^3*c^2*d^2)*i^2*x - (9*(4*A*B + B^2)*b^4*c^4 - 16*(3*A*B + B^2)*a*b^3*c^3*d)*i^2)
*log((b*e*x + a*e)/(d*x + c)))/((b^9*c^2 - 2*a*b^8*c*d + a^2*b^7*d^2)*g^5*x^4 + 4*(a*b^8*c^2 - 2*a^2*b^7*c*d +
a^3*b^6*d^2)*g^5*x^3 + 6*(a^2*b^7*c^2 - 2*a^3*b^6*c*d + a^4*b^5*d^2)*g^5*x^2 + 4*(a^3*b^6*c^2 - 2*a^4*b^5*c*d
+ a^5*b^4*d^2)*g^5*x + (a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*g^5)

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Sympy [B]  time = 133.626, size = 2054, normalized size = 6.87 \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)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**5,x)

[Out]

-B*d**4*i**2*(12*A + 7*B)*log(x + (12*A*B*a*d**5*i**2 + 12*A*B*b*c*d**4*i**2 + 7*B**2*a*d**5*i**2 + 7*B**2*b*c
*d**4*i**2 - B*a**3*d**7*i**2*(12*A + 7*B)/(a*d - b*c)**2 + 3*B*a**2*b*c*d**6*i**2*(12*A + 7*B)/(a*d - b*c)**2
- 3*B*a*b**2*c**2*d**5*i**2*(12*A + 7*B)/(a*d - b*c)**2 + B*b**3*c**3*d**4*i**2*(12*A + 7*B)/(a*d - b*c)**2)/
(24*A*B*b*d**5*i**2 + 14*B**2*b*d**5*i**2))/(72*b**3*g**5*(a*d - b*c)**2) + B*d**4*i**2*(12*A + 7*B)*log(x + (
12*A*B*a*d**5*i**2 + 12*A*B*b*c*d**4*i**2 + 7*B**2*a*d**5*i**2 + 7*B**2*b*c*d**4*i**2 + B*a**3*d**7*i**2*(12*A
+ 7*B)/(a*d - b*c)**2 - 3*B*a**2*b*c*d**6*i**2*(12*A + 7*B)/(a*d - b*c)**2 + 3*B*a*b**2*c**2*d**5*i**2*(12*A
+ 7*B)/(a*d - b*c)**2 - B*b**3*c**3*d**4*i**2*(12*A + 7*B)/(a*d - b*c)**2)/(24*A*B*b*d**5*i**2 + 14*B**2*b*d**
5*i**2))/(72*b**3*g**5*(a*d - b*c)**2) + (4*B**2*a*c**3*d*i**2 + 12*B**2*a*c**2*d**2*i**2*x + 12*B**2*a*c*d**3
*i**2*x**2 + 4*B**2*a*d**4*i**2*x**3 - 3*B**2*b*c**4*i**2 - 8*B**2*b*c**3*d*i**2*x - 6*B**2*b*c**2*d**2*i**2*x
**2 + B**2*b*d**4*i**2*x**4)*log(e*(a + b*x)/(c + d*x))**2/(12*a**6*d**2*g**5 - 24*a**5*b*c*d*g**5 + 48*a**5*b
*d**2*g**5*x + 12*a**4*b**2*c**2*g**5 - 96*a**4*b**2*c*d*g**5*x + 72*a**4*b**2*d**2*g**5*x**2 + 48*a**3*b**3*c
**2*g**5*x - 144*a**3*b**3*c*d*g**5*x**2 + 48*a**3*b**3*d**2*g**5*x**3 + 72*a**2*b**4*c**2*g**5*x**2 - 96*a**2
*b**4*c*d*g**5*x**3 + 12*a**2*b**4*d**2*g**5*x**4 + 48*a*b**5*c**2*g**5*x**3 - 24*a*b**5*c*d*g**5*x**4 + 12*b*
*6*c**2*g**5*x**4) + (-12*A*B*a**3*d**3*i**2 - 12*A*B*a**2*b*c*d**2*i**2 - 48*A*B*a**2*b*d**3*i**2*x - 12*A*B*
a*b**2*c**2*d*i**2 - 48*A*B*a*b**2*c*d**2*i**2*x - 72*A*B*a*b**2*d**3*i**2*x**2 + 36*A*B*b**3*c**3*i**2 + 96*A
*B*b**3*c**2*d*i**2*x + 72*A*B*b**3*c*d**2*i**2*x**2 - 7*B**2*a**3*d**3*i**2 - 7*B**2*a**2*b*c*d**2*i**2 - 28*
B**2*a**2*b*d**3*i**2*x - 7*B**2*a*b**2*c**2*d*i**2 - 28*B**2*a*b**2*c*d**2*i**2*x - 42*B**2*a*b**2*d**3*i**2*
x**2 + 9*B**2*b**3*c**3*i**2 + 20*B**2*b**3*c**2*d*i**2*x + 6*B**2*b**3*c*d**2*i**2*x**2 - 12*B**2*b**3*d**3*i
**2*x**3)*log(e*(a + b*x)/(c + d*x))/(72*a**5*b**3*d*g**5 - 72*a**4*b**4*c*g**5 + 288*a**4*b**4*d*g**5*x - 288
*a**3*b**5*c*g**5*x + 432*a**3*b**5*d*g**5*x**2 - 432*a**2*b**6*c*g**5*x**2 + 288*a**2*b**6*d*g**5*x**3 - 288*
a*b**7*c*g**5*x**3 + 72*a*b**7*d*g**5*x**4 - 72*b**8*c*g**5*x**4) - (72*A**2*a**3*d**3*i**2 + 72*A**2*a**2*b*c
*d**2*i**2 + 72*A**2*a*b**2*c**2*d*i**2 - 216*A**2*b**3*c**3*i**2 + 84*A*B*a**3*d**3*i**2 + 84*A*B*a**2*b*c*d*
*2*i**2 + 84*A*B*a*b**2*c**2*d*i**2 - 108*A*B*b**3*c**3*i**2 + 37*B**2*a**3*d**3*i**2 + 37*B**2*a**2*b*c*d**2*
i**2 + 37*B**2*a*b**2*c**2*d*i**2 - 27*B**2*b**3*c**3*i**2 + x**3*(144*A*B*b**3*d**3*i**2 + 84*B**2*b**3*d**3*
i**2) + x**2*(432*A**2*a*b**2*d**3*i**2 - 432*A**2*b**3*c*d**2*i**2 + 504*A*B*a*b**2*d**3*i**2 - 72*A*B*b**3*c
*d**2*i**2 + 222*B**2*a*b**2*d**3*i**2 + 30*B**2*b**3*c*d**2*i**2) + x*(288*A**2*a**2*b*d**3*i**2 + 288*A**2*a
*b**2*c*d**2*i**2 - 576*A**2*b**3*c**2*d*i**2 + 336*A*B*a**2*b*d**3*i**2 + 336*A*B*a*b**2*c*d**2*i**2 - 240*A*
B*b**3*c**2*d*i**2 + 148*B**2*a**2*b*d**3*i**2 + 148*B**2*a*b**2*c*d**2*i**2 - 44*B**2*b**3*c**2*d*i**2))/(864
*a**5*b**3*d*g**5 - 864*a**4*b**4*c*g**5 + x**4*(864*a*b**7*d*g**5 - 864*b**8*c*g**5) + x**3*(3456*a**2*b**6*d
*g**5 - 3456*a*b**7*c*g**5) + x**2*(5184*a**3*b**5*d*g**5 - 5184*a**2*b**6*c*g**5) + x*(3456*a**4*b**4*d*g**5
- 3456*a**3*b**5*c*g**5))

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