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

Optimal. Leaf size=445 \[ -\frac{b^2 i (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^3}-\frac{b^2 B i (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^3}-\frac{d^2 i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^5 (a+b x)^2 (b c-a d)^3}-\frac{B d^2 i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^3}+\frac{2 b d i (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^5 (a+b x)^3 (b c-a d)^3}+\frac{4 b B d i (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{9 g^5 (a+b x)^3 (b c-a d)^3}-\frac{b^2 B^2 i (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^3}-\frac{B^2 d^2 i (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^3}+\frac{4 b B^2 d i (c+d x)^3}{27 g^5 (a+b x)^3 (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 2.61133, antiderivative size = 826, normalized size of antiderivative = 1.86, number of steps used = 74, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B^2 i \log ^2(a+b x) d^4}{12 b^2 (b c-a d)^3 g^5}+\frac{B^2 i \log ^2(c+d x) d^4}{12 b^2 (b c-a d)^3 g^5}-\frac{13 B^2 i \log (a+b x) d^4}{72 b^2 (b c-a d)^3 g^5}-\frac{B i \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^4}{6 b^2 (b c-a d)^3 g^5}+\frac{13 B^2 i \log (c+d x) d^4}{72 b^2 (b c-a d)^3 g^5}-\frac{B^2 i \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{6 b^2 (b c-a d)^3 g^5}+\frac{B i \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x) d^4}{6 b^2 (b c-a d)^3 g^5}-\frac{B^2 i \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^4}{6 b^2 (b c-a d)^3 g^5}-\frac{B^2 i \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^4}{6 b^2 (b c-a d)^3 g^5}-\frac{B^2 i \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^4}{6 b^2 (b c-a d)^3 g^5}-\frac{B i \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^3}{6 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{13 B^2 i d^3}{72 b^2 (b c-a d)^2 g^5 (a+b x)}+\frac{B i \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^2}{12 b^2 (b c-a d) g^5 (a+b x)^2}+\frac{B^2 i d^2}{144 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{i \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 d}{3 b^2 g^5 (a+b x)^3}-\frac{B i \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d}{18 b^2 g^5 (a+b x)^3}+\frac{5 B^2 i d}{216 b^2 g^5 (a+b x)^3}-\frac{(b c-a d) i \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{B (b c-a d) i \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac{B^2 (b c-a d) i}{32 b^2 g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B^2*(b*c - a*d)*i)/(32*b^2*g^5*(a + b*x)^4) + (5*B^2*d*i)/(216*b^2*g^5*(a + b*x)^3) + (B^2*d^2*i)/(144*b^2*(
b*c - a*d)*g^5*(a + b*x)^2) - (13*B^2*d^3*i)/(72*b^2*(b*c - a*d)^2*g^5*(a + b*x)) - (13*B^2*d^4*i*Log[a + b*x]
)/(72*b^2*(b*c - a*d)^3*g^5) + (B^2*d^4*i*Log[a + b*x]^2)/(12*b^2*(b*c - a*d)^3*g^5) - (B*(b*c - a*d)*i*(A + B
*Log[(e*(a + b*x))/(c + d*x)]))/(8*b^2*g^5*(a + b*x)^4) - (B*d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(18*b^2
*g^5*(a + b*x)^3) + (B*d^2*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^2*(b*c - a*d)*g^5*(a + b*x)^2) - (B*d
^3*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^2*(b*c - a*d)^2*g^5*(a + b*x)) - (B*d^4*i*Log[a + b*x]*(A + B*
Log[(e*(a + b*x))/(c + d*x)]))/(6*b^2*(b*c - a*d)^3*g^5) - ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])
^2)/(4*b^2*g^5*(a + b*x)^4) - (d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*b^2*g^5*(a + b*x)^3) + (13*B^2*d
^4*i*Log[c + d*x])/(72*b^2*(b*c - a*d)^3*g^5) - (B^2*d^4*i*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(6*
b^2*(b*c - a*d)^3*g^5) + (B*d^4*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(6*b^2*(b*c - a*d)^3*g^5)
+ (B^2*d^4*i*Log[c + d*x]^2)/(12*b^2*(b*c - a*d)^3*g^5) - (B^2*d^4*i*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*
d)])/(6*b^2*(b*c - a*d)^3*g^5) - (B^2*d^4*i*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(6*b^2*(b*c - a*d)^3*g^5
) - (B^2*d^4*i*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(6*b^2*(b*c - a*d)^3*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{(63 c+63 d x) \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{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g^5 (a+b x)^5}+\frac{63 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g^5 (a+b x)^4}\right ) \, dx\\ &=\frac{(63 d) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{b g^5}+\frac{(63 (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)^5} \, dx}{b g^5}\\ &=-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{(42 B d) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{b^2 g^5}+\frac{(63 B (b c-a d)) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b^2 g^5}\\ &=-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{(42 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 (c+d x)} \, dx}{b^2 g^5}+\frac{\left (63 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 (c+d x)} \, dx}{2 b^2 g^5}\\ &=-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{(42 B d (b c-a d)) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^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^2 g^5}+\frac{\left (63 B (b c-a d)^2\right ) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^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^2 g^5}\\ &=-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}-\frac{(63 B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{2 b g^5}+\frac{(42 B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b g^5}+\frac{\left (63 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 (b c-a d)^3 g^5}-\frac{\left (42 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d)^3 g^5}-\frac{\left (63 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^2 (b c-a d)^3 g^5}+\frac{\left (42 B d^5\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d)^3 g^5}-\frac{\left (63 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 (b c-a d)^2 g^5}+\frac{\left (42 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 (b c-a d)^2 g^5}+\frac{\left (63 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{2 b (b c-a d) g^5}-\frac{\left (42 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 (b c-a d) g^5}+\frac{(63 B (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{2 b g^5}\\ &=-\frac{63 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac{7 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^5 (a+b x)^3}+\frac{21 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{21 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{21 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{21 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}-\frac{\left (21 B^2 d\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{2 b^2 g^5}+\frac{\left (14 B^2 d\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^2 g^5}-\frac{\left (63 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^2 (b c-a d)^3 g^5}+\frac{\left (63 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^2 (b c-a d)^3 g^5}+\frac{\left (42 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^2 (b c-a d)^3 g^5}-\frac{\left (42 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^2 (b c-a d)^3 g^5}-\frac{\left (63 B^2 d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2 b^2 (b c-a d)^2 g^5}+\frac{\left (42 B^2 d^3\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 (b c-a d)^2 g^5}+\frac{\left (63 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{4 b^2 (b c-a d) g^5}-\frac{\left (21 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^2 (b c-a d) g^5}+\frac{\left (63 B^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{8 b^2 g^5}\\ &=-\frac{63 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac{7 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^5 (a+b x)^3}+\frac{21 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{21 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{21 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{21 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}+\frac{\left (63 B^2 d^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{4 b^2 g^5}-\frac{\left (21 B^2 d^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^5}-\frac{\left (63 B^2 d^3\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{2 b^2 (b c-a d) g^5}+\frac{\left (42 B^2 d^3\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^2 (b c-a d) g^5}-\frac{\left (21 B^2 d (b c-a d)\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{2 b^2 g^5}+\frac{\left (14 B^2 d (b c-a d)\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^2 g^5}+\frac{\left (63 B^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{8 b^2 g^5}-\frac{\left (63 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^2 (b c-a d)^3 e g^5}+\frac{\left (63 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^2 (b c-a d)^3 e g^5}+\frac{\left (42 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^2 (b c-a d)^3 e g^5}-\frac{\left (42 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^2 (b c-a d)^3 e g^5}\\ &=-\frac{63 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac{7 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^5 (a+b x)^3}+\frac{21 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{21 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{21 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{21 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}+\frac{\left (63 B^2 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^2 g^5}-\frac{\left (21 B^2 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^2 g^5}-\frac{\left (63 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}{2 b^2 (b c-a d) g^5}+\frac{\left (42 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^2 (b c-a d) g^5}-\frac{\left (21 B^2 d (b c-a d)\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^2 g^5}+\frac{\left (14 B^2 d (b c-a d)\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^2 g^5}+\frac{\left (63 B^2 (b c-a d)^2\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^2 g^5}-\frac{\left (63 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^2 (b c-a d)^3 e g^5}+\frac{\left (63 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^2 (b c-a d)^3 e g^5}+\frac{\left (42 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^2 (b c-a d)^3 e g^5}-\frac{\left (42 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^2 (b c-a d)^3 e g^5}\\ &=-\frac{63 B^2 (b c-a d)}{32 b^2 g^5 (a+b x)^4}+\frac{35 B^2 d}{24 b^2 g^5 (a+b x)^3}+\frac{7 B^2 d^2}{16 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{91 B^2 d^3}{8 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{91 B^2 d^4 \log (a+b x)}{8 b^2 (b c-a d)^3 g^5}-\frac{63 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac{7 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^5 (a+b x)^3}+\frac{21 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{21 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{21 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{91 B^2 d^4 \log (c+d x)}{8 b^2 (b c-a d)^3 g^5}+\frac{21 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}-\frac{\left (63 B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{2 b (b c-a d)^3 g^5}+\frac{\left (63 B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2 b (b c-a d)^3 g^5}+\frac{\left (42 B^2 d^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b (b c-a d)^3 g^5}-\frac{\left (42 B^2 d^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b (b c-a d)^3 g^5}+\frac{\left (63 B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{2 b^2 (b c-a d)^3 g^5}-\frac{\left (63 B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{2 b^2 (b c-a d)^3 g^5}-\frac{\left (42 B^2 d^5\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d)^3 g^5}+\frac{\left (42 B^2 d^5\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d)^3 g^5}\\ &=-\frac{63 B^2 (b c-a d)}{32 b^2 g^5 (a+b x)^4}+\frac{35 B^2 d}{24 b^2 g^5 (a+b x)^3}+\frac{7 B^2 d^2}{16 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{91 B^2 d^3}{8 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{91 B^2 d^4 \log (a+b x)}{8 b^2 (b c-a d)^3 g^5}-\frac{63 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac{7 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^5 (a+b x)^3}+\frac{21 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{21 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{21 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{91 B^2 d^4 \log (c+d x)}{8 b^2 (b c-a d)^3 g^5}-\frac{21 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}+\frac{21 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}-\frac{21 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{\left (63 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{\left (63 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2 b^2 (b c-a d)^3 g^5}+\frac{\left (42 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d)^3 g^5}+\frac{\left (42 B^2 d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d)^3 g^5}-\frac{\left (63 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 (b c-a d)^3 g^5}+\frac{\left (42 B^2 d^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d)^3 g^5}-\frac{\left (63 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^2 (b c-a d)^3 g^5}+\frac{\left (42 B^2 d^5\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d)^3 g^5}\\ &=-\frac{63 B^2 (b c-a d)}{32 b^2 g^5 (a+b x)^4}+\frac{35 B^2 d}{24 b^2 g^5 (a+b x)^3}+\frac{7 B^2 d^2}{16 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{91 B^2 d^3}{8 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{91 B^2 d^4 \log (a+b x)}{8 b^2 (b c-a d)^3 g^5}+\frac{21 B^2 d^4 \log ^2(a+b x)}{4 b^2 (b c-a d)^3 g^5}-\frac{63 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac{7 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^5 (a+b x)^3}+\frac{21 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{21 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{21 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{91 B^2 d^4 \log (c+d x)}{8 b^2 (b c-a d)^3 g^5}-\frac{21 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}+\frac{21 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}+\frac{21 B^2 d^4 \log ^2(c+d x)}{4 b^2 (b c-a d)^3 g^5}-\frac{21 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{\left (63 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^2 (b c-a d)^3 g^5}-\frac{\left (63 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^2 (b c-a d)^3 g^5}+\frac{\left (42 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^2 (b c-a d)^3 g^5}+\frac{\left (42 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^2 (b c-a d)^3 g^5}\\ &=-\frac{63 B^2 (b c-a d)}{32 b^2 g^5 (a+b x)^4}+\frac{35 B^2 d}{24 b^2 g^5 (a+b x)^3}+\frac{7 B^2 d^2}{16 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{91 B^2 d^3}{8 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{91 B^2 d^4 \log (a+b x)}{8 b^2 (b c-a d)^3 g^5}+\frac{21 B^2 d^4 \log ^2(a+b x)}{4 b^2 (b c-a d)^3 g^5}-\frac{63 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{8 b^2 g^5 (a+b x)^4}-\frac{7 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^5 (a+b x)^3}+\frac{21 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{21 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{21 B d^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{63 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{4 b^2 g^5 (a+b x)^4}-\frac{21 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}+\frac{91 B^2 d^4 \log (c+d x)}{8 b^2 (b c-a d)^3 g^5}-\frac{21 B^2 d^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}+\frac{21 B d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b^2 (b c-a d)^3 g^5}+\frac{21 B^2 d^4 \log ^2(c+d x)}{4 b^2 (b c-a d)^3 g^5}-\frac{21 B^2 d^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{21 B^2 d^4 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b^2 (b c-a d)^3 g^5}-\frac{21 B^2 d^4 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2 b^2 (b c-a d)^3 g^5}\\ \end{align*}

Mathematica [C]  time = 1.71301, size = 1340, normalized size = 3.01 \[ -\frac{i \left (216 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^4-288 d (a d-b c)^3 (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2+16 B d (a+b x) \left (12 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3-18 d (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2+36 d^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)+36 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-36 d^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+36 B d^2 (a+b x)^2 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-9 B d (a+b x) \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B \left (2 (b c-a d)^3-3 d (a+b x) (b c-a d)^2+6 d^2 (a+b x)^2 (b c-a d)+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )-18 B d^3 (a+b x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+18 B d^3 (a+b x)^3 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )+3 B \left (36 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^4+72 d^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2+144 d^3 (a d-b c) (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+48 d (a d-b c)^3 (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-144 d^4 (a+b x)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+144 d^4 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \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 \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )-8 B d (a+b x) \left (2 (b c-a d)^3-3 d (a+b x) (b c-a d)^2+6 d^2 (a+b x)^2 (b c-a d)+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )+3 B \left (3 (b c-a d)^4+6 d^2 (a+b x)^2 (b c-a d)^2+12 d^3 (a d-b c) (a+b x)^3+4 d (a d-b c)^3 (a+b x)-12 d^4 (a+b x)^4 \log (a+b x)+12 d^4 (a+b x)^4 \log (c+d x)\right )+72 B d^4 (a+b x)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-72 B d^4 (a+b x)^4 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )\right )}{864 b^2 (b c-a d)^3 g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(i*(216*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 288*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e
*(a + b*x))/(c + d*x)])^2 + 16*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*L
og[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)])*Lo
g[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*Lo
g[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^2*(b*c -
a*d)^3*g^5*(a + b*x)^4)

________________________________________________________________________________________

Maple [B]  time = 0.053, size = 2689, normalized size = 6. \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)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x)

[Out]

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

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Maxima [B]  time = 3.39295, size = 6491, normalized size = 14.59 \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)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

-1/12*(4*b*x + a)*B^2*d*i*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) + 1/288*(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) + 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 + 1
00*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*i - 1/864*(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 + 64
8*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*d*i - 1/72*A*B*d*i*(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*i*((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*i*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) - 1/12*(4*b*x + a)*A^2*d*i/(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*A^2*c*i/(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 = 0.582325, size = 2048, normalized size = 4.6 \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)*(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 + 13*B^2)*b^4*c*d^3 - (12*A*B + 13*B^2)*a*b^3*d^4)*i*x^3 - 6*((12*A*B + B^2)*b^4*c^2*d^2 -
16*(6*A*B + 5*B^2)*a*b^3*c*d^3 + (84*A*B + 79*B^2)*a^2*b^2*d^4)*i*x^2 + 4*((72*A^2 + 12*A*B - 5*B^2)*b^4*c^3*
d - 12*(18*A^2 + 6*A*B - B^2)*a*b^3*c^2*d^2 + 108*(2*A^2 + 2*A*B + B^2)*a^2*b^2*c*d^3 - (72*A^2 + 156*A*B + 11
5*B^2)*a^3*b*d^4)*i*x + 72*(B^2*b^4*d^4*i*x^4 + 4*B^2*a*b^3*d^4*i*x^3 + 6*B^2*a^2*b^2*d^4*i*x^2 + 4*(B^2*b^4*c
^3*d - 3*B^2*a*b^3*c^2*d^2 + 3*B^2*a^2*b^2*c*d^3)*i*x + (3*B^2*b^4*c^4 - 8*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2
*d^2)*i)*log((b*e*x + a*e)/(d*x + c))^2 + (27*(8*A^2 + 4*A*B + B^2)*b^4*c^4 - 64*(9*A^2 + 6*A*B + 2*B^2)*a*b^3
*c^3*d + 216*(2*A^2 + 2*A*B + B^2)*a^2*b^2*c^2*d^2 - (72*A^2 + 156*A*B + 115*B^2)*a^4*d^4)*i + 12*((12*A*B + 1
3*B^2)*b^4*d^4*i*x^4 + 4*(3*B^2*b^4*c*d^3 + 2*(6*A*B + 5*B^2)*a*b^3*d^4)*i*x^3 - 6*(B^2*b^4*c^2*d^2 - 8*B^2*a*
b^3*c*d^3 - 6*(2*A*B + B^2)*a^2*b^2*d^4)*i*x^2 + 4*((12*A*B + B^2)*b^4*c^3*d - 6*(6*A*B + B^2)*a*b^3*c^2*d^2 +
18*(2*A*B + B^2)*a^2*b^2*c*d^3)*i*x + (9*(4*A*B + B^2)*b^4*c^4 - 32*(3*A*B + B^2)*a*b^3*c^3*d + 36*(2*A*B + B
^2)*a^2*b^2*c^2*d^2)*i)*log((b*e*x + a*e)/(d*x + c)))/((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)

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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)*(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 )}{\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)*(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)*(B*log((b*x + a)*e/(d*x + c)) + A)^2/(b*g*x + a*g)^5, x)