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]

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

________________________________________________________________________________________

Rubi [C]  time = 2.61, 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 12

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

Rule 44

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

Rule 2301

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

Rule 2390

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

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

Rule 2394

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

Rule 2418

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

Rule 2524

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

Rule 2525

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

Rule 2528

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

Rubi steps

\begin {align*} \int \frac {(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 = 1.63, 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 {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+18 B d^3 (a+b x)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+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 {Li}_2\left (\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 {Li}_2\left (\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]

-1/864*(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*PolyL
og[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]) + 3
6*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)]))))/(b^2*(b*c
- a*d)^3*g^5*(a + b*x)^4)

________________________________________________________________________________________

fricas [B]  time = 1.00, size = 985, normalized size = 2.21 \[ -\frac {12 \, {\left ({\left (12 \, A B + 13 \, B^{2}\right )} b^{4} c d^{3} - {\left (12 \, A B + 13 \, B^{2}\right )} a b^{3} d^{4}\right )} i x^{3} - 6 \, {\left ({\left (12 \, A B + B^{2}\right )} b^{4} c^{2} d^{2} - 16 \, {\left (6 \, A B + 5 \, B^{2}\right )} a b^{3} c d^{3} + {\left (84 \, A B + 79 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} i x^{2} + 4 \, {\left ({\left (72 \, A^{2} + 12 \, A B - 5 \, B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (18 \, A^{2} + 6 \, A B - B^{2}\right )} a b^{3} c^{2} d^{2} + 108 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a^{2} b^{2} c d^{3} - {\left (72 \, A^{2} + 156 \, A B + 115 \, B^{2}\right )} a^{3} b d^{4}\right )} i x + 72 \, {\left (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 \, {\left (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}\right )} i x + {\left (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}\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + {\left (27 \, {\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{4} - 64 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{3} c^{3} d + 216 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a^{2} b^{2} c^{2} d^{2} - {\left (72 \, A^{2} + 156 \, A B + 115 \, B^{2}\right )} a^{4} d^{4}\right )} i + 12 \, {\left ({\left (12 \, A B + 13 \, B^{2}\right )} b^{4} d^{4} i x^{4} + 4 \, {\left (3 \, B^{2} b^{4} c d^{3} + 2 \, {\left (6 \, A B + 5 \, B^{2}\right )} a b^{3} d^{4}\right )} i x^{3} - 6 \, {\left (B^{2} b^{4} c^{2} d^{2} - 8 \, B^{2} a b^{3} c d^{3} - 6 \, {\left (2 \, A B + B^{2}\right )} a^{2} b^{2} d^{4}\right )} i x^{2} + 4 \, {\left ({\left (12 \, A B + B^{2}\right )} b^{4} c^{3} d - 6 \, {\left (6 \, A B + B^{2}\right )} a b^{3} c^{2} d^{2} + 18 \, {\left (2 \, A B + B^{2}\right )} a^{2} b^{2} c d^{3}\right )} i x + {\left (9 \, {\left (4 \, A B + B^{2}\right )} b^{4} c^{4} - 32 \, {\left (3 \, A B + B^{2}\right )} a b^{3} c^{3} d + 36 \, {\left (2 \, A B + B^{2}\right )} a^{2} b^{2} c^{2} d^{2}\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{864 \, {\left ({\left (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}\right )} g^{5} x^{4} + 4 \, {\left (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}\right )} g^{5} x^{3} + 6 \, {\left (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}\right )} g^{5} x^{2} + 4 \, {\left (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}\right )} g^{5} x + {\left (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}\right )} g^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

giac [A]  time = 1.67, size = 727, normalized size = 1.63 \[ -\frac {{\left (216 \, B^{2} b^{2} i e^{5} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} - \frac {576 \, {\left (b x e + a e\right )} B^{2} b d i e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} + \frac {432 \, {\left (b x e + a e\right )}^{2} B^{2} d^{2} i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{2}} + 432 \, A B b^{2} i e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) + 108 \, B^{2} b^{2} i e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {1152 \, {\left (b x e + a e\right )} A B b d i e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {384 \, {\left (b x e + a e\right )} B^{2} b d i e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {864 \, {\left (b x e + a e\right )}^{2} A B d^{2} i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {432 \, {\left (b x e + a e\right )}^{2} B^{2} d^{2} i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + 216 \, A^{2} b^{2} i e^{5} + 108 \, A B b^{2} i e^{5} + 27 \, B^{2} b^{2} i e^{5} - \frac {576 \, {\left (b x e + a e\right )} A^{2} b d i e^{4}}{d x + c} - \frac {384 \, {\left (b x e + a e\right )} A B b d i e^{4}}{d x + c} - \frac {128 \, {\left (b x e + a e\right )} B^{2} b d i e^{4}}{d x + c} + \frac {432 \, {\left (b x e + a e\right )}^{2} A^{2} d^{2} i e^{3}}{{\left (d x + c\right )}^{2}} + \frac {432 \, {\left (b x e + a e\right )}^{2} A B d^{2} i e^{3}}{{\left (d x + c\right )}^{2}} + \frac {216 \, {\left (b x e + a e\right )}^{2} B^{2} d^{2} i e^{3}}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{864 \, {\left (\frac {{\left (b x e + a e\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x e + a e\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x e + a e\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864*(216*B^2*b^2*i*e^5*log((b*x*e + a*e)/(d*x + c))^2 - 576*(b*x*e + a*e)*B^2*b*d*i*e^4*log((b*x*e + a*e)/(
d*x + c))^2/(d*x + c) + 432*(b*x*e + a*e)^2*B^2*d^2*i*e^3*log((b*x*e + a*e)/(d*x + c))^2/(d*x + c)^2 + 432*A*B
*b^2*i*e^5*log((b*x*e + a*e)/(d*x + c)) + 108*B^2*b^2*i*e^5*log((b*x*e + a*e)/(d*x + c)) - 1152*(b*x*e + a*e)*
A*B*b*d*i*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 384*(b*x*e + a*e)*B^2*b*d*i*e^4*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c) + 864*(b*x*e + a*e)^2*A*B*d^2*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 432*(b*x*e + a*
e)^2*B^2*d^2*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 216*A^2*b^2*i*e^5 + 108*A*B*b^2*i*e^5 + 27*B^2*b
^2*i*e^5 - 576*(b*x*e + a*e)*A^2*b*d*i*e^4/(d*x + c) - 384*(b*x*e + a*e)*A*B*b*d*i*e^4/(d*x + c) - 128*(b*x*e
+ a*e)*B^2*b*d*i*e^4/(d*x + c) + 432*(b*x*e + a*e)^2*A^2*d^2*i*e^3/(d*x + c)^2 + 432*(b*x*e + a*e)^2*A*B*d^2*i
*e^3/(d*x + c)^2 + 216*(b*x*e + a*e)^2*B^2*d^2*i*e^3/(d*x + c)^2)*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b
*c*e - a*d*e)*(b*c - a*d)))/((b*x*e + a*e)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*x*e + a*e)^4*a*b*c*d*g^5/(d*x + c)
^4 + (b*x*e + a*e)^4*a^2*d^2*g^5/(d*x + c)^4)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 2689, normalized size = 6.04 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 5.05, size = 4808, normalized size = 10.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

mupad [B]  time = 10.82, size = 1870, normalized size = 4.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((72*A^2*a^3*d^3*i + 216*A^2*b^3*c^3*i + 115*B^2*a^3*d^3*i + 27*B^2*b^3*c^3*i + 156*A*B*a^3*d^3*i + 108*A*B*b^
3*c^3*i - 360*A^2*a*b^2*c^2*d*i + 72*A^2*a^2*b*c*d^2*i - 101*B^2*a*b^2*c^2*d*i + 115*B^2*a^2*b*c*d^2*i - 276*A
*B*a*b^2*c^2*d*i + 156*A*B*a^2*b*c*d^2*i)/(12*(a*d - b*c)) + (x^2*(79*B^2*a*b^2*d^3*i - B^2*b^3*c*d^2*i + 84*A
*B*a*b^2*d^3*i - 12*A*B*b^3*c*d^2*i))/(2*(a*d - b*c)) + (x*(72*A^2*a^2*b*d^3*i + 115*B^2*a^2*b*d^3*i + 72*A^2*
b^3*c^2*d*i - 5*B^2*b^3*c^2*d*i + 156*A*B*a^2*b*d^3*i + 12*A*B*b^3*c^2*d*i - 144*A^2*a*b^2*c*d^2*i + 7*B^2*a*b
^2*c*d^2*i - 60*A*B*a*b^2*c*d^2*i))/(3*(a*d - b*c)) + (d*x^3*(13*B^2*b^3*d^2*i + 12*A*B*b^3*d^2*i))/(a*d - b*c
))/(x*(288*a^3*b^4*c*g^5 - 288*a^4*b^3*d*g^5) - x^3*(288*a^2*b^5*d*g^5 - 288*a*b^6*c*g^5) + x^4*(72*b^7*c*g^5
- 72*a*b^6*d*g^5) + x^2*(432*a^2*b^5*c*g^5 - 432*a^3*b^4*d*g^5) + 72*a^4*b^3*c*g^5 - 72*a^5*b^2*d*g^5) - log((
e*(a + b*x))/(c + d*x))^2*(((B^2*c*i)/(4*b^2*g^5) + (B^2*a*d*i)/(12*b^3*g^5) + (B^2*d*i*x)/(3*b^2*g^5))/(4*a^3
*x + a^4/b + b^3*x^4 + 6*a^2*b*x^2 + 4*a*b^2*x^3) - (B^2*d^4*i)/(12*b^2*g^5*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d
- 3*a^2*b*c*d^2))) - (log((e*(a + b*x))/(c + d*x))*(x*((2*A*B*i)/(3*b^2*g^5) + (B^2*d^4*i*(b*(a*((4*a^2*d^2 +
b^2*c^2 - 5*a*b*c*d)/(12*b*d^3) + (a*(a*d - b*c))/(4*b*d^2)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*
b*c*d^2)/(12*b*d^4)) + a*(b*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(12*b*d^3) + (a*(a*d - b*c))/(4*b*d^2)) + (4*a^
2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*d^3) + (a*(a*d - b*c))/(2*d^2)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^
2*b*c*d^2)/(4*d^4)))/(6*b^2*g^5*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) + (A*B*a*i)/(6*b^3*g^5)
+ (B*i*(3*A*b*c - B*a*d + B*b*c))/(6*b^3*d*g^5) + (B^2*d^4*i*(a*(a*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(12*b*d^
3) + (a*(a*d - b*c))/(4*b*d^2)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2)/(12*b*d^4)) + (4*a^4*
d^4 + b^4*c^4 + 10*a^2*b^2*c^2*d^2 - 5*a*b^3*c^3*d - 10*a^3*b*c*d^3)/(4*b*d^5)))/(6*b^2*g^5*(a^3*d^3 - b^3*c^3
+ 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (B^2*d^4*i*x^2*(b*(b*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(12*b*d^3) + (a*(
a*d - b*c))/(4*b*d^2)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*d^3) + (a*(a*d - b*c))/(2*d^2)) - a*((b^2*c - a*
b*d)/(4*d^2) - (b*(a*d - b*c))/(2*d^2)) + (b^3*c^2 + 4*a^2*b*d^2 - 5*a*b^2*c*d)/(4*d^3)))/(6*b^2*g^5*(a^3*d^3
- b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B^2*d^4*i*x^3*(b*((b^2*c - a*b*d)/(4*d^2) - (b*(a*d - b*c))/(2*
d^2)) + (b^3*c - a*b^2*d)/(4*d^2)))/(6*b^2*g^5*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((4*a^3*
x)/d + a^4/(b*d) + (b^3*x^4)/d + (6*a^2*b*x^2)/d + (4*a*b^2*x^3)/d) - (B*d^4*i*atan((B*d^4*i*(12*A + 13*B)*(72
*b^5*c^3*g^5 + 72*a^3*b^2*d^3*g^5 - 72*a*b^4*c^2*d*g^5 - 72*a^2*b^3*c*d^2*g^5)*1i)/(72*b^2*g^5*(13*B^2*d^4*i +
12*A*B*d^4*i)*(a*d - b*c)^3) + (B*d^5*i*x*(12*A + 13*B)*(b^4*c^2*g^5 + a^2*b^2*d^2*g^5 - 2*a*b^3*c*d*g^5)*2i)
/(b*g^5*(13*B^2*d^4*i + 12*A*B*d^4*i)*(a*d - b*c)^3))*(12*A + 13*B)*1i)/(36*b^2*g^5*(a*d - b*c)^3)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________