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

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

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 3.69, antiderivative size = 920, normalized size of antiderivative = 3.08, number of steps used = 104, number of rules used = 11, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac {B^2 i^2 \log ^2(a+b x) d^4}{12 b^3 (b c-a d)^2 g^5}-\frac {B^2 i^2 \log ^2(c+d x) d^4}{12 b^3 (b c-a d)^2 g^5}+\frac {7 B^2 i^2 \log (a+b x) d^4}{72 b^3 (b c-a d)^2 g^5}+\frac {B i^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d^4}{6 b^3 (b c-a d)^2 g^5}-\frac {7 B^2 i^2 \log (c+d x) d^4}{72 b^3 (b c-a d)^2 g^5}+\frac {B^2 i^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{6 b^3 (b c-a d)^2 g^5}-\frac {B i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac {B^2 i^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac {B^2 i^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac {B^2 i^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac {B i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d^3}{6 b^3 (b c-a d) g^5 (a+b x)}+\frac {7 B^2 i^2 d^3}{72 b^3 (b c-a d) g^5 (a+b x)}-\frac {i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 d^2}{2 b^3 g^5 (a+b x)^2}-\frac {B i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d^2}{12 b^3 g^5 (a+b x)^2}+\frac {5 B^2 i^2 d^2}{144 b^3 g^5 (a+b x)^2}-\frac {2 (b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 d}{3 b^3 g^5 (a+b x)^3}-\frac {5 B (b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d}{18 b^3 g^5 (a+b x)^3}-\frac {11 B^2 (b c-a d) i^2 d}{216 b^3 g^5 (a+b x)^3}-\frac {(b c-a d)^2 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b^3 g^5 (a+b x)^4}-\frac {B (b c-a d)^2 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 b^3 g^5 (a+b x)^4}-\frac {B^2 (b c-a d)^2 i^2}{32 b^3 g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B^2*(b*c - a*d)^2*i^2)/(32*b^3*g^5*(a + b*x)^4) - (11*B^2*d*(b*c - a*d)*i^2)/(216*b^3*g^5*(a + b*x)^3) + (5*
B^2*d^2*i^2)/(144*b^3*g^5*(a + b*x)^2) + (7*B^2*d^3*i^2)/(72*b^3*(b*c - a*d)*g^5*(a + b*x)) + (7*B^2*d^4*i^2*L
og[a + b*x])/(72*b^3*(b*c - a*d)^2*g^5) - (B^2*d^4*i^2*Log[a + b*x]^2)/(12*b^3*(b*c - a*d)^2*g^5) - (B*(b*c -
a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*b^3*g^5*(a + b*x)^4) - (5*B*d*(b*c - a*d)*i^2*(A + B*Log[(
e*(a + b*x))/(c + d*x)]))/(18*b^3*g^5*(a + b*x)^3) - (B*d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(12*b^3*
g^5*(a + b*x)^2) + (B*d^3*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3*(b*c - a*d)*g^5*(a + b*x)) + (B*d^4
*i^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3*(b*c - a*d)^2*g^5) - ((b*c - a*d)^2*i^2*(A + B*
Log[(e*(a + b*x))/(c + d*x)])^2)/(4*b^3*g^5*(a + b*x)^4) - (2*d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c +
d*x)])^2)/(3*b^3*g^5*(a + b*x)^3) - (d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^3*g^5*(a + b*x)^2) -
(7*B^2*d^4*i^2*Log[c + d*x])/(72*b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[
c + d*x])/(6*b^3*(b*c - a*d)^2*g^5) - (B*d^4*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(6*b^3*(b*
c - a*d)^2*g^5) - (B^2*d^4*i^2*Log[c + d*x]^2)/(12*b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^2*Log[a + b*x]*Log[(b*(
c + d*x))/(b*c - a*d)])/(6*b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(6*
b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(6*b^3*(b*c - a*d)^2*g^5)

Rule 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 {(72 c+72 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx &=\int \left (\frac {5184 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^5}+\frac {10368 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^4}+\frac {5184 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^5 (a+b x)^3}\right ) \, dx\\ &=\frac {\left (5184 d^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{b^2 g^5}+\frac {(10368 d (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{b^2 g^5}+\frac {\left (5184 (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (5184 B d^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {(6912 B d (b c-a d)) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (2592 B (b c-a d)^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (5184 B d^2 (b c-a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (6912 B d (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (2592 B (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (5184 B d^2 (b c-a d)\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (6912 B d (b c-a d)^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^4}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (2592 B (b c-a d)^3\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^5}\\ &=-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (2592 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^5}+\frac {\left (5184 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^5}-\frac {\left (6912 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^5}+\frac {\left (2592 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (5184 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (6912 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (2592 B d^5\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (5184 B d^5\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (6912 B d^5\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d) g^5}-\frac {\left (5184 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d) g^5}+\frac {\left (6912 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d) g^5}-\frac {(2592 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^2 g^5}+\frac {(6912 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^2 g^5}+\frac {\left (2592 B (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac {648 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {1440 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac {432 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac {864 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac {864 B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {864 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac {\left (1296 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (2592 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (3456 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (2592 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (2592 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (5184 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (5184 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (6912 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (6912 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^3\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}-\frac {\left (5184 B^2 d^3\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}+\frac {\left (6912 B^2 d^3\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}-\frac {\left (864 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (2304 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (648 B^2 (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {648 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {1440 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac {432 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac {864 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac {864 B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {864 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (5184 B^2 d^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (6912 B^2 d^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (1296 B^2 d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (2592 B^2 d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (3456 B^2 d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (864 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (2304 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (648 B^2 (b c-a d)^3\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (2592 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d)^2 e g^5}+\frac {\left (2592 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^2 e g^5}-\frac {\left (5184 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d)^2 e g^5}+\frac {\left (5184 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^2 e g^5}+\frac {\left (6912 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d)^2 e g^5}-\frac {\left (6912 B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^2 e g^5}\\ &=-\frac {648 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {1440 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac {432 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac {864 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac {864 B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {864 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^5}-\frac {\left (5184 B^2 d^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (6912 B^2 d^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (1296 B^2 d^2 (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (2592 B^2 d^2 (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}-\frac {\left (3456 B^2 d^2 (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}-\frac {\left (864 B^2 d (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (2304 B^2 d (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (648 B^2 (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^5}-\frac {\left (2592 B^2 d^4\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 e g^5}+\frac {\left (2592 B^2 d^4\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 e g^5}-\frac {\left (5184 B^2 d^4\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 e g^5}+\frac {\left (5184 B^2 d^4\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 e g^5}+\frac {\left (6912 B^2 d^4\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 e g^5}-\frac {\left (6912 B^2 d^4\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 e g^5}\\ &=-\frac {162 B^2 (b c-a d)^2}{b^3 g^5 (a+b x)^4}-\frac {264 B^2 d (b c-a d)}{b^3 g^5 (a+b x)^3}+\frac {180 B^2 d^2}{b^3 g^5 (a+b x)^2}+\frac {504 B^2 d^3}{b^3 (b c-a d) g^5 (a+b x)}+\frac {504 B^2 d^4 \log (a+b x)}{b^3 (b c-a d)^2 g^5}-\frac {648 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {1440 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac {432 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac {864 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac {864 B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {504 B^2 d^4 \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac {864 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^4\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (2592 B^2 d^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (5184 B^2 d^4\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (5184 B^2 d^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (6912 B^2 d^4\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (6912 B^2 d^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (2592 B^2 d^5\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^5\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (5184 B^2 d^5\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (5184 B^2 d^5\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (6912 B^2 d^5\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (6912 B^2 d^5\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}\\ &=-\frac {162 B^2 (b c-a d)^2}{b^3 g^5 (a+b x)^4}-\frac {264 B^2 d (b c-a d)}{b^3 g^5 (a+b x)^3}+\frac {180 B^2 d^2}{b^3 g^5 (a+b x)^2}+\frac {504 B^2 d^3}{b^3 (b c-a d) g^5 (a+b x)}+\frac {504 B^2 d^4 \log (a+b x)}{b^3 (b c-a d)^2 g^5}-\frac {648 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {1440 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac {432 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac {864 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac {864 B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {504 B^2 d^4 \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac {864 B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac {864 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac {864 B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (5184 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (5184 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}+\frac {\left (6912 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}+\frac {\left (6912 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^4\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (5184 B^2 d^4\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (6912 B^2 d^4\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^5\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (5184 B^2 d^5\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (6912 B^2 d^5\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}\\ &=-\frac {162 B^2 (b c-a d)^2}{b^3 g^5 (a+b x)^4}-\frac {264 B^2 d (b c-a d)}{b^3 g^5 (a+b x)^3}+\frac {180 B^2 d^2}{b^3 g^5 (a+b x)^2}+\frac {504 B^2 d^3}{b^3 (b c-a d) g^5 (a+b x)}+\frac {504 B^2 d^4 \log (a+b x)}{b^3 (b c-a d)^2 g^5}-\frac {432 B^2 d^4 \log ^2(a+b x)}{b^3 (b c-a d)^2 g^5}-\frac {648 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {1440 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac {432 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac {864 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac {864 B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {504 B^2 d^4 \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac {864 B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac {864 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac {432 B^2 d^4 \log ^2(c+d x)}{b^3 (b c-a d)^2 g^5}+\frac {864 B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (2592 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (5184 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (5184 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}+\frac {\left (6912 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}+\frac {\left (6912 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}\\ &=-\frac {162 B^2 (b c-a d)^2}{b^3 g^5 (a+b x)^4}-\frac {264 B^2 d (b c-a d)}{b^3 g^5 (a+b x)^3}+\frac {180 B^2 d^2}{b^3 g^5 (a+b x)^2}+\frac {504 B^2 d^3}{b^3 (b c-a d) g^5 (a+b x)}+\frac {504 B^2 d^4 \log (a+b x)}{b^3 (b c-a d)^2 g^5}-\frac {432 B^2 d^4 \log ^2(a+b x)}{b^3 (b c-a d)^2 g^5}-\frac {648 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {1440 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^3}-\frac {432 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^5 (a+b x)^2}+\frac {864 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d) g^5 (a+b x)}+\frac {864 B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 (b c-a d)^2 g^5}-\frac {1296 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {3456 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^3}-\frac {2592 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {504 B^2 d^4 \log (c+d x)}{b^3 (b c-a d)^2 g^5}+\frac {864 B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac {864 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^3 (b c-a d)^2 g^5}-\frac {432 B^2 d^4 \log ^2(c+d x)}{b^3 (b c-a d)^2 g^5}+\frac {864 B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}+\frac {864 B^2 d^4 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}+\frac {864 B^2 d^4 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 (b c-a d)^2 g^5}\\ \end {align*}

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Mathematica [C]  time = 3.10, size = 1788, normalized size = 5.98 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/864*(i^2*(216*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 576*d*(-(b*c) + a*d)^3*(a + b*x)*(A +
B*Log[(e*(a + b*x))/(c + d*x)])^2 + 432*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 +
216*B*d^2*(a + b*x)^2*(2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*d*(-(b*c) + a*d)*(a + b*x)*(A
+ B*Log[(e*(a + b*x))/(c + d*x)]) - 4*d^2*(a + b*x)^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*d
^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 4*B*d*(a + b*x)*(b*c - a*d + d*(a + b*x)*Lo
g[a + b*x] - d*(a + b*x)*Log[c + d*x]) + B*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x)^2*L
og[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*d^2*(a + b*x)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c
+ d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) - 2*B*d^2*(a + b*x)^2*((2*Log[(d*(a + b*x)
)/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + 32*B*d*(a + b*x)*
(12*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 18*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*(a + b*x))
/(c + d*x)]) + 36*d^2*(b*c - a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 36*d^3*(a + b*x)^3*Log[a
+ b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 36*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c +
d*x] + 36*B*d^2*(a + b*x)^2*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - 9*B*d*(a + b*x
)*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d
*x]) + 2*B*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d)^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*
Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]) - 18*B*d^3*(a + b*x)^3*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(
c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 18*B*d^3*(a + b*x)^3*((2*Log[(d*(a + b
*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + 3*B*(36*(b*c -
a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 48*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d
*x)]) + 72*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 144*d^3*(-(b*c) + a*d)*(a + b*
x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 144*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x
)]) + 144*d^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 144*B*d^3*(a + b*x)^3*(b*c - a*d
+ d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + 36*B*d^2*(a + b*x)^2*((b*c - a*d)^2 + 2*d*(-(b*c) +
a*d)*(a + b*x) - 2*d^2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) - 8*B*d*(a + b*x)*(2*(b*c -
a*d)^3 - 3*d*(b*c - a*d)^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*
(a + b*x)^3*Log[c + d*x]) + 3*B*(3*(b*c - a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b*c - a*d)^2*(a + b
*x)^2 + 12*d^3*(-(b*c) + a*d)*(a + b*x)^3 - 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x])
+ 72*B*d^4*(a + b*x)^4*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a +
b*x))/(-(b*c) + a*d)]) - 72*B*d^4*(a + b*x)^4*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d
*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))))/(b^3*(b*c - a*d)^2*g^5*(a + b*x)^4)

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fricas [B]  time = 0.89, size = 837, normalized size = 2.80 \[ \frac {12 \, {\left ({\left (12 \, A B + 7 \, B^{2}\right )} b^{4} c d^{3} - {\left (12 \, A B + 7 \, B^{2}\right )} a b^{3} d^{4}\right )} i^{2} x^{3} - 6 \, {\left ({\left (72 \, A^{2} + 12 \, A B - 5 \, B^{2}\right )} b^{4} c^{2} d^{2} - 16 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{3} c d^{3} + {\left (72 \, A^{2} + 84 \, A B + 37 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} i^{2} x^{2} - 4 \, {\left ({\left (144 \, A^{2} + 60 \, A B + 11 \, B^{2}\right )} b^{4} c^{3} d - 24 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{3} c^{2} d^{2} + {\left (72 \, A^{2} + 84 \, A B + 37 \, B^{2}\right )} a^{3} b d^{4}\right )} i^{2} x - {\left (27 \, {\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{4} - 32 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{3} c^{3} d + {\left (72 \, A^{2} + 84 \, A B + 37 \, B^{2}\right )} a^{4} d^{4}\right )} i^{2} + 72 \, {\left (B^{2} b^{4} d^{4} i^{2} x^{4} + 4 \, B^{2} a b^{3} d^{4} i^{2} x^{3} - 6 \, {\left (B^{2} b^{4} c^{2} d^{2} - 2 \, B^{2} a b^{3} c d^{3}\right )} i^{2} x^{2} - 4 \, {\left (2 \, B^{2} b^{4} c^{3} d - 3 \, B^{2} a b^{3} c^{2} d^{2}\right )} i^{2} x - {\left (3 \, B^{2} b^{4} c^{4} - 4 \, B^{2} a b^{3} c^{3} d\right )} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 12 \, {\left ({\left (12 \, A B + 7 \, B^{2}\right )} b^{4} d^{4} i^{2} x^{4} + 4 \, {\left (3 \, B^{2} b^{4} c d^{3} + 4 \, {\left (3 \, A B + B^{2}\right )} a b^{3} d^{4}\right )} i^{2} x^{3} - 6 \, {\left ({\left (12 \, A B + B^{2}\right )} b^{4} c^{2} d^{2} - 8 \, {\left (3 \, A B + B^{2}\right )} a b^{3} c d^{3}\right )} i^{2} x^{2} - 4 \, {\left ({\left (24 \, A B + 5 \, B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (3 \, A B + B^{2}\right )} a b^{3} c^{2} d^{2}\right )} i^{2} x - {\left (9 \, {\left (4 \, A B + B^{2}\right )} b^{4} c^{4} - 16 \, {\left (3 \, A B + B^{2}\right )} a b^{3} c^{3} d\right )} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{864 \, {\left ({\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{2} - 2 \, a^{2} b^{7} c d + a^{3} b^{6} d^{2}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{2} - 2 \, a^{3} b^{6} c d + a^{4} b^{5} d^{2}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{2} - 2 \, a^{4} b^{5} c d + a^{5} b^{4} d^{2}\right )} g^{5} x + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} g^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 2.82, size = 425, normalized size = 1.42 \[ \frac {{\left (216 \, B^{2} b e^{5} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} - \frac {288 \, {\left (b x e + a e\right )} B^{2} d e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} + 432 \, A B b e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) + 108 \, B^{2} b e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {576 \, {\left (b x e + a e\right )} A B d e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {192 \, {\left (b x e + a e\right )} B^{2} d e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + 216 \, A^{2} b e^{5} + 108 \, A B b e^{5} + 27 \, B^{2} b e^{5} - \frac {288 \, {\left (b x e + a e\right )} A^{2} d e^{4}}{d x + c} - \frac {192 \, {\left (b x e + a e\right )} A B d e^{4}}{d x + c} - \frac {64 \, {\left (b x e + a e\right )} B^{2} d e^{4}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{864 \, {\left (\frac {{\left (b x e + a e\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x e + a e\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(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*e^5*log((b*x*e + a*e)/(d*x + c))^2 - 288*(b*x*e + a*e)*B^2*d*e^4*log((b*x*e + a*e)/(d*x + c))
^2/(d*x + c) + 432*A*B*b*e^5*log((b*x*e + a*e)/(d*x + c)) + 108*B^2*b*e^5*log((b*x*e + a*e)/(d*x + c)) - 576*(
b*x*e + a*e)*A*B*d*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 192*(b*x*e + a*e)*B^2*d*e^4*log((b*x*e + a*e)/
(d*x + c))/(d*x + c) + 216*A^2*b*e^5 + 108*A*B*b*e^5 + 27*B^2*b*e^5 - 288*(b*x*e + a*e)*A^2*d*e^4/(d*x + c) -
192*(b*x*e + a*e)*A*B*d*e^4/(d*x + c) - 64*(b*x*e + a*e)*B^2*d*e^4/(d*x + c))*(b*c/((b*c*e - a*d*e)*(b*c - a*d
)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/((b*x*e + a*e)^4*b*c*g^5/(d*x + c)^4 - (b*x*e + a*e)^4*a*d*g^5/(d*x +
c)^4)

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maple [B]  time = 0.05, size = 1814, normalized size = 6.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 8.09, size = 8031, normalized size = 26.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(4*b*x + a)*B^2*c*d*i^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4
*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/12*(6*b^2*x^2 + 4*a*b*x + a^2)*B^2*d^2*i^2*log(b*e*x/(d*x + c) +
a*e/(d*x + c))^2/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) + 1/288*
(12*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*
x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^
3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*
b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*
b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^
5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4
- 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*log(b*e*x/(d*x + c) + a*e/(d*x + c))
- (9*b^4*c^4 - 64*a*b^3*c^3*d + 216*a^2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d
^4)*x^3 + 6*(13*b^4*c^2*d^2 - 176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a
^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x
^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 324*a^2*b^2*c*d^3 - 271*a^3
*b*d^4)*x - 300*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 1
2*(25*b^4*d^4*x^4 + 100*a*b^3*d^4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*x + 25*a^4*d^4 - 12*(b^4*d^4*x^4 +
4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a))*log(d*x + c))/(a^4*b^5*c^4*g^5 -
4*a^5*b^4*c^3*d*g^5 + 6*a^6*b^3*c^2*d^2*g^5 - 4*a^7*b^2*c*d^3*g^5 + a^8*b*d^4*g^5 + (b^9*c^4*g^5 - 4*a*b^8*c^
3*d*g^5 + 6*a^2*b^7*c^2*d^2*g^5 - 4*a^3*b^6*c*d^3*g^5 + a^4*b^5*d^4*g^5)*x^4 + 4*(a*b^8*c^4*g^5 - 4*a^2*b^7*c^
3*d*g^5 + 6*a^3*b^6*c^2*d^2*g^5 - 4*a^4*b^5*c*d^3*g^5 + a^5*b^4*d^4*g^5)*x^3 + 6*(a^2*b^7*c^4*g^5 - 4*a^3*b^6*
c^3*d*g^5 + 6*a^4*b^5*c^2*d^2*g^5 - 4*a^5*b^4*c*d^3*g^5 + a^6*b^3*d^4*g^5)*x^2 + 4*(a^3*b^6*c^4*g^5 - 4*a^4*b^
5*c^3*d*g^5 + 6*a^5*b^4*c^2*d^2*g^5 - 4*a^6*b^3*c*d^3*g^5 + a^7*b^2*d^4*g^5)*x))*B^2*c^2*i^2 - 1/432*(12*((7*a
*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d
- 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((
b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*
c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(
a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*
b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*
d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*
a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (37*a*b^4*c^4 -
304*a^2*b^3*c^3*d + 1512*a^3*b^2*c^2*d^2 - 1360*a^4*b*c*d^3 + 115*a^5*d^4 + 12*(88*b^5*c^2*d^2 - 101*a*b^4*c*d
^3 + 13*a^2*b^3*d^4)*x^3 - 6*(40*b^5*c^3*d - 609*a*b^4*c^2*d^2 + 648*a^2*b^3*c*d^3 - 79*a^3*b^2*d^4)*x^2 - 72*
(4*a^4*b*c*d^3 - a^5*d^4 + (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^3 + 6*(4*a^2*b^3*
c*d^3 - a^3*b^2*d^4)*x^2 + 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(b*x + a)^2 - 72*(4*a^4*b*c*d^3 - a^5*d^4 + (
4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^3 + 6*(4*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2 + 4
*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(d*x + c)^2 + 4*(16*b^5*c^4 - 163*a*b^4*c^3*d + 1068*a^2*b^3*c^2*d^2 - 10
36*a^3*b^2*c*d^3 + 115*a^4*b*d^4)*x + 12*(88*a^4*b*c*d^3 - 13*a^5*d^4 + (88*b^5*c*d^3 - 13*a*b^4*d^4)*x^4 + 4*
(88*a*b^4*c*d^3 - 13*a^2*b^3*d^4)*x^3 + 6*(88*a^2*b^3*c*d^3 - 13*a^3*b^2*d^4)*x^2 + 4*(88*a^3*b^2*c*d^3 - 13*a
^4*b*d^4)*x)*log(b*x + a) - 12*(88*a^4*b*c*d^3 - 13*a^5*d^4 + (88*b^5*c*d^3 - 13*a*b^4*d^4)*x^4 + 4*(88*a*b^4*
c*d^3 - 13*a^2*b^3*d^4)*x^3 + 6*(88*a^2*b^3*c*d^3 - 13*a^3*b^2*d^4)*x^2 + 4*(88*a^3*b^2*c*d^3 - 13*a^4*b*d^4)*
x - 12*(4*a^4*b*c*d^3 - a^5*d^4 + (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^3 + 6*(4*a
^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2 + 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(b*x + a))*log(d*x + c))/(a^4*b^6*c^4*
g^5 - 4*a^5*b^5*c^3*d*g^5 + 6*a^6*b^4*c^2*d^2*g^5 - 4*a^7*b^3*c*d^3*g^5 + a^8*b^2*d^4*g^5 + (b^10*c^4*g^5 - 4*
a*b^9*c^3*d*g^5 + 6*a^2*b^8*c^2*d^2*g^5 - 4*a^3*b^7*c*d^3*g^5 + a^4*b^6*d^4*g^5)*x^4 + 4*(a*b^9*c^4*g^5 - 4*a^
2*b^8*c^3*d*g^5 + 6*a^3*b^7*c^2*d^2*g^5 - 4*a^4*b^6*c*d^3*g^5 + a^5*b^5*d^4*g^5)*x^3 + 6*(a^2*b^8*c^4*g^5 - 4*
a^3*b^7*c^3*d*g^5 + 6*a^4*b^6*c^2*d^2*g^5 - 4*a^5*b^5*c*d^3*g^5 + a^6*b^4*d^4*g^5)*x^2 + 4*(a^3*b^7*c^4*g^5 -
4*a^4*b^6*c^3*d*g^5 + 6*a^5*b^5*c^2*d^2*g^5 - 4*a^6*b^4*c*d^3*g^5 + a^7*b^3*d^4*g^5)*x))*B^2*c*d*i^2 - 1/864*(
12*((13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*d^2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^
3*d^3)*x^3 + 6*(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 + 4*(10*a*b^4*c^3 - 63*a^2*
b^3*c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5
*x^4 + 4*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a^2*b^8*c^3 - 3*a^3*b^7*c^
2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^3*b^7*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^
3)*g^5*x + (a^4*b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*(6*b^2*c^2*d^2 - 4*a*b*c*
d^3 + a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5
) + 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(d*x + c)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*
a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (115*a^2*b^4*c^4 - 1360*a^3*b^3*c^3*
d + 1512*a^4*b^2*c^2*d^2 - 304*a^5*b*c*d^3 + 37*a^6*d^4 - 12*(108*b^6*c^3*d - 148*a*b^5*c^2*d^2 + 47*a^2*b^4*c
*d^3 - 7*a^3*b^3*d^4)*x^3 + 6*(36*b^6*c^4 - 712*a*b^5*c^3*d + 903*a^2*b^4*c^2*d^2 - 264*a^3*b^3*c*d^3 + 37*a^4
*b^2*d^4)*x^2 + 72*(6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4 + (6*b^6*c^2*d^2 - 4*a*b^5*c*d^3 + a^2*b^4*d^4
)*x^4 + 4*(6*a*b^5*c^2*d^2 - 4*a^2*b^4*c*d^3 + a^3*b^3*d^4)*x^3 + 6*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4
*b^2*d^4)*x^2 + 4*(6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4)*x)*log(b*x + a)^2 + 72*(6*a^4*b^2*c^2*d^2
- 4*a^5*b*c*d^3 + a^6*d^4 + (6*b^6*c^2*d^2 - 4*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 + 4*(6*a*b^5*c^2*d^2 - 4*a^2*b^4
*c*d^3 + a^3*b^3*d^4)*x^3 + 6*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^2 + 4*(6*a^3*b^3*c^2*d^2 -
4*a^4*b^2*c*d^3 + a^5*b*d^4)*x)*log(d*x + c)^2 + 4*(76*a*b^5*c^4 - 1057*a^2*b^4*c^3*d + 1248*a^3*b^3*c^2*d^2
- 304*a^4*b^2*c*d^3 + 37*a^5*b*d^4)*x - 12*(108*a^4*b^2*c^2*d^2 - 40*a^5*b*c*d^3 + 7*a^6*d^4 + (108*b^6*c^2*d^
2 - 40*a*b^5*c*d^3 + 7*a^2*b^4*d^4)*x^4 + 4*(108*a*b^5*c^2*d^2 - 40*a^2*b^4*c*d^3 + 7*a^3*b^3*d^4)*x^3 + 6*(10
8*a^2*b^4*c^2*d^2 - 40*a^3*b^3*c*d^3 + 7*a^4*b^2*d^4)*x^2 + 4*(108*a^3*b^3*c^2*d^2 - 40*a^4*b^2*c*d^3 + 7*a^5*
b*d^4)*x)*log(b*x + a) + 12*(108*a^4*b^2*c^2*d^2 - 40*a^5*b*c*d^3 + 7*a^6*d^4 + (108*b^6*c^2*d^2 - 40*a*b^5*c*
d^3 + 7*a^2*b^4*d^4)*x^4 + 4*(108*a*b^5*c^2*d^2 - 40*a^2*b^4*c*d^3 + 7*a^3*b^3*d^4)*x^3 + 6*(108*a^2*b^4*c^2*d
^2 - 40*a^3*b^3*c*d^3 + 7*a^4*b^2*d^4)*x^2 + 4*(108*a^3*b^3*c^2*d^2 - 40*a^4*b^2*c*d^3 + 7*a^5*b*d^4)*x - 12*(
6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4 + (6*b^6*c^2*d^2 - 4*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 + 4*(6*a*b^5*c
^2*d^2 - 4*a^2*b^4*c*d^3 + a^3*b^3*d^4)*x^3 + 6*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^2 + 4*(6
*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4)*x)*log(b*x + a))*log(d*x + c))/(a^4*b^7*c^4*g^5 - 4*a^5*b^6*c^
3*d*g^5 + 6*a^6*b^5*c^2*d^2*g^5 - 4*a^7*b^4*c*d^3*g^5 + a^8*b^3*d^4*g^5 + (b^11*c^4*g^5 - 4*a*b^10*c^3*d*g^5 +
6*a^2*b^9*c^2*d^2*g^5 - 4*a^3*b^8*c*d^3*g^5 + a^4*b^7*d^4*g^5)*x^4 + 4*(a*b^10*c^4*g^5 - 4*a^2*b^9*c^3*d*g^5
+ 6*a^3*b^8*c^2*d^2*g^5 - 4*a^4*b^7*c*d^3*g^5 + a^5*b^6*d^4*g^5)*x^3 + 6*(a^2*b^9*c^4*g^5 - 4*a^3*b^8*c^3*d*g^
5 + 6*a^4*b^7*c^2*d^2*g^5 - 4*a^5*b^6*c*d^3*g^5 + a^6*b^5*d^4*g^5)*x^2 + 4*(a^3*b^8*c^4*g^5 - 4*a^4*b^7*c^3*d*
g^5 + 6*a^5*b^6*c^2*d^2*g^5 - 4*a^6*b^5*c*d^3*g^5 + a^7*b^4*d^4*g^5)*x))*B^2*d^2*i^2 - 1/72*A*B*d^2*i^2*(12*(6
*b^2*x^2 + 4*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*
x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) + (13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*d^2 - 7*a^5*d^3 - 12*(6
*b^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 6*(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d
^3)*x^2 + 4*(10*a*b^4*c^3 - 63*a^2*b^3*c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^10*c^3 - 3*a*b^9*c^2*d +
3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5*x^4 + 4*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*
x^3 + 6*(a^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^3*b^7*c^3 - 3*a^4*b^6*c
^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*g^5*x + (a^4*b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*
g^5) - 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 -
4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5) + 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(d*x + c)/((b^7*c^4 - 4*a
*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5)) - 1/36*A*B*c*d*i^2*(12*(4*b*x + a)*log(b
*e*x/(d*x + c) + a*e/(d*x + c))/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2
*g^5) + (7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(
4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*
b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d
+ 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g
^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^
2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*
a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d*x + c)/((b^6*c^4 - 4*a*b^
5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)) + 1/24*A*B*c^2*i^2*((12*b^3*d^3*x^3 - 3*b^3
*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2
*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 -
3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2
- a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^
3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) - 12*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^5*x^4
+ 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^
4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*
d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 1/4*B^2*c^2*i^2*log(b*e*x/(d*x + c) + a*e/(d*x +
c))^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) - 1/6*(4*b*x + a)*A^2*
c*d*i^2/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/12*(6*b^2*x^2
+ 4*a*b*x + a^2)*A^2*d^2*i^2/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^
5) - 1/4*A^2*c^2*i^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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mupad [B]  time = 11.20, size = 1940, normalized size = 6.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- log((e*(a + b*x))/(c + d*x))^2*((x*(b*((B^2*c*d*i^2)/(6*b^3*g^5) + (B^2*a*d^2*i^2)/(12*b^4*g^5)) + (B^2*c*d*
i^2)/(2*b^2*g^5) + (B^2*a*d^2*i^2)/(4*b^3*g^5)) + a*((B^2*c*d*i^2)/(6*b^3*g^5) + (B^2*a*d^2*i^2)/(12*b^4*g^5))
+ (B^2*c^2*i^2)/(4*b^2*g^5) + (B^2*d^2*i^2*x^2)/(2*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^2)/(12*b^3*g^5*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - ((72*A^2*a^3*d^3*i^2 - 216*A^2*b^3*c^3
*i^2 + 37*B^2*a^3*d^3*i^2 - 27*B^2*b^3*c^3*i^2 + 84*A*B*a^3*d^3*i^2 - 108*A*B*b^3*c^3*i^2 + 72*A^2*a*b^2*c^2*d
*i^2 + 72*A^2*a^2*b*c*d^2*i^2 + 37*B^2*a*b^2*c^2*d*i^2 + 37*B^2*a^2*b*c*d^2*i^2 + 84*A*B*a*b^2*c^2*d*i^2 + 84*
A*B*a^2*b*c*d^2*i^2)/(12*(a*d - b*c)) + (x^3*(7*B^2*b^3*d^3*i^2 + 12*A*B*b^3*d^3*i^2))/(a*d - b*c) + (x*(72*A^
2*a^2*b*d^3*i^2 + 37*B^2*a^2*b*d^3*i^2 - 144*A^2*b^3*c^2*d*i^2 - 11*B^2*b^3*c^2*d*i^2 + 72*A^2*a*b^2*c*d^2*i^2
+ 37*B^2*a*b^2*c*d^2*i^2 + 84*A*B*a^2*b*d^3*i^2 - 60*A*B*b^3*c^2*d*i^2 + 84*A*B*a*b^2*c*d^2*i^2))/(3*(a*d - b
*c)) + (x^2*(72*A^2*a*b^2*d^3*i^2 + 37*B^2*a*b^2*d^3*i^2 - 72*A^2*b^3*c*d^2*i^2 + 5*B^2*b^3*c*d^2*i^2 + 84*A*B
*a*b^2*d^3*i^2 - 12*A*B*b^3*c*d^2*i^2))/(2*(a*d - b*c)))/(72*a^4*b^3*g^5 + 72*b^7*g^5*x^4 + 288*a^3*b^4*g^5*x
+ 288*a*b^6*g^5*x^3 + 432*a^2*b^5*g^5*x^2) - (log((e*(a + b*x))/(c + d*x))*(x^2*((A*B*d*i^2)/(b^2*g^5) + (B^2*
d^4*i^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^3*g^5*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + a*((B*i^2*(4*A*b*
c - B*a*d + B*b*c))/(12*b^4*g^5) + (A*B*a*d*i^2)/(6*b^4*g^5)) + x*(b*((B*i^2*(4*A*b*c - B*a*d + B*b*c))/(12*b^
4*g^5) + (A*B*a*d*i^2)/(6*b^4*g^5)) + (B*i^2*(4*A*b*c - B*a*d + B*b*c))/(4*b^3*g^5) + (B^2*d^4*i^2*(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^3*g^5*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (A*B*a*d*i^2)/(2*b^3*g^5)) + (B*i^2
*(6*A*b^2*c^2 - 2*B*a^2*d^2 + B*b^2*c^2 + B*a*b*c*d))/(12*b^4*d*g^5) + (B^2*d^4*i^2*(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^3
*g^5*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (B^2*d^4*i^2*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^3*g^5*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))))/((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^2*atan(((2*b*d*x - (72*b^5*c^2*g^5 - 72*a^2*b^3*d^2*g^5
)/(72*b^3*g^5*(a*d - b*c)))*1i)/(a*d - b*c))*(12*A + 7*B)*1i)/(36*b^3*g^5*(a*d - b*c)^2)

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sympy [B]  time = 134.61, size = 2055, normalized size = 6.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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