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

Optimal. Leaf size=268 \[ -\frac {b B (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)^2}+\frac {2 B d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 (a+b x) (b c-a d)^2}-\frac {b (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)^2}+\frac {d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^3 (a+b x) (b c-a d)^2}-\frac {b B^2 (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)^2}+\frac {2 B^2 d (c+d x)}{g^3 (a+b x) (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.91, antiderivative size = 577, normalized size of antiderivative = 2.15, number of steps used = 30, number of rules used = 11, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B^2 d^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}+\frac {B^2 d^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}+\frac {B d^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g^3 (b c-a d)^2}-\frac {B d^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g^3 (b c-a d)^2}+\frac {B d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g^3 (a+b x) (b c-a d)}-\frac {B \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b g^3 (a+b x)^2}-\frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 b g^3 (a+b x)^2}-\frac {B^2 d^2 \log ^2(a+b x)}{2 b g^3 (b c-a d)^2}-\frac {B^2 d^2 \log ^2(c+d x)}{2 b g^3 (b c-a d)^2}+\frac {3 B^2 d^2 \log (a+b x)}{2 b g^3 (b c-a d)^2}+\frac {B^2 d^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}-\frac {3 B^2 d^2 \log (c+d x)}{2 b g^3 (b c-a d)^2}+\frac {B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g^3 (b c-a d)^2}+\frac {3 B^2 d}{2 b g^3 (a+b x) (b c-a d)}-\frac {B^2}{4 b g^3 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {B \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{g^2 (a+b x)^3 (c+d x)} \, dx}{b g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {(B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{b g^3}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {(B (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)^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 g^3}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}+\frac {B \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{g^3}+\frac {\left (B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{(b c-a d)^2 g^3}-\frac {\left (B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b (b c-a d)^2 g^3}-\frac {(B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{(b c-a d) g^3}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}+\frac {B^2 \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b g^3}-\frac {\left (B^2 d^2\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 (b c-a d)^2 g^3}+\frac {\left (B^2 d^2\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 (b c-a d)^2 g^3}-\frac {\left (B^2 d\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b (b c-a d) g^3}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b g^3}+\frac {\left (B^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b g^3}-\frac {\left (B^2 d^2\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 (b c-a d)^2 e g^3}+\frac {\left (B^2 d^2\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 (b c-a d)^2 e g^3}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b g^3}+\frac {\left (B^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}{2 b g^3}-\frac {\left (B^2 d^2\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 (b c-a d)^2 e g^3}+\frac {\left (B^2 d^2\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 (b c-a d)^2 e g^3}\\ &=-\frac {B^2}{4 b g^3 (a+b x)^2}+\frac {3 B^2 d}{2 b (b c-a d) g^3 (a+b x)}+\frac {3 B^2 d^2 \log (a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {3 B^2 d^2 \log (c+d x)}{2 b (b c-a d)^2 g^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{(b c-a d)^2 g^3}+\frac {\left (B^2 d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{(b c-a d)^2 g^3}+\frac {\left (B^2 d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b (b c-a d)^2 g^3}\\ &=-\frac {B^2}{4 b g^3 (a+b x)^2}+\frac {3 B^2 d}{2 b (b c-a d) g^3 (a+b x)}+\frac {3 B^2 d^2 \log (a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {3 B^2 d^2 \log (c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}+\frac {B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d)^2 g^3}-\frac {\left (B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d)^2 g^3}\\ &=-\frac {B^2}{4 b g^3 (a+b x)^2}+\frac {3 B^2 d}{2 b (b c-a d) g^3 (a+b x)}+\frac {3 B^2 d^2 \log (a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B^2 d^2 \log ^2(a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {3 B^2 d^2 \log (c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B^2 d^2 \log ^2(c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}-\frac {\left (B^2 d^2\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 (b c-a d)^2 g^3}-\frac {\left (B^2 d^2\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 (b c-a d)^2 g^3}\\ &=-\frac {B^2}{4 b g^3 (a+b x)^2}+\frac {3 B^2 d}{2 b (b c-a d) g^3 (a+b x)}+\frac {3 B^2 d^2 \log (a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B^2 d^2 \log ^2(a+b x)}{2 b (b c-a d)^2 g^3}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g^3 (a+b x)^2}-\frac {3 B^2 d^2 \log (c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b (b c-a d)^2 g^3}-\frac {B^2 d^2 \log ^2(c+d x)}{2 b (b c-a d)^2 g^3}+\frac {B^2 d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac {B^2 d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}+\frac {B^2 d^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 g^3}\\ \end {align*}

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Mathematica [C]  time = 0.46, size = 443, normalized size = 1.65 \[ -\frac {\frac {B \left (-4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+4 d (a+b x) (a d-b c) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-2 B d^2 (a+b x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+B \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )-4 B d (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )}{(b c-a d)^2}+2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b g^3 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(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)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + B*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a
+ b*x) - 2*d^2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*d^2*(a + b*x)^2*(Log[a + b*x]
*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) - 2*B*d^2*(a
+ b*x)^2*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c
- a*d)])))/(b*c - a*d)^2)/(b*g^3*(a + b*x)^2)

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fricas [A]  time = 0.57, size = 367, normalized size = 1.37 \[ -\frac {{\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - 4 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b c d + {\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x - B^{2} b^{2} c^{2} + 2 \, B^{2} a b c d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} c d - {\left (2 \, A B + 3 \, B^{2}\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - {\left (2 \, A B + B^{2}\right )} b^{2} c^{2} + 4 \, {\left (A B + B^{2}\right )} a b c d + 2 \, {\left (B^{2} b^{2} c d + 2 \, {\left (A B + B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.75, size = 424, normalized size = 1.58 \[ -\frac {{\left (2 \, B^{2} b e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} - \frac {4 \, {\left (b x e + a e\right )} B^{2} d e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} + 4 \, A B b e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) + 2 \, B^{2} b e^{3} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {8 \, {\left (b x e + a e\right )} A B d e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {8 \, {\left (b x e + a e\right )} B^{2} d e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + 2 \, A^{2} b e^{3} + 2 \, A B b e^{3} + B^{2} b e^{3} - \frac {4 \, {\left (b x e + a e\right )} A^{2} d e^{2}}{d x + c} - \frac {8 \, {\left (b x e + a e\right )} A B d e^{2}}{d x + c} - \frac {8 \, {\left (b x e + a e\right )} B^{2} d e^{2}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{4 \, {\left (\frac {{\left (b x e + a e\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x e + a e\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 1715, normalized size = 6.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.81, size = 848, normalized size = 3.16 \[ \frac {1}{4} \, {\left (2 \, {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {b^{2} c^{2} - 8 \, a b c d + 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d - a b d^{2}\right )} x - 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, a b d^{2} x + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a^{2} b^{3} c^{2} g^{3} - 2 \, a^{3} b^{2} c d g^{3} + a^{4} b d^{2} g^{3} + {\left (b^{5} c^{2} g^{3} - 2 \, a b^{4} c d g^{3} + a^{2} b^{3} d^{2} g^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{2} g^{3} - 2 \, a^{2} b^{3} c d g^{3} + a^{3} b^{2} d^{2} g^{3}\right )} x}\right )} B^{2} + \frac {1}{2} \, A B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {B^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} - \frac {A^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.85, size = 507, normalized size = 1.89 \[ -\frac {\frac {2\,A^2\,a\,d-2\,A^2\,b\,c+7\,B^2\,a\,d-B^2\,b\,c+6\,A\,B\,a\,d-2\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (3\,b\,d\,B^2+2\,A\,b\,d\,B\right )}{a\,d-b\,c}}{2\,a^2\,b\,g^3+4\,a\,b^2\,g^3\,x+2\,b^3\,g^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{2\,b^2\,g^3\,\left (2\,a\,x+b\,x^2+\frac {a^2}{b}\right )}-\frac {B^2\,d^2}{2\,b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {A\,B}{b^2\,d\,g^3}+\frac {B^2\,x\,\left (a\,d-b\,c\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B^2\,d^2\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,\mathrm {atan}\left (\frac {B\,d^2\,\left (2\,b\,d\,x-\frac {b^3\,c^2\,g^3-a^2\,b\,d^2\,g^3}{b\,g^3\,\left (a\,d-b\,c\right )}\right )\,\left (2\,A+3\,B\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (3\,B^2\,d^2+2\,A\,B\,d^2\right )}\right )\,\left (2\,A+3\,B\right )\,1{}\mathrm {i}}{b\,g^3\,{\left (a\,d-b\,c\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*A^2*a*d - 2*A^2*b*c + 7*B^2*a*d - B^2*b*c + 6*A*B*a*d - 2*A*B*b*c)/(2*(a*d - b*c)) + (x*(3*B^2*b*d + 2*A
*B*b*d))/(a*d - b*c))/(2*a^2*b*g^3 + 2*b^3*g^3*x^2 + 4*a*b^2*g^3*x) - log((e*(a + b*x))/(c + d*x))^2*(B^2/(2*b
^2*g^3*(2*a*x + b*x^2 + a^2/b)) - (B^2*d^2)/(2*b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - (log((e*(a + b*x))/(c
+ d*x))*((A*B)/(b^2*d*g^3) + (B^2*x*(a*d - b*c))/(b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B^2*d^2*((2*a^2*d
^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)))/(b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))))/((
b*x^2)/d + a^2/(b*d) + (2*a*x)/d) - (B*d^2*atan((B*d^2*(2*b*d*x - (b^3*c^2*g^3 - a^2*b*d^2*g^3)/(b*g^3*(a*d -
b*c)))*(2*A + 3*B)*1i)/((a*d - b*c)*(3*B^2*d^2 + 2*A*B*d^2)))*(2*A + 3*B)*1i)/(b*g^3*(a*d - b*c)^2)

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sympy [B]  time = 6.55, size = 894, normalized size = 3.34 \[ - \frac {B d^{2} \left (2 A + 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} + 3 B^{2} a d^{3} + 3 B^{2} b c d^{2} - \frac {B a^{3} d^{5} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{4} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{3} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {B b^{3} c^{3} d^{2} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} + 6 B^{2} b d^{3}} \right )}}{2 b g^{3} \left (a d - b c\right )^{2}} + \frac {B d^{2} \left (2 A + 3 B\right ) \log {\left (x + \frac {2 A B a d^{3} + 2 A B b c d^{2} + 3 B^{2} a d^{3} + 3 B^{2} b c d^{2} + \frac {B a^{3} d^{5} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{4} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{3} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}} - \frac {B b^{3} c^{3} d^{2} \left (2 A + 3 B\right )}{\left (a d - b c\right )^{2}}}{4 A B b d^{3} + 6 B^{2} b d^{3}} \right )}}{2 b g^{3} \left (a d - b c\right )^{2}} + \frac {\left (2 B^{2} a c d + 2 B^{2} a d^{2} x - B^{2} b c^{2} + B^{2} b d^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{4} d^{2} g^{3} - 4 a^{3} b c d g^{3} + 4 a^{3} b d^{2} g^{3} x + 2 a^{2} b^{2} c^{2} g^{3} - 8 a^{2} b^{2} c d g^{3} x + 2 a^{2} b^{2} d^{2} g^{3} x^{2} + 4 a b^{3} c^{2} g^{3} x - 4 a b^{3} c d g^{3} x^{2} + 2 b^{4} c^{2} g^{3} x^{2}} + \frac {\left (- 2 A B a d + 2 A B b c - 3 B^{2} a d + B^{2} b c - 2 B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{3} b d g^{3} - 2 a^{2} b^{2} c g^{3} + 4 a^{2} b^{2} d g^{3} x - 4 a b^{3} c g^{3} x + 2 a b^{3} d g^{3} x^{2} - 2 b^{4} c g^{3} x^{2}} + \frac {- 2 A^{2} a d + 2 A^{2} b c - 6 A B a d + 2 A B b c - 7 B^{2} a d + B^{2} b c + x \left (- 4 A B b d - 6 B^{2} b d\right )}{4 a^{3} b d g^{3} - 4 a^{2} b^{2} c g^{3} + x^{2} \left (4 a b^{3} d g^{3} - 4 b^{4} c g^{3}\right ) + x \left (8 a^{2} b^{2} d g^{3} - 8 a b^{3} c g^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*d**2*(2*A + 3*B)*log(x + (2*A*B*a*d**3 + 2*A*B*b*c*d**2 + 3*B**2*a*d**3 + 3*B**2*b*c*d**2 - B*a**3*d**5*(2*
A + 3*B)/(a*d - b*c)**2 + 3*B*a**2*b*c*d**4*(2*A + 3*B)/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**3*(2*A + 3*B)/(a*d
- b*c)**2 + B*b**3*c**3*d**2*(2*A + 3*B)/(a*d - b*c)**2)/(4*A*B*b*d**3 + 6*B**2*b*d**3))/(2*b*g**3*(a*d - b*c
)**2) + B*d**2*(2*A + 3*B)*log(x + (2*A*B*a*d**3 + 2*A*B*b*c*d**2 + 3*B**2*a*d**3 + 3*B**2*b*c*d**2 + B*a**3*d
**5*(2*A + 3*B)/(a*d - b*c)**2 - 3*B*a**2*b*c*d**4*(2*A + 3*B)/(a*d - b*c)**2 + 3*B*a*b**2*c**2*d**3*(2*A + 3*
B)/(a*d - b*c)**2 - B*b**3*c**3*d**2*(2*A + 3*B)/(a*d - b*c)**2)/(4*A*B*b*d**3 + 6*B**2*b*d**3))/(2*b*g**3*(a*
d - b*c)**2) + (2*B**2*a*c*d + 2*B**2*a*d**2*x - B**2*b*c**2 + B**2*b*d**2*x**2)*log(e*(a + b*x)/(c + d*x))**2
/(2*a**4*d**2*g**3 - 4*a**3*b*c*d*g**3 + 4*a**3*b*d**2*g**3*x + 2*a**2*b**2*c**2*g**3 - 8*a**2*b**2*c*d*g**3*x
+ 2*a**2*b**2*d**2*g**3*x**2 + 4*a*b**3*c**2*g**3*x - 4*a*b**3*c*d*g**3*x**2 + 2*b**4*c**2*g**3*x**2) + (-2*A
*B*a*d + 2*A*B*b*c - 3*B**2*a*d + B**2*b*c - 2*B**2*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a**3*b*d*g**3 - 2*a**
2*b**2*c*g**3 + 4*a**2*b**2*d*g**3*x - 4*a*b**3*c*g**3*x + 2*a*b**3*d*g**3*x**2 - 2*b**4*c*g**3*x**2) + (-2*A*
*2*a*d + 2*A**2*b*c - 6*A*B*a*d + 2*A*B*b*c - 7*B**2*a*d + B**2*b*c + x*(-4*A*B*b*d - 6*B**2*b*d))/(4*a**3*b*d
*g**3 - 4*a**2*b**2*c*g**3 + x**2*(4*a*b**3*d*g**3 - 4*b**4*c*g**3) + x*(8*a**2*b**2*d*g**3 - 8*a*b**3*c*g**3)
)

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