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

Optimal. Leaf size=126 \[ -\frac {2 B (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 (a+b x) (b c-a d)}-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^2 (a+b x) (b c-a d)}-\frac {2 B^2 (c+d x)}{g^2 (a+b x) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.77, antiderivative size = 470, normalized size of antiderivative = 3.73, number of steps used = 26, 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} \[ -\frac {2 B^2 d \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac {2 B^2 d \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac {2 B d \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g^2 (b c-a d)}-\frac {2 B \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g^2 (a+b x)}+\frac {2 B d \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g^2 (b c-a d)}-\frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b g^2 (a+b x)}+\frac {B^2 d \log ^2(a+b x)}{b g^2 (b c-a d)}+\frac {B^2 d \log ^2(c+d x)}{b g^2 (b c-a d)}-\frac {2 B^2 d \log (a+b x)}{b g^2 (b c-a d)}-\frac {2 B^2 d \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}+\frac {2 B^2 d \log (c+d x)}{b g^2 (b c-a d)}-\frac {2 B^2 d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac {2 B^2}{b g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [C]  time = 0.45, size = 314, normalized size = 2.49 \[ -\frac {\frac {B \left (2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 d (a+b x) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-2 d (a+b x) \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 B (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )}{b c-a d}+\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{b g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.10, size = 150, normalized size = 1.19 \[ -\frac {{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b c - {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a d + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left ({\left (A B + B^{2}\right )} b d x + {\left (A B + B^{2}\right )} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \]

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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 2.08, size = 176, normalized size = 1.40 \[ -\frac {{\left (B^{2} e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 2 \, A B e^{2} \log \left (\frac {b x e + a e}{d x + c}\right ) + 2 \, B^{2} e^{2} \log \left (\frac {b x e + a e}{d x + c}\right ) + A^{2} e^{2} + 2 \, A B e^{2} + 2 \, B^{2} e^{2}\right )} {\left (d x + c\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{{\left (b x e + a e\right )} g^{2}} \]

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)^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.05, size = 828, normalized size = 6.57 \[ \frac {B^{2} a d e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}-\frac {B^{2} b c e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}+\frac {2 A B a d e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}-\frac {2 A B b c e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}+\frac {2 B^{2} a d e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}-\frac {2 B^{2} b c e \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}+\frac {A^{2} a d e}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}-\frac {A^{2} b c e}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}+\frac {2 A B a d e}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}-\frac {2 A B b c e}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}+\frac {2 B^{2} a d e}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}-\frac {2 B^{2} b c e}{\left (a d -b c \right )^{2} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}} \]

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)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.40, size = 416, normalized size = 3.30 \[ -{\left (2 \, {\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {{\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c g^{2} - a^{2} b d g^{2} + {\left (b^{3} c g^{2} - a b^{2} d g^{2}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {B^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac {A^{2}}{b^{2} g^{2} x + a b g^{2}} \]

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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 5.26, size = 222, normalized size = 1.76 \[ -\frac {A^2+2\,A\,B+2\,B^2}{x\,b^2\,g^2+a\,b\,g^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B^2\,d}{b\,g^2\,\left (a\,d-b\,c\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {2\,B^2}{b^2\,d\,g^2}+\frac {2\,A\,B}{b^2\,d\,g^2}\right )}{\frac {x}{d}+\frac {a}{b\,d}}-\frac {B\,d\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {c\,b^2\,g^2+a\,d\,b\,g^2}{b\,g^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A+B\right )\,4{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \]

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)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 3.59, size = 434, normalized size = 3.44 \[ - \frac {2 B d \left (A + B\right ) \log {\left (x + \frac {2 A B a d^{2} + 2 A B b c d + 2 B^{2} a d^{2} + 2 B^{2} b c d - \frac {2 B a^{2} d^{3} \left (A + B\right )}{a d - b c} + \frac {4 B a b c d^{2} \left (A + B\right )}{a d - b c} - \frac {2 B b^{2} c^{2} d \left (A + B\right )}{a d - b c}}{4 A B b d^{2} + 4 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {2 B d \left (A + B\right ) \log {\left (x + \frac {2 A B a d^{2} + 2 A B b c d + 2 B^{2} a d^{2} + 2 B^{2} b c d + \frac {2 B a^{2} d^{3} \left (A + B\right )}{a d - b c} - \frac {4 B a b c d^{2} \left (A + B\right )}{a d - b c} + \frac {2 B b^{2} c^{2} d \left (A + B\right )}{a d - b c}}{4 A B b d^{2} + 4 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B - 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a b g^{2} + b^{2} g^{2} x} + \frac {\left (B^{2} c + B^{2} d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{2} d g^{2} - a b c g^{2} + a b d g^{2} x - b^{2} c g^{2} x} + \frac {- A^{2} - 2 A B - 2 B^{2}}{a b g^{2} + b^{2} g^{2} x} \]

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)**2,x)

[Out]

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

________________________________________________________________________________________