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

Optimal. Leaf size=261 \[ -\frac{b^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (a+b x) (b c-a d)^3}+\frac{d^2 (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (c+d x) (b c-a d)^3}-\frac{2 b d \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (b c-a d)^3}-\frac{b^2 B (c+d x)}{g^2 i^2 (a+b x) (b c-a d)^3}-\frac{B d^2 (a+b x)}{g^2 i^2 (c+d x) (b c-a d)^3}+\frac{b B d \log ^2\left (\frac{a+b x}{c+d x}\right )}{g^2 i^2 (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.867815, antiderivative size = 462, normalized size of antiderivative = 1.77, number of steps used = 28, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{2 b B d \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^2 i^2 (b c-a d)^3}-\frac{2 b B d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^2 i^2 (b c-a d)^3}-\frac{2 b d \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (b c-a d)^3}-\frac{b \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (a+b x) (b c-a d)^2}-\frac{d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (c+d x) (b c-a d)^2}+\frac{2 b d \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (b c-a d)^3}-\frac{b B}{g^2 i^2 (a+b x) (b c-a d)^2}+\frac{B d}{g^2 i^2 (c+d x) (b c-a d)^2}+\frac{b B d \log ^2(a+b x)}{g^2 i^2 (b c-a d)^3}+\frac{b B d \log ^2(c+d x)}{g^2 i^2 (b c-a d)^3}-\frac{2 b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^2 i^2 (b c-a d)^3}-\frac{2 b B d \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^2 i^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2528

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

Rule 2525

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

Rule 12

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

Rule 44

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

Rule 2524

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

Rule 2418

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

Rule 2390

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

Rule 2301

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

Rule 2394

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

Rule 2393

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

Rule 2391

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

Rubi steps

\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(44 c+44 d x)^2 (a g+b g x)^2} \, dx &=\int \left (\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)^2}-\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)^2}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2 (c+d x)}\right ) \, dx\\ &=-\frac{\left (b^2 d\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{968 (b c-a d)^3 g^2}+\frac{\left (b d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{968 (b c-a d)^3 g^2}+\frac{b^2 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{1936 (b c-a d)^2 g^2}+\frac{d^2 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1936 (b c-a d)^2 g^2}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac{b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac{(b B d) \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}{968 (b c-a d)^3 g^2}-\frac{(b B d) \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}{968 (b c-a d)^3 g^2}+\frac{(b B) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{1936 (b c-a d)^2 g^2}+\frac{(B d) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{1936 (b c-a d)^2 g^2}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac{b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac{(b B) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{1936 (b c-a d) g^2}+\frac{(B d) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1936 (b c-a d) g^2}+\frac{(b B d) \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}{968 (b c-a d)^3 e g^2}-\frac{(b B d) \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}{968 (b c-a d)^3 e g^2}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac{b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac{(b B) \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}{1936 (b c-a d) g^2}+\frac{(B d) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1936 (b c-a d) g^2}+\frac{(b B d) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{968 (b c-a d)^3 e g^2}-\frac{(b B d) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{968 (b c-a d)^3 e g^2}\\ &=-\frac{b B}{1936 (b c-a d)^2 g^2 (a+b x)}+\frac{B d}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac{b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac{\left (b^2 B d\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{968 (b c-a d)^3 g^2}-\frac{\left (b^2 B d\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{968 (b c-a d)^3 g^2}-\frac{\left (b B d^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{968 (b c-a d)^3 g^2}+\frac{\left (b B d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{968 (b c-a d)^3 g^2}\\ &=-\frac{b B}{1936 (b c-a d)^2 g^2 (a+b x)}+\frac{B d}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac{b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}-\frac{b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}-\frac{b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}+\frac{(b B d) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{968 (b c-a d)^3 g^2}+\frac{(b B d) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{968 (b c-a d)^3 g^2}+\frac{\left (b^2 B d\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{968 (b c-a d)^3 g^2}+\frac{\left (b B d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{968 (b c-a d)^3 g^2}\\ &=-\frac{b B}{1936 (b c-a d)^2 g^2 (a+b x)}+\frac{B d}{1936 (b c-a d)^2 g^2 (c+d x)}+\frac{b B d \log ^2(a+b x)}{1936 (b c-a d)^3 g^2}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac{b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}-\frac{b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac{b B d \log ^2(c+d x)}{1936 (b c-a d)^3 g^2}-\frac{b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}+\frac{(b B d) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{968 (b c-a d)^3 g^2}+\frac{(b B d) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{968 (b c-a d)^3 g^2}\\ &=-\frac{b B}{1936 (b c-a d)^2 g^2 (a+b x)}+\frac{B d}{1936 (b c-a d)^2 g^2 (c+d x)}+\frac{b B d \log ^2(a+b x)}{1936 (b c-a d)^3 g^2}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1936 (b c-a d)^2 g^2 (c+d x)}-\frac{b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{968 (b c-a d)^3 g^2}-\frac{b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{968 (b c-a d)^3 g^2}+\frac{b B d \log ^2(c+d x)}{1936 (b c-a d)^3 g^2}-\frac{b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}-\frac{b B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}-\frac{b B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{968 (b c-a d)^3 g^2}\\ \end{align*}

Mathematica [C]  time = 0.452001, size = 324, normalized size = 1.24 \[ \frac{b B d \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-b B d \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 b d \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{b (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+2 b d \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{d (a d-b c) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-\frac{b^2 B c}{a+b x}+\frac{a b B d}{a+b x}-\frac{a B d^2}{c+d x}+\frac{b B c d}{c+d x}}{g^2 i^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.053, size = 1187, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.42633, size = 1160, normalized size = 4.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.515332, size = 686, normalized size = 2.63 \begin{align*} -\frac{{\left (A + B\right )} b^{2} c^{2} - 2 \, B a b c d -{\left (A - B\right )} a^{2} d^{2} +{\left (B b^{2} d^{2} x^{2} + B a b c d +{\left (B b^{2} c d + B a b d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left (A b^{2} c d - A a b d^{2}\right )} x +{\left (2 \, A b^{2} d^{2} x^{2} + B b^{2} c^{2} + 2 \, A a b c d - B a^{2} d^{2} + 2 \,{\left ({\left (A + B\right )} b^{2} c d +{\left (A - B\right )} a b d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} i^{2} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} i^{2} x +{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} g^{2} i^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 7.05368, size = 828, normalized size = 3.17 \begin{align*} - \frac{2 A b d \log{\left (x + \frac{- \frac{2 A a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac{8 A a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac{12 A a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac{8 A a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 A a b d^{2} - \frac{2 A b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 A b^{2} c d}{4 A b^{2} d^{2}} \right )}}{g^{2} i^{2} \left (a d - b c\right )^{3}} + \frac{2 A b d \log{\left (x + \frac{\frac{2 A a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac{8 A a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac{12 A a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac{8 A a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 A a b d^{2} + \frac{2 A b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 A b^{2} c d}{4 A b^{2} d^{2}} \right )}}{g^{2} i^{2} \left (a d - b c\right )^{3}} + \frac{B b d \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{3} d^{3} g^{2} i^{2} - 3 a^{2} b c d^{2} g^{2} i^{2} + 3 a b^{2} c^{2} d g^{2} i^{2} - b^{3} c^{3} g^{2} i^{2}} + \frac{\left (- B a d - B b c - 2 B b d x\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{a^{3} c d^{2} g^{2} i^{2} + a^{3} d^{3} g^{2} i^{2} x - 2 a^{2} b c^{2} d g^{2} i^{2} - a^{2} b c d^{2} g^{2} i^{2} x + a^{2} b d^{3} g^{2} i^{2} x^{2} + a b^{2} c^{3} g^{2} i^{2} - a b^{2} c^{2} d g^{2} i^{2} x - 2 a b^{2} c d^{2} g^{2} i^{2} x^{2} + b^{3} c^{3} g^{2} i^{2} x + b^{3} c^{2} d g^{2} i^{2} x^{2}} - \frac{A a d + A b c + 2 A b d x - B a d + B b c}{a^{3} c d^{2} g^{2} i^{2} - 2 a^{2} b c^{2} d g^{2} i^{2} + a b^{2} c^{3} g^{2} i^{2} + x^{2} \left (a^{2} b d^{3} g^{2} i^{2} - 2 a b^{2} c d^{2} g^{2} i^{2} + b^{3} c^{2} d g^{2} i^{2}\right ) + x \left (a^{3} d^{3} g^{2} i^{2} - a^{2} b c d^{2} g^{2} i^{2} - a b^{2} c^{2} d g^{2} i^{2} + b^{3} c^{3} g^{2} i^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*b*d*log(x + (-2*A*a**4*b*d**5/(a*d - b*c)**3 + 8*A*a**3*b**2*c*d**4/(a*d - b*c)**3 - 12*A*a**2*b**3*c**2*
d**3/(a*d - b*c)**3 + 8*A*a*b**4*c**3*d**2/(a*d - b*c)**3 + 2*A*a*b*d**2 - 2*A*b**5*c**4*d/(a*d - b*c)**3 + 2*
A*b**2*c*d)/(4*A*b**2*d**2))/(g**2*i**2*(a*d - b*c)**3) + 2*A*b*d*log(x + (2*A*a**4*b*d**5/(a*d - b*c)**3 - 8*
A*a**3*b**2*c*d**4/(a*d - b*c)**3 + 12*A*a**2*b**3*c**2*d**3/(a*d - b*c)**3 - 8*A*a*b**4*c**3*d**2/(a*d - b*c)
**3 + 2*A*a*b*d**2 + 2*A*b**5*c**4*d/(a*d - b*c)**3 + 2*A*b**2*c*d)/(4*A*b**2*d**2))/(g**2*i**2*(a*d - b*c)**3
) + B*b*d*log(e*(a + b*x)/(c + d*x))**2/(a**3*d**3*g**2*i**2 - 3*a**2*b*c*d**2*g**2*i**2 + 3*a*b**2*c**2*d*g**
2*i**2 - b**3*c**3*g**2*i**2) + (-B*a*d - B*b*c - 2*B*b*d*x)*log(e*(a + b*x)/(c + d*x))/(a**3*c*d**2*g**2*i**2
+ a**3*d**3*g**2*i**2*x - 2*a**2*b*c**2*d*g**2*i**2 - a**2*b*c*d**2*g**2*i**2*x + a**2*b*d**3*g**2*i**2*x**2
+ a*b**2*c**3*g**2*i**2 - a*b**2*c**2*d*g**2*i**2*x - 2*a*b**2*c*d**2*g**2*i**2*x**2 + b**3*c**3*g**2*i**2*x +
b**3*c**2*d*g**2*i**2*x**2) - (A*a*d + A*b*c + 2*A*b*d*x - B*a*d + B*b*c)/(a**3*c*d**2*g**2*i**2 - 2*a**2*b*c
**2*d*g**2*i**2 + a*b**2*c**3*g**2*i**2 + x**2*(a**2*b*d**3*g**2*i**2 - 2*a*b**2*c*d**2*g**2*i**2 + b**3*c**2*
d*g**2*i**2) + x*(a**3*d**3*g**2*i**2 - a**2*b*c*d**2*g**2*i**2 - a*b**2*c**2*d*g**2*i**2 + b**3*c**3*g**2*i**
2))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A}{{\left (b g x + a g\right )}^{2}{\left (d i x + c i\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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