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

Optimal. Leaf size=173 \[ -\frac{d \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (b c-a d)^2}-\frac{b (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (a+b x) (b c-a d)^2}-\frac{b B (c+d x)}{g^2 i (a+b x) (b c-a d)^2}+\frac{B d \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g^2 i (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.703046, antiderivative size = 437, normalized size of antiderivative = 2.53, number of steps used = 24, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{B d \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^2 i (b c-a d)^2}-\frac{B d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^2 i (b c-a d)^2}-\frac{d \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (b c-a d)^2}-\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{g^2 i (a+b x) (b c-a d)}+\frac{d \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (b c-a d)^2}-\frac{B}{g^2 i (a+b x) (b c-a d)}+\frac{B d \log ^2(a+b x)}{2 g^2 i (b c-a d)^2}+\frac{B d \log ^2(c+d x)}{2 g^2 i (b c-a d)^2}-\frac{B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^2 i (b c-a d)^2}-\frac{B d \log (a+b x)}{g^2 i (b c-a d)^2}-\frac{B d \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^2 i (b c-a d)^2}+\frac{B d \log (c+d x)}{g^2 i (b c-a d)^2} \]

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)),x]

[Out]

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

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 )}{(36 c+36 d x) (a g+b g x)^2} \, dx &=\int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d) g^2 (a+b x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2 (c+d x)}\right ) \, dx\\ &=-\frac{(b d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{36 (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} \, dx}{36 (b c-a d)^2 g^2}+\frac{b \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{36 (b c-a d) g^2}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{(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}{36 (b c-a d)^2 g^2}-\frac{(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}{36 (b c-a d)^2 g^2}+\frac{B \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{36 (b c-a d) g^2}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{B \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{36 g^2}+\frac{(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}{36 (b c-a d)^2 e g^2}-\frac{(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}{36 (b c-a d)^2 e g^2}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{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}{36 g^2}+\frac{(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}{36 (b c-a d)^2 e g^2}-\frac{(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}{36 (b c-a d)^2 e g^2}\\ &=-\frac{B}{36 (b c-a d) g^2 (a+b x)}-\frac{B d \log (a+b x)}{36 (b c-a d)^2 g^2}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac{B d \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{(b B d) \int \frac{\log (a+b x)}{a+b x} \, dx}{36 (b c-a d)^2 g^2}-\frac{(b B d) \int \frac{\log (c+d x)}{a+b x} \, dx}{36 (b c-a d)^2 g^2}-\frac{\left (B d^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{36 (b c-a d)^2 g^2}+\frac{\left (B d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{36 (b c-a d)^2 g^2}\\ &=-\frac{B}{36 (b c-a d) g^2 (a+b x)}-\frac{B d \log (a+b x)}{36 (b c-a d)^2 g^2}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac{B d \log (c+d x)}{36 (b c-a d)^2 g^2}-\frac{B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}-\frac{B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}+\frac{(B d) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{36 (b c-a d)^2 g^2}+\frac{(B d) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{36 (b c-a d)^2 g^2}+\frac{(b B d) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{36 (b c-a d)^2 g^2}+\frac{\left (B d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{36 (b c-a d)^2 g^2}\\ &=-\frac{B}{36 (b c-a d) g^2 (a+b x)}-\frac{B d \log (a+b x)}{36 (b c-a d)^2 g^2}+\frac{B d \log ^2(a+b x)}{72 (b c-a d)^2 g^2}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac{B d \log (c+d x)}{36 (b c-a d)^2 g^2}-\frac{B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{B d \log ^2(c+d x)}{72 (b c-a d)^2 g^2}-\frac{B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}+\frac{(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 )}{36 (b c-a d)^2 g^2}+\frac{(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 )}{36 (b c-a d)^2 g^2}\\ &=-\frac{B}{36 (b c-a d) g^2 (a+b x)}-\frac{B d \log (a+b x)}{36 (b c-a d)^2 g^2}+\frac{B d \log ^2(a+b x)}{72 (b c-a d)^2 g^2}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac{B d \log (c+d x)}{36 (b c-a d)^2 g^2}-\frac{B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac{B d \log ^2(c+d x)}{72 (b c-a d)^2 g^2}-\frac{B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}-\frac{B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}-\frac{B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}\\ \end{align*}

Mathematica [C]  time = 0.298984, size = 292, normalized size = 1.69 \[ -\frac{-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 d (a+b x) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 d (a+b x) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 B (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)}{2 g^2 i (a+b x) (b c-a d)^2} \]

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)),x]

[Out]

-(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)])*Log[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)] - L
og[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))/(2*(b*c - a*d)^2*g^2*i*(a + b*x))

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Maple [B]  time = 0.056, size = 605, normalized size = 3.5 \begin{align*} -{\frac{{d}^{2}Aa}{i \left ( ad-bc \right ) ^{3}{g}^{2}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) }+{\frac{dAbc}{i \left ( ad-bc \right ) ^{3}{g}^{2}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) }-{\frac{deAba}{i \left ( ad-bc \right ) ^{3}{g}^{2}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}+{\frac{eA{b}^{2}c}{i \left ( ad-bc \right ) ^{3}{g}^{2}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}-{\frac{{d}^{2}Ba}{2\,i \left ( ad-bc \right ) ^{3}{g}^{2}} \left ( \ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \right ) ^{2}}+{\frac{dBbc}{2\,i \left ( ad-bc \right ) ^{3}{g}^{2}} \left ( \ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \right ) ^{2}}-{\frac{deBba}{i \left ( ad-bc \right ) ^{3}{g}^{2}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}+{\frac{eB{b}^{2}c}{i \left ( ad-bc \right ) ^{3}{g}^{2}}\ln \left ({\frac{be}{d}}+{\frac{e \left ( ad-bc \right ) }{ \left ( dx+c \right ) d}} \right ) \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}-{\frac{deBba}{i \left ( ad-bc \right ) ^{3}{g}^{2}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}}+{\frac{eB{b}^{2}c}{i \left ( ad-bc \right ) ^{3}{g}^{2}} \left ({\frac{be}{d}}+{\frac{ae}{dx+c}}-{\frac{bec}{ \left ( dx+c \right ) d}} \right ) ^{-1}} \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),x)

[Out]

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

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Maxima [B]  time = 1.30179, size = 572, normalized size = 3.31 \begin{align*} -B{\left (\frac{1}{{\left (b^{2} c - a b d\right )} g^{2} i x +{\left (a b c - a^{2} d\right )} g^{2} i} + \frac{d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac{d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) - A{\left (\frac{1}{{\left (b^{2} c - a b d\right )} g^{2} i x +{\left (a b c - a^{2} d\right )} g^{2} i} + \frac{d \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i} - \frac{d \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g^{2} i}\right )} + \frac{{\left ({\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 )\right )} B}{2 \,{\left (a b^{2} c^{2} g^{2} i - 2 \, a^{2} b c d g^{2} i + a^{3} d^{2} g^{2} i +{\left (b^{3} c^{2} g^{2} i - 2 \, a b^{2} c d g^{2} i + a^{2} b d^{2} g^{2} i\right )} x\right )}} \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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.509202, size = 329, normalized size = 1.9 \begin{align*} -\frac{2 \,{\left (A + B\right )} b c - 2 \,{\left (A + B\right )} a d +{\left (B b d x + B a d\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left ({\left (A + B\right )} b d x + B b c + A a d\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{2 \,{\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{2} i x +{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} g^{2} i\right )}} \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),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 3.0551, size = 386, normalized size = 2.23 \begin{align*} - \frac{B d \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{2} d^{2} g^{2} i - 4 a b c d g^{2} i + 2 b^{2} c^{2} g^{2} i} + \frac{B \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{a^{2} d g^{2} i - a b c g^{2} i + a b d g^{2} i x - b^{2} c g^{2} i x} + \left (A + B\right ) \left (\frac{d \log{\left (x + \frac{- \frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} - \frac{d \log{\left (x + \frac{\frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} + \frac{1}{a^{2} d g^{2} i - a b c g^{2} i + x \left (a b d g^{2} i - b^{2} c g^{2} i\right )}\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),x)

[Out]

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

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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 )}}\,{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),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)), x)