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

Optimal. Leaf size=243 \[ \frac{b^2 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (b c-a d)^3}+\frac{d^2 (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac{2 b d (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (c+d x) (b c-a d)^3}-\frac{b^2 B \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g i^3 (b c-a d)^3}-\frac{B \left (4 b-\frac{d (a+b x)}{c+d x}\right )^2}{4 g i^3 (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.89849, antiderivative size = 535, normalized size of antiderivative = 2.2, 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, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 44} \[ \frac{b^2 B \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}+\frac{b^2 B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}+\frac{b^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (b c-a d)^3}-\frac{b^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (b c-a d)^3}+\frac{b \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (c+d x) (b c-a d)^2}+\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{2 g i^3 (c+d x)^2 (b c-a d)}-\frac{b^2 B \log ^2(a+b x)}{2 g i^3 (b c-a d)^3}-\frac{b^2 B \log ^2(c+d x)}{2 g i^3 (b c-a d)^3}-\frac{3 b^2 B \log (a+b x)}{2 g i^3 (b c-a d)^3}+\frac{b^2 B \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}+\frac{3 b^2 B \log (c+d x)}{2 g i^3 (b c-a d)^3}+\frac{b^2 B \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g i^3 (b c-a d)^3}-\frac{3 b B}{2 g i^3 (c+d x) (b c-a d)^2}-\frac{B}{4 g i^3 (c+d x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 12

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

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]

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 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])

Rubi steps

\begin{align*} \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(51 c+51 d x)^3 (a g+b g x)} \, dx &=\int \left (\frac{b^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d) g (c+d x)^3}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)^2}-\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g (c+d x)}\right ) \, dx\\ &=\frac{b^3 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{132651 (b c-a d)^3 g}-\frac{\left (b^2 d\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{132651 (b c-a d)^3 g}-\frac{(b d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{132651 (b c-a d)^2 g}-\frac{d \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{132651 (b c-a d) g}\\ &=\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac{b^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac{\left (b^2 B\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}{132651 (b c-a d)^3 g}+\frac{\left (b^2 B\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}{132651 (b c-a d)^3 g}-\frac{(b B) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{132651 (b c-a d)^2 g}-\frac{B \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{265302 (b c-a d) g}\\ &=\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac{b^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac{B \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{265302 g}-\frac{(b B) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{132651 (b c-a d) g}-\frac{\left (b^2 B\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}{132651 (b c-a d)^3 e g}+\frac{\left (b^2 B\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}{132651 (b c-a d)^3 e g}\\ &=\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac{b^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac{B \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{265302 g}-\frac{(b B) \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}{132651 (b c-a d) g}-\frac{\left (b^2 B\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}{132651 (b c-a d)^3 e g}+\frac{\left (b^2 B\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}{132651 (b c-a d)^3 e g}\\ &=-\frac{B}{530604 (b c-a d) g (c+d x)^2}-\frac{b B}{88434 (b c-a d)^2 g (c+d x)}-\frac{b^2 B \log (a+b x)}{88434 (b c-a d)^3 g}+\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac{b^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}+\frac{b^2 B \log (c+d x)}{88434 (b c-a d)^3 g}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac{\left (b^3 B\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{132651 (b c-a d)^3 g}+\frac{\left (b^3 B\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{132651 (b c-a d)^3 g}+\frac{\left (b^2 B d\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{132651 (b c-a d)^3 g}-\frac{\left (b^2 B d\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{132651 (b c-a d)^3 g}\\ &=-\frac{B}{530604 (b c-a d) g (c+d x)^2}-\frac{b B}{88434 (b c-a d)^2 g (c+d x)}-\frac{b^2 B \log (a+b x)}{88434 (b c-a d)^3 g}+\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac{b^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}+\frac{b^2 B \log (c+d x)}{88434 (b c-a d)^3 g}+\frac{b^2 B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}+\frac{b^2 B \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}-\frac{\left (b^2 B\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{132651 (b c-a d)^3 g}-\frac{\left (b^2 B\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{132651 (b c-a d)^3 g}-\frac{\left (b^3 B\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{132651 (b c-a d)^3 g}-\frac{\left (b^2 B d\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{132651 (b c-a d)^3 g}\\ &=-\frac{B}{530604 (b c-a d) g (c+d x)^2}-\frac{b B}{88434 (b c-a d)^2 g (c+d x)}-\frac{b^2 B \log (a+b x)}{88434 (b c-a d)^3 g}-\frac{b^2 B \log ^2(a+b x)}{265302 (b c-a d)^3 g}+\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac{b^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}+\frac{b^2 B \log (c+d x)}{88434 (b c-a d)^3 g}+\frac{b^2 B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac{b^2 B \log ^2(c+d x)}{265302 (b c-a d)^3 g}+\frac{b^2 B \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}-\frac{\left (b^2 B\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{132651 (b c-a d)^3 g}-\frac{\left (b^2 B\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{132651 (b c-a d)^3 g}\\ &=-\frac{B}{530604 (b c-a d) g (c+d x)^2}-\frac{b B}{88434 (b c-a d)^2 g (c+d x)}-\frac{b^2 B \log (a+b x)}{88434 (b c-a d)^3 g}-\frac{b^2 B \log ^2(a+b x)}{265302 (b c-a d)^3 g}+\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{265302 (b c-a d) g (c+d x)^2}+\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^2 g (c+d x)}+\frac{b^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{132651 (b c-a d)^3 g}+\frac{b^2 B \log (c+d x)}{88434 (b c-a d)^3 g}+\frac{b^2 B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{132651 (b c-a d)^3 g}-\frac{b^2 B \log ^2(c+d x)}{265302 (b c-a d)^3 g}+\frac{b^2 B \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}+\frac{b^2 B \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}+\frac{b^2 B \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{132651 (b c-a d)^3 g}\\ \end{align*}

Mathematica [C]  time = 0.46679, size = 418, normalized size = 1.72 \[ \frac{-2 b^2 B (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+2 b^2 B (c+d x)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+4 b^2 (c+d x)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-4 b^2 (c+d x)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+4 b (c+d x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-B \left (2 b^2 (c+d x)^2 \log (a+b x)+2 b (c+d x) (b c-a d)+(b c-a d)^2-2 b^2 (c+d x)^2 \log (c+d x)\right )-4 b B (c+d x) (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)}{4 g i^3 (c+d x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*b*(b*c - a*d)*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]) + 4*b^2*(c + d*x)^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 4*b^2*(c + d*x)^2*(A + B*Log[
(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 4*b*B*(c + d*x)*(b*c - a*d + b*(c + d*x)*Log[a + b*x] - b*(c + d*x)*L
og[c + d*x]) - B*((b*c - a*d)^2 + 2*b*(b*c - a*d)*(c + d*x) + 2*b^2*(c + d*x)^2*Log[a + b*x] - 2*b^2*(c + d*x)
^2*Log[c + d*x]) - 2*b^2*B*(c + d*x)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*Pol
yLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 2*b^2*B*(c + d*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)]))/(4*(b*c - a*d)^3*g*i^3*(c + d*x)^2)

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Maple [B]  time = 0.054, size = 1287, normalized size = 5.3 \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)/(d*i*x+c*i)^3,x)

[Out]

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

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Maxima [B]  time = 1.42318, size = 1195, normalized size = 4.92 \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)/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.534056, size = 730, normalized size = 3. \begin{align*} \frac{{\left (6 \, A - 7 \, B\right )} b^{2} c^{2} - 8 \,{\left (A - B\right )} a b c d +{\left (2 \, A - B\right )} a^{2} d^{2} + 2 \,{\left (B b^{2} d^{2} x^{2} + 2 \, B b^{2} c d x + B b^{2} c^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 2 \,{\left ({\left (2 \, A - 3 \, B\right )} b^{2} c d -{\left (2 \, A - 3 \, B\right )} a b d^{2}\right )} x + 2 \,{\left ({\left (2 \, A - 3 \, B\right )} b^{2} d^{2} x^{2} + 2 \, A b^{2} c^{2} - 4 \, B a b c d + B a^{2} d^{2} + 2 \,{\left (2 \,{\left (A - B\right )} b^{2} c d - B a b d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{4 \,{\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g i^{3} x^{2} + 2 \,{\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g i^{3} x +{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g i^{3}\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)/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 8.0485, size = 889, normalized size = 3.66 \begin{align*} \text{result too large to display} \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)/(d*i*x+c*i)**3,x)

[Out]

-B*b**2*log(e*(a + b*x)/(c + d*x))**2/(2*a**3*d**3*g*i**3 - 6*a**2*b*c*d**2*g*i**3 + 6*a*b**2*c**2*d*g*i**3 -
2*b**3*c**3*g*i**3) + b**2*(2*A - 3*B)*log(x + (2*A*a*b**2*d + 2*A*b**3*c - 3*B*a*b**2*d - 3*B*b**3*c - a**4*b
**2*d**4*(2*A - 3*B)/(a*d - b*c)**3 + 4*a**3*b**3*c*d**3*(2*A - 3*B)/(a*d - b*c)**3 - 6*a**2*b**4*c**2*d**2*(2
*A - 3*B)/(a*d - b*c)**3 + 4*a*b**5*c**3*d*(2*A - 3*B)/(a*d - b*c)**3 - b**6*c**4*(2*A - 3*B)/(a*d - b*c)**3)/
(4*A*b**3*d - 6*B*b**3*d))/(2*g*i**3*(a*d - b*c)**3) - b**2*(2*A - 3*B)*log(x + (2*A*a*b**2*d + 2*A*b**3*c - 3
*B*a*b**2*d - 3*B*b**3*c + a**4*b**2*d**4*(2*A - 3*B)/(a*d - b*c)**3 - 4*a**3*b**3*c*d**3*(2*A - 3*B)/(a*d - b
*c)**3 + 6*a**2*b**4*c**2*d**2*(2*A - 3*B)/(a*d - b*c)**3 - 4*a*b**5*c**3*d*(2*A - 3*B)/(a*d - b*c)**3 + b**6*
c**4*(2*A - 3*B)/(a*d - b*c)**3)/(4*A*b**3*d - 6*B*b**3*d))/(2*g*i**3*(a*d - b*c)**3) + (-B*a*d + 3*B*b*c + 2*
B*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a**2*c**2*d**2*g*i**3 + 4*a**2*c*d**3*g*i**3*x + 2*a**2*d**4*g*i**3*x**
2 - 4*a*b*c**3*d*g*i**3 - 8*a*b*c**2*d**2*g*i**3*x - 4*a*b*c*d**3*g*i**3*x**2 + 2*b**2*c**4*g*i**3 + 4*b**2*c*
*3*d*g*i**3*x + 2*b**2*c**2*d**2*g*i**3*x**2) + (-2*A*a*d + 6*A*b*c + B*a*d - 7*B*b*c + x*(4*A*b*d - 6*B*b*d))
/(4*a**2*c**2*d**2*g*i**3 - 8*a*b*c**3*d*g*i**3 + 4*b**2*c**4*g*i**3 + x**2*(4*a**2*d**4*g*i**3 - 8*a*b*c*d**3
*g*i**3 + 4*b**2*c**2*d**2*g*i**3) + x*(8*a**2*c*d**3*g*i**3 - 16*a*b*c**2*d**2*g*i**3 + 8*b**2*c**3*d*g*i**3)
)

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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 )}{\left (d i x + c i\right )}^{3}}\,{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)/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

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