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

Optimal. Leaf size=141 \[ -\frac{i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)}-\frac{B i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac{B^2 i (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.94, antiderivative size = 639, normalized size of antiderivative = 4.53, number of steps used = 58, 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} \[ -\frac{B^2 d^2 i \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac{B^2 d^2 i \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac{B d^2 i \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (b c-a d)}+\frac{B d^2 i \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (b c-a d)}-\frac{B d i \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac{B i (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^3 (a+b x)^2}-\frac{d i \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{b^2 g^3 (a+b x)}-\frac{i (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 b^2 g^3 (a+b x)^2}+\frac{B^2 d^2 i \log ^2(a+b x)}{2 b^2 g^3 (b c-a d)}+\frac{B^2 d^2 i \log ^2(c+d x)}{2 b^2 g^3 (b c-a d)}-\frac{B^2 d^2 i \log (a+b x)}{2 b^2 g^3 (b c-a d)}+\frac{B^2 d^2 i \log (c+d x)}{2 b^2 g^3 (b c-a d)}-\frac{B^2 d^2 i \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac{B^2 d^2 i \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g^3 (b c-a d)}-\frac{B^2 i (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac{B^2 d i}{2 b^2 g^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B^2*(b*c - a*d)*i)/(4*b^2*g^3*(a + b*x)^2) - (B^2*d*i)/(2*b^2*g^3*(a + b*x)) - (B^2*d^2*i*Log[a + b*x])/(2*b
^2*(b*c - a*d)*g^3) + (B^2*d^2*i*Log[a + b*x]^2)/(2*b^2*(b*c - a*d)*g^3) - (B*(b*c - a*d)*i*(A + B*Log[(e*(a +
b*x))/(c + d*x)]))/(2*b^2*g^3*(a + b*x)^2) - (B*d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g^3*(a + b*x))
- (B*d^2*i*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*(b*c - a*d)*g^3) - ((b*c - a*d)*i*(A + B*L
og[(e*(a + b*x))/(c + d*x)])^2)/(2*b^2*g^3*(a + b*x)^2) - (d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(b^2*g^
3*(a + b*x)) + (B^2*d^2*i*Log[c + d*x])/(2*b^2*(b*c - a*d)*g^3) - (B^2*d^2*i*Log[-((d*(a + b*x))/(b*c - a*d))]
*Log[c + d*x])/(b^2*(b*c - a*d)*g^3) + (B*d^2*i*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(b^2*(b*c -
a*d)*g^3) + (B^2*d^2*i*Log[c + d*x]^2)/(2*b^2*(b*c - a*d)*g^3) - (B^2*d^2*i*Log[a + b*x]*Log[(b*(c + d*x))/(b
*c - a*d)])/(b^2*(b*c - a*d)*g^3) - (B^2*d^2*i*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^2*(b*c - a*d)*g^3)
- (B^2*d^2*i*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(b^2*(b*c - a*d)*g^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 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{(61 c+61 d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx &=\int \left (\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g^3 (a+b x)^3}+\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b g^3 (a+b x)^2}\right ) \, dx\\ &=\frac{(61 d) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b g^3}+\frac{(61 (b c-a d)) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{(122 B d) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac{(61 B (b c-a d)) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{(122 B d (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^2 g^3}+\frac{\left (61 B (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^3}\\ &=-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{(122 B d (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^2 g^3}+\frac{\left (61 B (b c-a d)^2\right ) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^3}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^3}\\ &=-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}-\frac{(61 B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac{(122 B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^3}+\frac{\left (61 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac{\left (122 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac{\left (61 B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac{\left (122 B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac{(61 B (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^3}\\ &=-\frac{61 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{61 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac{61 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{61 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac{\left (122 B^2 d\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}-\frac{\left (61 B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^2 (b c-a d) g^3}+\frac{\left (61 B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^2 (b c-a d) g^3}+\frac{\left (122 B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^2 (b c-a d) g^3}-\frac{\left (122 B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^2 (b c-a d) g^3}+\frac{\left (61 B^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}\\ &=-\frac{61 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{61 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac{61 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{61 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac{\left (122 B^2 d (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^3}+\frac{\left (61 B^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b^2 g^3}-\frac{\left (61 B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d) e g^3}+\frac{\left (61 B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d) e g^3}+\frac{\left (122 B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d) e g^3}-\frac{\left (122 B^2 d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d) e g^3}\\ &=-\frac{61 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{61 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac{61 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{61 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d (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^2 g^3}+\frac{\left (122 B^2 d (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^2 g^3}+\frac{\left (61 B^2 (b c-a d)^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^2 g^3}-\frac{\left (61 B^2 d^2\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 (b c-a d) e g^3}+\frac{\left (61 B^2 d^2\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{b^2 (b c-a d) e g^3}+\frac{\left (122 B^2 d^2\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 (b c-a d) e g^3}-\frac{\left (122 B^2 d^2\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{b^2 (b c-a d) e g^3}\\ &=-\frac{61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac{61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac{61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac{61 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{61 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac{61 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}+\frac{61 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac{\left (61 B^2 d^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac{\left (122 B^2 d^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b (b c-a d) g^3}-\frac{\left (122 B^2 d^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b (b c-a d) g^3}+\frac{\left (61 B^2 d^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}-\frac{\left (122 B^2 d^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac{\left (122 B^2 d^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac{61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac{61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac{61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}-\frac{61 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{61 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac{61 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac{61 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac{61 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}-\frac{61 B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}+\frac{\left (122 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}+\frac{\left (122 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}+\frac{\left (122 B^2 d^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b (b c-a d) g^3}-\frac{\left (61 B^2 d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}+\frac{\left (122 B^2 d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 (b c-a d) g^3}\\ &=-\frac{61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac{61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac{61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac{61 B^2 d^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac{61 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{61 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac{61 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac{61 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac{61 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac{61 B^2 d^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac{61 B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}-\frac{\left (61 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}+\frac{\left (122 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 (b c-a d) g^3}+\frac{\left (122 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2 (b c-a d) g^3}\\ &=-\frac{61 B^2 (b c-a d)}{4 b^2 g^3 (a+b x)^2}-\frac{61 B^2 d}{2 b^2 g^3 (a+b x)}-\frac{61 B^2 d^2 \log (a+b x)}{2 b^2 (b c-a d) g^3}+\frac{61 B^2 d^2 \log ^2(a+b x)}{2 b^2 (b c-a d) g^3}-\frac{61 B (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac{61 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac{61 B d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 (b c-a d) g^3}-\frac{61 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2 b^2 g^3 (a+b x)^2}-\frac{61 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^3 (a+b x)}+\frac{61 B^2 d^2 \log (c+d x)}{2 b^2 (b c-a d) g^3}-\frac{61 B^2 d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac{61 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{b^2 (b c-a d) g^3}+\frac{61 B^2 d^2 \log ^2(c+d x)}{2 b^2 (b c-a d) g^3}-\frac{61 B^2 d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac{61 B^2 d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}-\frac{61 B^2 d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 (b c-a d) g^3}\\ \end{align*}

Mathematica [C]  time = 0.930415, size = 765, normalized size = 5.43 \[ -\frac{i \left (B \left (2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-2 B d^2 (a+b x)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+4 d^2 (a+b x)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+4 d (a+b x) (a d-b c) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+B \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )-4 B d (a+b x) (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )+4 B d (a+b x) \left (-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 d (a+b x) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 d (a+b x) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 B (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )+2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2-4 d (a+b x) (a d-b c) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2\right )}{4 b^2 g^3 (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.053, size = 865, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.76029, size = 2682, normalized size = 19.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 0.525851, size = 602, normalized size = 4.27 \begin{align*} -\frac{2 \,{\left ({\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c d -{\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b d^{2}\right )} i x + 2 \,{\left (B^{2} b^{2} d^{2} i x^{2} + 2 \, B^{2} b^{2} c d i x + B^{2} b^{2} c^{2} i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} +{\left ({\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} -{\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a^{2} d^{2}\right )} i + 2 \,{\left ({\left (2 \, A B + B^{2}\right )} b^{2} d^{2} i x^{2} + 2 \,{\left (2 \, A B + B^{2}\right )} b^{2} c d i x +{\left (2 \, A B + B^{2}\right )} b^{2} c^{2} i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{4 \,{\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \,{\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x +{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 13.8162, size = 712, normalized size = 5.05 \begin{align*} - \frac{B d^{2} i \left (2 A + B\right ) \log{\left (x + \frac{2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i - \frac{B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} + \frac{2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} - \frac{B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac{B d^{2} i \left (2 A + B\right ) \log{\left (x + \frac{2 A B a d^{3} i + 2 A B b c d^{2} i + B^{2} a d^{3} i + B^{2} b c d^{2} i + \frac{B a^{2} d^{4} i \left (2 A + B\right )}{a d - b c} - \frac{2 B a b c d^{3} i \left (2 A + B\right )}{a d - b c} + \frac{B b^{2} c^{2} d^{2} i \left (2 A + B\right )}{a d - b c}}{4 A B b d^{3} i + 2 B^{2} b d^{3} i} \right )}}{2 b^{2} g^{3} \left (a d - b c\right )} + \frac{\left (B^{2} c^{2} i + 2 B^{2} c d i x + B^{2} d^{2} i x^{2}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{3} d g^{3} - 2 a^{2} b c g^{3} + 4 a^{2} b d g^{3} x - 4 a b^{2} c g^{3} x + 2 a b^{2} d g^{3} x^{2} - 2 b^{3} c g^{3} x^{2}} - \frac{2 A^{2} a d i + 2 A^{2} b c i + 2 A B a d i + 2 A B b c i + B^{2} a d i + B^{2} b c i + x \left (4 A^{2} b d i + 4 A B b d i + 2 B^{2} b d i\right )}{4 a^{2} b^{2} g^{3} + 8 a b^{3} g^{3} x + 4 b^{4} g^{3} x^{2}} + \frac{\left (- 2 A B a d i - 2 A B b c i - 4 A B b d i x - B^{2} a d i - B^{2} b c i - 2 B^{2} b d i x\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} b^{2} g^{3} + 4 a b^{3} g^{3} x + 2 b^{4} g^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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