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

Optimal. Leaf size=281 \[ \frac{4 B^2 (b c-a d) (-a d g-b c g+2 b d f) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}+\frac{2 B (b c-a d) (-a d g-b c g+2 b d f) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b^2 d^2}-\frac{(b f-a g)^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 b^2 g}-\frac{2 B g (a+b x) (b c-a d) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b^2 d}+\frac{(f+g x)^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 g}+\frac{4 B^2 g (b c-a d)^2 \log (c+d x)}{b^2 d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.959411, antiderivative size = 450, normalized size of antiderivative = 1.6, number of steps used = 25, number of rules used = 12, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.414, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{4 B^2 (b f-a g)^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}-\frac{4 B^2 (d f-c g)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^2 g}-\frac{2 B (b f-a g)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b^2 g}+\frac{2 B (d f-c g)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{d^2 g}+\frac{(f+g x)^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 g}-\frac{2 A B g x (b c-a d)}{b d}+\frac{4 B^2 g (b c-a d)^2 \log (c+d x)}{b^2 d^2}-\frac{2 B^2 g (a+b x) (b c-a d) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^2 d}-\frac{4 B^2 (b f-a g)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{2 B^2 (b f-a g)^2 \log ^2(a+b x)}{b^2 g}-\frac{4 B^2 (d f-c g)^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d^2 g}+\frac{2 B^2 (d f-c g)^2 \log ^2(c+d x)}{d^2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 (f+g x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx &=\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}-\frac{B \int \frac{2 (b c-a d) (f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x) (c+d x)} \, dx}{g}\\ &=\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}-\frac{(2 B (b c-a d)) \int \frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x) (c+d x)} \, dx}{g}\\ &=\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}-\frac{(2 B (b c-a d)) \int \left (\frac{g^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b d}+\frac{(b f-a g)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) (a+b x)}+\frac{(d f-c g)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{d (-b c+a d) (c+d x)}\right ) \, dx}{g}\\ &=\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}-\frac{(2 B (b c-a d) g) \int \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{b d}-\frac{\left (2 B (b f-a g)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{b g}+\frac{\left (2 B (d f-c g)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{d g}\\ &=-\frac{2 A B (b c-a d) g x}{b d}-\frac{2 B (b f-a g)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^2 g}+\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}+\frac{2 B (d f-c g)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^2 g}-\frac{\left (2 B^2 (b c-a d) g\right ) \int \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right ) \, dx}{b d}+\frac{\left (2 B^2 (b f-a g)^2\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{e (a+b x)^2} \, dx}{b^2 g}-\frac{\left (2 B^2 (d f-c g)^2\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{e (a+b x)^2} \, dx}{d^2 g}\\ &=-\frac{2 A B (b c-a d) g x}{b d}-\frac{2 B^2 (b c-a d) g (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^2 d}-\frac{2 B (b f-a g)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^2 g}+\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}+\frac{2 B (d f-c g)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^2 g}+\frac{\left (4 B^2 (b c-a d)^2 g\right ) \int \frac{1}{c+d x} \, dx}{b^2 d}+\frac{\left (2 B^2 (b f-a g)^2\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{(a+b x)^2} \, dx}{b^2 e g}-\frac{\left (2 B^2 (d f-c g)^2\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{(a+b x)^2} \, dx}{d^2 e g}\\ &=-\frac{2 A B (b c-a d) g x}{b d}-\frac{2 B^2 (b c-a d) g (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^2 d}-\frac{2 B (b f-a g)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^2 g}+\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}+\frac{4 B^2 (b c-a d)^2 g \log (c+d x)}{b^2 d^2}+\frac{2 B (d f-c g)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^2 g}+\frac{\left (2 B^2 (b f-a g)^2\right ) \int \left (\frac{2 b e \log (a+b x)}{a+b x}-\frac{2 d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 e g}-\frac{\left (2 B^2 (d f-c g)^2\right ) \int \left (\frac{2 b e \log (c+d x)}{a+b x}-\frac{2 d e \log (c+d x)}{c+d x}\right ) \, dx}{d^2 e g}\\ &=-\frac{2 A B (b c-a d) g x}{b d}-\frac{2 B^2 (b c-a d) g (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^2 d}-\frac{2 B (b f-a g)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^2 g}+\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}+\frac{4 B^2 (b c-a d)^2 g \log (c+d x)}{b^2 d^2}+\frac{2 B (d f-c g)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^2 g}+\frac{\left (4 B^2 (b f-a g)^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b g}-\frac{\left (4 B^2 d (b f-a g)^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 g}-\frac{\left (4 b B^2 (d f-c g)^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d^2 g}+\frac{\left (4 B^2 (d f-c g)^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{d g}\\ &=-\frac{2 A B (b c-a d) g x}{b d}-\frac{2 B^2 (b c-a d) g (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^2 d}-\frac{2 B (b f-a g)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^2 g}+\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}+\frac{4 B^2 (b c-a d)^2 g \log (c+d x)}{b^2 d^2}-\frac{4 B^2 (d f-c g)^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 g}+\frac{2 B (d f-c g)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^2 g}-\frac{4 B^2 (b f-a g)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{\left (4 B^2 (b f-a g)^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}+\frac{\left (4 B^2 (b f-a g)^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g}+\frac{\left (4 B^2 (d f-c g)^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{d^2 g}+\frac{\left (4 B^2 (d f-c g)^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{d g}\\ &=-\frac{2 A B (b c-a d) g x}{b d}+\frac{2 B^2 (b f-a g)^2 \log ^2(a+b x)}{b^2 g}-\frac{2 B^2 (b c-a d) g (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^2 d}-\frac{2 B (b f-a g)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^2 g}+\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}+\frac{4 B^2 (b c-a d)^2 g \log (c+d x)}{b^2 d^2}-\frac{4 B^2 (d f-c g)^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 g}+\frac{2 B (d f-c g)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^2 g}+\frac{2 B^2 (d f-c g)^2 \log ^2(c+d x)}{d^2 g}-\frac{4 B^2 (b f-a g)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{\left (4 B^2 (b f-a g)^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 g}+\frac{\left (4 B^2 (d f-c g)^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 )}{d^2 g}\\ &=-\frac{2 A B (b c-a d) g x}{b d}+\frac{2 B^2 (b f-a g)^2 \log ^2(a+b x)}{b^2 g}-\frac{2 B^2 (b c-a d) g (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^2 d}-\frac{2 B (b f-a g)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^2 g}+\frac{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 g}+\frac{4 B^2 (b c-a d)^2 g \log (c+d x)}{b^2 d^2}-\frac{4 B^2 (d f-c g)^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^2 g}+\frac{2 B (d f-c g)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^2 g}+\frac{2 B^2 (d f-c g)^2 \log ^2(c+d x)}{d^2 g}-\frac{4 B^2 (b f-a g)^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{4 B^2 (b f-a g)^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}-\frac{4 B^2 (d f-c g)^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d^2 g}\\ \end{align*}

Mathematica [A]  time = 0.306174, size = 351, normalized size = 1.25 \[ \frac{(f+g x)^2 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2-\frac{4 B \left (b^2 B (d f-c g)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-B d^2 (b f-a g)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-b^2 (d f-c g)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+d^2 (b f-a g)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+A b d g^2 x (b c-a d)+B d g^2 (a+b x) (b c-a d) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )-2 B g^2 (b c-a d)^2 \log (c+d x)\right )}{b^2 d^2}}{2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.416, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.59674, size = 1061, normalized size = 3.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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