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

Optimal. Leaf size=869 \[ \frac{2 B^2 g^3 \log \left (\frac{a+b x}{c+d x}\right ) (b c-a d)^4}{3 b^4 d^4}+\frac{2 B^2 g^3 \log (c+d x) (b c-a d)^4}{3 b^4 d^4}+\frac{2 B^2 g^3 x (b c-a d)^3}{3 b^3 d^3}+\frac{B^2 g^2 (4 b d f-3 b c g-a d g) \log \left (\frac{a+b x}{c+d x}\right ) (b c-a d)^3}{b^4 d^4}+\frac{B^2 g^2 (4 b d f-3 b c g-a d g) \log (c+d x) (b c-a d)^3}{b^4 d^4}+\frac{B^2 g^3 (c+d x)^2 (b c-a d)^2}{3 b^2 d^4}+\frac{B^2 g^2 (4 b d f-3 b c g-a d g) x (b c-a d)^2}{b^3 d^3}+\frac{2 B^2 g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) \log (c+d x) (b c-a d)^2}{b^4 d^4}-\frac{B g^3 (c+d x)^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) (b c-a d)}{3 b d^4}-\frac{B g^2 (4 b d f-3 b c g-a d g) (c+d x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) (b c-a d)}{2 b^2 d^4}-\frac{B g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) (b c-a d)}{b^4 d^3}-\frac{B (2 b d f-b c g-a d g) \left (-\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2+2 a d^2 f g b-a^2 d^2 g^2\right ) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac{b c-a d}{b (c+d x)}\right ) (b c-a d)}{b^4 d^4}-\frac{2 B^2 (2 b d f-b c g-a d g) \left (-\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2+2 a d^2 f g b-a^2 d^2 g^2\right ) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) (b c-a d)}{b^4 d^4}-\frac{(b f-a g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g} \]

[Out]

(2*B^2*(b*c - a*d)^3*g^3*x)/(3*b^3*d^3) + (B^2*(b*c - a*d)^2*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*x)/(b^3*d^3) + (B
^2*(b*c - a*d)^2*g^3*(c + d*x)^2)/(3*b^2*d^4) - (B*(b*c - a*d)*g*(a^2*d^2*g^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*
(6*d^2*f^2 - 8*c*d*f*g + 3*c^2*g^2))*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(b^4*d^3) - (B*(b*c -
a*d)*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(2*b^2*d^4) - (B*(
b*c - a*d)*g^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*b*d^4) - ((b*f - a*g)^4*(A + B*Log[(e*
(a + b*x)^2)/(c + d*x)^2])^2)/(4*b^4*g) + ((f + g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(4*g) - (B*
(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*(A
+ B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[(b*c - a*d)/(b*(c + d*x))])/(b^4*d^4) + (2*B^2*(b*c - a*d)^4*g^3*Lo
g[(a + b*x)/(c + d*x)])/(3*b^4*d^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - 3*b*c*g - a*d*g)*Log[(a + b*x)/(c + d*
x)])/(b^4*d^4) + (2*B^2*(b*c - a*d)^4*g^3*Log[c + d*x])/(3*b^4*d^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - 3*b*c*
g - a*d*g)*Log[c + d*x])/(b^4*d^4) + (2*B^2*(b*c - a*d)^2*g*(a^2*d^2*g^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*(6*d^
2*f^2 - 8*c*d*f*g + 3*c^2*g^2))*Log[c + d*x])/(b^4*d^4) - (2*B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*
d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^4*d
^4)

________________________________________________________________________________________

Rubi [A]  time = 1.80806, antiderivative size = 973, normalized size of antiderivative = 1.12, number of steps used = 33, number of rules used = 13, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.419, Rules used = {2525, 12, 2528, 2486, 31, 72, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B^2 \log ^2(a+b x) (b f-a g)^4}{b^4 g}-\frac{B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) (b f-a g)^4}{b^4 g}-\frac{2 B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) (b f-a g)^4}{b^4 g}-\frac{2 B^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) (b f-a g)^4}{b^4 g}+\frac{B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac{B^2 (d f-c g)^4 \log ^2(c+d x)}{d^4 g}-\frac{2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac{A B (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) x}{b^3 d^3}-\frac{2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac{a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}-\frac{B^2 (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}+\frac{2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac{B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac{2 B^2 (b c-a d)^2 g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) \log (c+d x)}{b^4 d^4}-\frac{2 B^2 (d f-c g)^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}-\frac{2 B^2 (d f-c g)^4 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d^4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2*(b*c - a*d)^2*(b*c + a*d)*g^3*x)/(3*b^3*d^3) + (B^2*(b*c - a*d)^2*g^2*(4*b*d*f - b*c*g - a*d*g)*x)/(b^
3*d^3) - (A*B*(b*c - a*d)*g*(a^2*d^2*g^2 - a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*x)/(
b^3*d^3) + (B^2*(b*c - a*d)^2*g^3*x^2)/(3*b^2*d^2) - (2*a^3*B^2*(b*c - a*d)*g^3*Log[a + b*x])/(3*b^4*d) + (a^2
*B^2*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - a*d*g)*Log[a + b*x])/(b^4*d^2) + (B^2*(b*f - a*g)^4*Log[a + b*x]^2)/(b
^4*g) - (B^2*(b*c - a*d)*g*(a^2*d^2*g^2 - a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*(a +
b*x)*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(b^4*d^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - a*d*g)*x^2*(A + B*Log
[(e*(a + b*x)^2)/(c + d*x)^2]))/(2*b^2*d^2) - (B*(b*c - a*d)*g^3*x^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))
/(3*b*d) - (B*(b*f - a*g)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(b^4*g) + ((f + g*x)^4*(A +
B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(4*g) + (2*B^2*c^3*(b*c - a*d)*g^3*Log[c + d*x])/(3*b*d^4) - (B^2*c^2*
(b*c - a*d)*g^2*(4*b*d*f - b*c*g - a*d*g)*Log[c + d*x])/(b^2*d^4) + (2*B^2*(b*c - a*d)^2*g*(a^2*d^2*g^2 - a*b*
d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*Log[c + d*x])/(b^4*d^4) - (2*B^2*(d*f - c*g)^4*Log[
-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^4*g) + (B*(d*f - c*g)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]
)*Log[c + d*x])/(d^4*g) + (B^2*(d*f - c*g)^4*Log[c + d*x]^2)/(d^4*g) - (2*B^2*(b*f - a*g)^4*Log[a + b*x]*Log[(
b*(c + d*x))/(b*c - a*d)])/(b^4*g) - (2*B^2*(b*f - a*g)^4*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^4*g) -
(2*B^2*(d*f - c*g)^4*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^4*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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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)^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx &=\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}-\frac{B \int \frac{2 (b c-a d) (f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x) (c+d x)} \, dx}{2 g}\\ &=\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}-\frac{(B (b c-a d)) \int \frac{(f+g x)^4 \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)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}-\frac{(B (b c-a d)) \int \left (\frac{g^2 \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^3 d^3}+\frac{g^3 (4 b d f-b c g-a d g) x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^2 d^2}+\frac{g^4 x^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)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^3 (b c-a d) (a+b x)}+\frac{(d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{d^3 (-b c+a d) (c+d x)}\right ) \, dx}{g}\\ &=\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}-\frac{\left (B (b c-a d) g^3\right ) \int x^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{b d}-\frac{\left (B (b f-a g)^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{b^3 g}+\frac{\left (B (d f-c g)^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{d^3 g}-\frac{\left (B (b c-a d) g^2 (4 b d f-b c g-a d g)\right ) \int x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{b^2 d^2}-\frac{\left (B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right )\right ) \int \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx}{b^3 d^3}\\ &=-\frac{A B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x}{b^3 d^3}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac{\left (B^2 (b c-a d) g^3\right ) \int \frac{2 (b c-a d) x^3}{(a+b x) (c+d x)} \, dx}{3 b d}+\frac{\left (B^2 (b f-a g)^4\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^4 g}-\frac{\left (B^2 (d f-c g)^4\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^4 g}+\frac{\left (B^2 (b c-a d) g^2 (4 b d f-b c g-a d g)\right ) \int \frac{2 (b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{2 b^2 d^2}-\frac{\left (B^2 (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right )\right ) \int \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right ) \, dx}{b^3 d^3}\\ &=-\frac{A B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x}{b^3 d^3}-\frac{B^2 (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac{\left (2 B^2 (b c-a d)^2 g^3\right ) \int \frac{x^3}{(a+b x) (c+d x)} \, dx}{3 b d}+\frac{\left (B^2 (b f-a g)^4\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^4 e g}-\frac{\left (B^2 (d f-c g)^4\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^4 e g}+\frac{\left (B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g)\right ) \int \frac{x^2}{(a+b x) (c+d x)} \, dx}{b^2 d^2}+\frac{\left (2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right )\right ) \int \frac{1}{c+d x} \, dx}{b^4 d^3}\\ &=-\frac{A B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x}{b^3 d^3}-\frac{B^2 (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac{2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) \log (c+d x)}{b^4 d^4}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac{\left (2 B^2 (b c-a d)^2 g^3\right ) \int \left (\frac{-b c-a d}{b^2 d^2}+\frac{x}{b d}-\frac{a^3}{b^2 (b c-a d) (a+b x)}-\frac{c^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{3 b d}+\frac{\left (B^2 (b f-a g)^4\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^4 e g}-\frac{\left (B^2 (d f-c g)^4\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^4 e g}+\frac{\left (B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g)\right ) \int \left (\frac{1}{b d}+\frac{a^2}{b (b c-a d) (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)}\right ) \, dx}{b^2 d^2}\\ &=-\frac{2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac{A B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x}{b^3 d^3}+\frac{B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}-\frac{2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac{a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}-\frac{B^2 (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac{2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac{B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac{2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) \log (c+d x)}{b^4 d^4}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac{\left (2 B^2 (b f-a g)^4\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g}-\frac{\left (2 B^2 d (b f-a g)^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g}-\frac{\left (2 b B^2 (d f-c g)^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d^4 g}+\frac{\left (2 B^2 (d f-c g)^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{d^3 g}\\ &=-\frac{2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac{A B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x}{b^3 d^3}+\frac{B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}-\frac{2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac{a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}-\frac{B^2 (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac{2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac{B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac{2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) \log (c+d x)}{b^4 d^4}-\frac{2 B^2 (d f-c g)^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}-\frac{2 B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g}+\frac{\left (2 B^2 (b f-a g)^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g}+\frac{\left (2 B^2 (b f-a g)^4\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g}+\frac{\left (2 B^2 (d f-c g)^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{d^4 g}+\frac{\left (2 B^2 (d f-c g)^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{d^3 g}\\ &=-\frac{2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac{A B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x}{b^3 d^3}+\frac{B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}-\frac{2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac{a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}+\frac{B^2 (b f-a g)^4 \log ^2(a+b x)}{b^4 g}-\frac{B^2 (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac{2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac{B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac{2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) \log (c+d x)}{b^4 d^4}-\frac{2 B^2 (d f-c g)^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac{B^2 (d f-c g)^4 \log ^2(c+d x)}{d^4 g}-\frac{2 B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g}+\frac{\left (2 B^2 (b f-a g)^4\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^4 g}+\frac{\left (2 B^2 (d f-c g)^4\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^4 g}\\ &=-\frac{2 B^2 (b c-a d)^2 (b c+a d) g^3 x}{3 b^3 d^3}+\frac{B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{b^3 d^3}-\frac{A B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x}{b^3 d^3}+\frac{B^2 (b c-a d)^2 g^3 x^2}{3 b^2 d^2}-\frac{2 a^3 B^2 (b c-a d) g^3 \log (a+b x)}{3 b^4 d}+\frac{a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{b^4 d^2}+\frac{B^2 (b f-a g)^4 \log ^2(a+b x)}{b^4 g}-\frac{B^2 (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) (a+b x) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{b^4 d^3}-\frac{B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b^2 d^2}-\frac{B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b d}-\frac{B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b^4 g}+\frac{(f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 g}+\frac{2 B^2 c^3 (b c-a d) g^3 \log (c+d x)}{3 b d^4}-\frac{B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{b^2 d^4}+\frac{2 B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) \log (c+d x)}{b^4 d^4}-\frac{2 B^2 (d f-c g)^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d^4 g}+\frac{B (d f-c g)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d^4 g}+\frac{B^2 (d f-c g)^4 \log ^2(c+d x)}{d^4 g}-\frac{2 B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g}-\frac{2 B^2 (b f-a g)^4 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g}-\frac{2 B^2 (d f-c g)^4 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d^4 g}\\ \end{align*}

Mathematica [A]  time = 0.966501, size = 746, normalized size = 0.86 \[ \frac{(f+g x)^4 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2-\frac{2 B \left (6 b^4 B (d f-c g)^4 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-6 B d^4 (b f-a g)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+6 A b d g^2 x (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )+6 B d g^2 (a+b x) (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right ) \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )-12 B g^2 (b c-a d)^2 \log (c+d x) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )+2 B g^4 (b c-a d) \left (2 a^3 d^3 \log (a+b x)+b d x (b c-a d) (2 a d+2 b c-b d x)-2 b^3 c^3 \log (c+d x)\right )-6 B g^3 (b c-a d) (a d g+b c g-4 b d f) \left (b \left (d x (a d-b c)+b c^2 \log (c+d x)\right )-a^2 d^2 \log (a+b x)\right )+3 b^2 d^2 g^3 x^2 (b c-a d) (-a d g-b c g+4 b d f) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+2 b^3 d^3 g^4 x^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )-6 b^4 (d f-c g)^4 \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+6 d^4 (b f-a g)^4 \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )\right )}{3 b^4 d^4}}{4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((f + g*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 - (2*B*(6*A*b*d*(b*c - a*d)*g^2*(a^2*d^2*g^2 + a*b*d*g
*(-4*d*f + c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*x + 6*B*d*(b*c - a*d)*g^2*(a^2*d^2*g^2 + a*b*d*g*(-4*
d*f + c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*(a + b*x)*Log[(e*(a + b*x)^2)/(c + d*x)^2] + 3*b^2*d^2*(b*
c - a*d)*g^3*(4*b*d*f - b*c*g - a*d*g)*x^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 2*b^3*d^3*(b*c - a*d)*g^
4*x^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 6*d^4*(b*f - a*g)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x)^2)/(
c + d*x)^2]) - 12*B*(b*c - a*d)^2*g^2*(a^2*d^2*g^2 + a*b*d*g*(-4*d*f + c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2
*g^2))*Log[c + d*x] - 6*b^4*(d*f - c*g)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[c + d*x] + 2*B*(b*c - a
*d)*g^4*(b*d*(b*c - a*d)*x*(2*b*c + 2*a*d - b*d*x) + 2*a^3*d^3*Log[a + b*x] - 2*b^3*c^3*Log[c + d*x]) - 6*B*(b
*c - a*d)*g^3*(-4*b*d*f + b*c*g + a*d*g)*(-(a^2*d^2*Log[a + b*x]) + b*(d*(-(b*c) + a*d)*x + b*c^2*Log[c + d*x]
)) - 6*B*d^4*(b*f - a*g)^4*(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)]) + 6*b^4*B*(d*f - c*g)^4*((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)])))/(3*b^4*d^4))/(4*g)

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Maple [F]  time = 1.77, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) ^{3} \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)^3*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2,x)

[Out]

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

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Maxima [B]  time = 1.9495, size = 3174, normalized size = 3.65 \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)^3*(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="maxima")

[Out]

1/4*A^2*g^3*x^4 + A^2*f*g^2*x^3 + 3/2*A^2*f^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
^3 + 3*(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*f^2*g + 2*(x^3*l
og(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^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2
*d^2))*A*B*f*g^2 + 1/6*(3*x^4*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)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3
)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*A*B*g^3 + A^2*f^3*x - 1/3*(6*a^3*c
*d^3*g^3 - 3*(8*c*d^3*f*g^2 - c^2*d^2*g^3)*a^2*b + 2*(18*c*d^3*f^2*g - 6*c^2*d^2*f*g^2 + c^3*d*g^3)*a*b^2 + (1
2*c*d^3*f^3*log(e) - (3*g^3*log(e) + 11*g^3)*c^4 + 12*(f*g^2*log(e) + 3*f*g^2)*c^3*d - 18*(f^2*g*log(e) + 2*f^
2*g)*c^2*d^2)*b^3)*B^2*log(d*x + c)/(b^3*d^4) + 2*(4*a*b^3*d^4*f^3 - 6*a^2*b^2*d^4*f^2*g + 4*a^3*b*d^4*f*g^2 -
a^4*d^4*g^3 - (4*c*d^3*f^3 - 6*c^2*d^2*f^2*g + 4*c^3*d*f*g^2 - c^4*g^3)*b^4)*(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^4*d^4) + 1/12*(3*B^2*b^4*d^4*g^3*x^4*log(e)^2 + 4
*(a*b^3*d^4*g^3*log(e) + (3*d^4*f*g^2*log(e)^2 - c*d^3*g^3*log(e))*b^4)*B^2*x^3 - 2*((3*g^3*log(e) - 2*g^3)*a^
2*b^2*d^4 - 4*(3*d^4*f*g^2*log(e) - c*d^3*g^3)*a*b^3 - (9*d^4*f^2*g*log(e)^2 - 12*c*d^3*f*g^2*log(e) + (3*g^3*
log(e) + 2*g^3)*c^2*d^2)*b^4)*B^2*x^2 + 4*((3*g^3*log(e) - 5*g^3)*a^3*b*d^4 + (5*c*d^3*g^3 - 12*(f*g^2*log(e)
- f*g^2)*d^4)*a^2*b^2 + (18*d^4*f^2*g*log(e) - 24*c*d^3*f*g^2 + 5*c^2*d^2*g^3)*a*b^3 + (3*d^4*f^3*log(e)^2 - 1
8*c*d^3*f^2*g*log(e) - (3*g^3*log(e) + 5*g^3)*c^3*d + 12*(f*g^2*log(e) + f*g^2)*c^2*d^2)*b^4)*B^2*x + 12*(B^2*
b^4*d^4*g^3*x^4 + 4*B^2*b^4*d^4*f*g^2*x^3 + 6*B^2*b^4*d^4*f^2*g*x^2 + 4*B^2*b^4*d^4*f^3*x + (4*a*b^3*d^4*f^3 -
6*a^2*b^2*d^4*f^2*g + 4*a^3*b*d^4*f*g^2 - a^4*d^4*g^3)*B^2)*log(b*x + a)^2 + 12*(B^2*b^4*d^4*g^3*x^4 + 4*B^2*
b^4*d^4*f*g^2*x^3 + 6*B^2*b^4*d^4*f^2*g*x^2 + 4*B^2*b^4*d^4*f^3*x + (4*c*d^3*f^3 - 6*c^2*d^2*f^2*g + 4*c^3*d*f
*g^2 - c^4*g^3)*B^2*b^4)*log(d*x + c)^2 + 4*(3*B^2*b^4*d^4*g^3*x^4*log(e) + 2*(a*b^3*d^4*g^3 + (6*d^4*f*g^2*lo
g(e) - c*d^3*g^3)*b^4)*B^2*x^3 + 3*(4*a*b^3*d^4*f*g^2 - a^2*b^2*d^4*g^3 + (6*d^4*f^2*g*log(e) - 4*c*d^3*f*g^2
+ c^2*d^2*g^3)*b^4)*B^2*x^2 + 6*(6*a*b^3*d^4*f^2*g - 4*a^2*b^2*d^4*f*g^2 + a^3*b*d^4*g^3 + (2*d^4*f^3*log(e) -
6*c*d^3*f^2*g + 4*c^2*d^2*f*g^2 - c^3*d*g^3)*b^4)*B^2*x - ((3*g^3*log(e) - 11*g^3)*a^4*d^4 + 2*(c*d^3*g^3 - 6
*(f*g^2*log(e) - 3*f*g^2)*d^4)*a^3*b - 3*(4*c*d^3*f*g^2 - c^2*d^2*g^3 - 6*(f^2*g*log(e) - 2*f^2*g)*d^4)*a^2*b^
2 - 6*(2*d^4*f^3*log(e) - 6*c*d^3*f^2*g + 4*c^2*d^2*f*g^2 - c^3*d*g^3)*a*b^3)*B^2)*log(b*x + a) - 4*(3*B^2*b^4
*d^4*g^3*x^4*log(e) + 2*(a*b^3*d^4*g^3 + (6*d^4*f*g^2*log(e) - c*d^3*g^3)*b^4)*B^2*x^3 + 3*(4*a*b^3*d^4*f*g^2
- a^2*b^2*d^4*g^3 + (6*d^4*f^2*g*log(e) - 4*c*d^3*f*g^2 + c^2*d^2*g^3)*b^4)*B^2*x^2 + 6*(6*a*b^3*d^4*f^2*g - 4
*a^2*b^2*d^4*f*g^2 + a^3*b*d^4*g^3 + (2*d^4*f^3*log(e) - 6*c*d^3*f^2*g + 4*c^2*d^2*f*g^2 - c^3*d*g^3)*b^4)*B^2
*x + 6*(B^2*b^4*d^4*g^3*x^4 + 4*B^2*b^4*d^4*f*g^2*x^3 + 6*B^2*b^4*d^4*f^2*g*x^2 + 4*B^2*b^4*d^4*f^3*x + (4*a*b
^3*d^4*f^3 - 6*a^2*b^2*d^4*f^2*g + 4*a^3*b*d^4*f*g^2 - a^4*d^4*g^3)*B^2)*log(b*x + a))*log(d*x + c))/(b^4*d^4)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} g^{3} x^{3} + 3 \, A^{2} f g^{2} x^{2} + 3 \, A^{2} f^{2} g x + A^{2} f^{3} +{\left (B^{2} g^{3} x^{3} + 3 \, B^{2} f g^{2} x^{2} + 3 \, B^{2} f^{2} g x + B^{2} f^{3}\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^{3} x^{3} + 3 \, A B f g^{2} x^{2} + 3 \, A B f^{2} g x + A B f^{3}\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)^3*(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="fricas")

[Out]

integral(A^2*g^3*x^3 + 3*A^2*f*g^2*x^2 + 3*A^2*f^2*g*x + A^2*f^3 + (B^2*g^3*x^3 + 3*B^2*f*g^2*x^2 + 3*B^2*f^2*
g*x + B^2*f^3)*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^3*x^3 + 3*A*B*f*g^2
*x^2 + 3*A*B*f^2*g*x + A*B*f^3)*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)**3*(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 )}^{3}{\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)^3*(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="giac")

[Out]

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