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

Optimal. Leaf size=874 \[ \frac {B^2 g^3 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^4}{6 b^4 d^4}+\frac {B^2 g^3 \log (c+d x) (b c-a d)^4}{6 b^4 d^4}+\frac {B^2 g^3 x (b c-a d)^3}{6 b^3 d^3}+\frac {B^2 g^2 (4 b d f-3 b c g-a d g) \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^3}{4 b^4 d^4}+\frac {B^2 g^2 (4 b d f-3 b c g-a d g) \log (c+d x) (b c-a d)^3}{4 b^4 d^4}+\frac {B^2 g^3 (c+d x)^2 (b c-a d)^2}{12 b^2 d^4}+\frac {B^2 g^2 (4 b d f-3 b c g-a d g) x (b c-a d)^2}{4 b^3 d^3}+\frac {B^2 g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) \log (c+d x) (b c-a d)^2}{2 b^4 d^4}-\frac {B g^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{6 b d^4}-\frac {B g^2 (4 b d f-3 b c g-a d g) (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{4 b^2 d^4}-\frac {B g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{2 b^4 d^3}-\frac {B (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b c-a d)}{2 b^4 d^4}-\frac {B^2 (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)}{2 b^4 d^4}-\frac {(b f-a g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g} \]

[Out]

1/6*B^2*(-a*d+b*c)^3*g^3*x/b^3/d^3+1/4*B^2*(-a*d+b*c)^2*g^2*(-a*d*g-3*b*c*g+4*b*d*f)*x/b^3/d^3+1/12*B^2*(-a*d+
b*c)^2*g^3*(d*x+c)^2/b^2/d^4+1/6*B^2*(-a*d+b*c)^4*g^3*ln((b*x+a)/(d*x+c))/b^4/d^4+1/4*B^2*(-a*d+b*c)^3*g^2*(-a
*d*g-3*b*c*g+4*b*d*f)*ln((b*x+a)/(d*x+c))/b^4/d^4-1/2*B*(-a*d+b*c)*g*(a^2*d^2*g^2-2*a*b*d*g*(-c*g+2*d*f)+b^2*(
3*c^2*g^2-8*c*d*f*g+6*d^2*f^2))*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^4/d^3-1/4*B*(-a*d+b*c)*g^2*(-a*d*g-3*b*c
*g+4*b*d*f)*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^2/d^4-1/6*B*(-a*d+b*c)*g^3*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*
x+c)))/b/d^4-1/2*B*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*(2*a*b*d^2*f*g-a^2*d^2*g^2-b^2*(c^2*g^2-2*c*d*f*g+2*d^2*f
^2))*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^4/d^4-1/4*(-a*g+b*f)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))
^2/b^4/g+1/4*(g*x+f)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/g+1/6*B^2*(-a*d+b*c)^4*g^3*ln(d*x+c)/b^4/d^4+1/4*B^2*(-a*
d+b*c)^3*g^2*(-a*d*g-3*b*c*g+4*b*d*f)*ln(d*x+c)/b^4/d^4+1/2*B^2*(-a*d+b*c)^2*g*(a^2*d^2*g^2-2*a*b*d*g*(-c*g+2*
d*f)+b^2*(3*c^2*g^2-8*c*d*f*g+6*d^2*f^2))*ln(d*x+c)/b^4/d^4-1/2*B^2*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*(2*a*b*d
^2*f*g-a^2*d^2*g^2-b^2*(c^2*g^2-2*c*d*f*g+2*d^2*f^2))*polylog(2,d*(b*x+a)/b/(d*x+c))/b^4/d^4

________________________________________________________________________________________

Rubi [A]  time = 1.74, antiderivative size = 994, normalized size of antiderivative = 1.14, number of steps used = 33, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {2525, 12, 2528, 2486, 31, 72, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B^2 \log ^2(a+b x) (b f-a g)^4}{4 b^4 g}-\frac {B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) (b f-a g)^4}{2 b^4 g}-\frac {B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) (b f-a g)^4}{2 b^4 g}-\frac {B^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) (b f-a g)^4}{2 b^4 g}+\frac {B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac {B^2 (d f-c g)^4 \log ^2(c+d x)}{4 d^4 g}-\frac {B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-a d g) x}{4 b^3 d^3}-\frac {A B (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) x}{2 b^3 d^3}-\frac {a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 b^4 d}+\frac {a^2 B^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (a+b x)}{4 b^4 d^2}-\frac {B^2 (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{2 b^4 d^3}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 b^2 d^2}+\frac {B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 b d^4}-\frac {B^2 c^2 (b c-a d) g^2 (4 b d f-b c g-a d g) \log (c+d x)}{4 b^2 d^4}+\frac {B^2 (b c-a d)^2 g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-a d g (4 d f-c g) b+a^2 d^2 g^2\right ) \log (c+d x)}{2 b^4 d^4}-\frac {B^2 (d f-c g)^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 d^4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}-\frac {B^2 (d f-c g)^4 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{2 d^4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

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 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 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 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 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 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 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 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 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 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 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]

Rubi steps

\begin {align*} \int (f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx &=\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g}-\frac {B \int \frac {(b c-a d) (f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\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)}{c+d x}\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)}{c+d x}\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)}{c+d x}\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)}{c+d x}\right )\right )}{b^2 d^2}+\frac {g^4 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b d}+\frac {(b f-a g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\right )\right )}{d^3 (-b c+a d) (c+d x)}\right ) \, dx}{2 g}\\ &=\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\right )\right ) \, dx}{2 b d}-\frac {\left (B (b f-a g)^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{2 b^3 g}+\frac {\left (B (d f-c g)^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{2 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)}{c+d x}\right )\right ) \, dx}{2 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)}{c+d x}\right )\right ) \, dx}{2 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}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac {\left (B^2 (b c-a d) g^3\right ) \int \frac {(b c-a d) x^3}{(a+b x) (c+d x)} \, dx}{6 b d}+\frac {\left (B^2 (b f-a g)^4\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}{2 b^4 g}-\frac {\left (B^2 (d f-c g)^4\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}{2 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 {(b c-a d) x^2}{(a+b x) (c+d x)} \, dx}{4 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)}{c+d x}\right ) \, dx}{2 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}{2 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac {\left (B^2 (b c-a d)^2 g^3\right ) \int \frac {x^3}{(a+b x) (c+d x)} \, dx}{6 b d}+\frac {\left (B^2 (b f-a g)^4\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}{2 b^4 e g}-\frac {\left (B^2 (d f-c g)^4\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}{2 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}{4 b^2 d^2}+\frac {\left (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}{2 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}{2 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac {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)}{2 b^4 d^4}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac {\left (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}{6 b d}+\frac {\left (B^2 (b f-a g)^4\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}{2 b^4 e g}-\frac {\left (B^2 (d f-c g)^4\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}{2 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}{4 b^2 d^2}\\ &=-\frac {B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 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}{4 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}{2 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}-\frac {a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 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)}{4 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac {B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 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)}{4 b^2 d^4}+\frac {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)}{2 b^4 d^4}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac {\left (B^2 (b f-a g)^4\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 b^3 g}-\frac {\left (B^2 d (b f-a g)^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 b^4 g}-\frac {\left (b B^2 (d f-c g)^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 d^4 g}+\frac {\left (B^2 (d f-c g)^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 d^3 g}\\ &=-\frac {B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 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}{4 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}{2 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}-\frac {a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 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)}{4 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac {B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 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)}{4 b^2 d^4}+\frac {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)}{2 b^4 d^4}-\frac {B^2 (d f-c g)^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 d^4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}-\frac {B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 g}+\frac {\left (B^2 (b f-a g)^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 b^4 g}+\frac {\left (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}{2 b^3 g}+\frac {\left (B^2 (d f-c g)^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2 d^4 g}+\frac {\left (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}{2 d^3 g}\\ &=-\frac {B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 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}{4 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}{2 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}-\frac {a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 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)}{4 b^4 d^2}+\frac {B^2 (b f-a g)^4 \log ^2(a+b x)}{4 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac {B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 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)}{4 b^2 d^4}+\frac {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)}{2 b^4 d^4}-\frac {B^2 (d f-c g)^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 d^4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac {B^2 (d f-c g)^4 \log ^2(c+d x)}{4 d^4 g}-\frac {B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 g}+\frac {\left (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 )}{2 b^4 g}+\frac {\left (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 )}{2 d^4 g}\\ &=-\frac {B^2 (b c-a d)^2 (b c+a d) g^3 x}{6 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}{4 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}{2 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 x^2}{12 b^2 d^2}-\frac {a^3 B^2 (b c-a d) g^3 \log (a+b x)}{6 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)}{4 b^4 d^2}+\frac {B^2 (b f-a g)^4 \log ^2(a+b x)}{4 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)}{c+d x}\right )}{2 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)}{c+d x}\right )\right )}{4 b^2 d^2}-\frac {B (b c-a d) g^3 x^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d}-\frac {B (b f-a g)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 g}+\frac {B^2 c^3 (b c-a d) g^3 \log (c+d x)}{6 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)}{4 b^2 d^4}+\frac {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)}{2 b^4 d^4}-\frac {B^2 (d f-c g)^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 d^4 g}+\frac {B (d f-c g)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 d^4 g}+\frac {B^2 (d f-c g)^4 \log ^2(c+d x)}{4 d^4 g}-\frac {B^2 (b f-a g)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 g}-\frac {B^2 (b f-a g)^4 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2 b^4 g}-\frac {B^2 (d f-c g)^4 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 d^4 g}\\ \end {align*}

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Mathematica [A]  time = 0.98, size = 733, normalized size = 0.84 \[ \frac {(f+g x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2-\frac {B \left (B g^4 (b c-a d) \left (2 a^3 d^3 \log (a+b x)+b d x (b c-a d) (2 a d+2 b c-b d x)-2 b^3 c^3 \log (c+d x)\right )+6 A b d g^2 x (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )+6 B d g^2 (a+b x) (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )-6 B g^2 (b c-a d)^2 \log (c+d x) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )-3 B g^3 (b c-a d) (a d g+b c g-4 b d f) \left (b \left (d x (a d-b c)+b c^2 \log (c+d x)\right )-a^2 d^2 \log (a+b x)\right )-6 b^4 (d f-c g)^4 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+2 b^3 d^3 g^4 x^3 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+3 b^2 d^2 g^3 x^2 (b c-a d) (-a d g-b c g+4 b d f) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+6 d^4 (b f-a g)^4 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+3 b^4 B (d f-c g)^4 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-3 B d^4 (b f-a g)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )\right )}{3 b^4 d^4}}{4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((f + g*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^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))/(c + d*x)] + 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))/(c + d*x)]) + 2*b^3*d^3*(b*c - a*d)*g^4*x^3*(A + B*L
og[(e*(a + b*x))/(c + d*x)]) + 6*d^4*(b*f - a*g)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*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))/(c + d*x)])*Log[c + d*x] + 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]) - 3*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])) - 3*B*d^4*(b*f - a*g)^4*(L
og[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)]) +
3*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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fricas [F]  time = 1.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} g^{3} x^{3} + 3 \, A^{2} f g^{2} x^{2} + 3 \, A^{2} f^{2} g x + A^{2} f^{3} + {\left (B^{2} g^{3} x^{3} + 3 \, B^{2} f g^{2} x^{2} + 3 \, B^{2} f^{2} g x + B^{2} f^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A B g^{3} x^{3} + 3 \, A B f g^{2} x^{2} + 3 \, A B f^{2} g x + A B f^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 2.58, size = 0, normalized size = 0.00 \[ \int \left (g x +f \right )^{3} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.91, size = 2140, normalized size = 2.45 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^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*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x +
a)/b - c*log(d*x + c)/d)*A*B*f^3 + 3*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*lo
g(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*f^2*g + (2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 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 +
1/12*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 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 - 1/12*(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 + (24*c*d^3*f^3*log(e) - (6*g^3*log(e) + 11*g^3)*c^4 + 12*(2*f*g^2*log(e) + 3*f*g^2)*c^3*d - 36*
(f^2*g*log(e) + f^2*g)*c^2*d^2)*b^3)*B^2*log(d*x + c)/(b^3*d^4) + 1/2*(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 + 2*(a*b^3*d^4*g^3*log(e) + (6*d^4*f*g^2*log(e)^2 - c*d^3*g^3*log(e))*b^4)*B^2*x^3 - ((3*g^3*
log(e) - g^3)*a^2*b^2*d^4 - 2*(6*d^4*f*g^2*log(e) - c*d^3*g^3)*a*b^3 - (18*d^4*f^2*g*log(e)^2 - 12*c*d^3*f*g^2
*log(e) + (3*g^3*log(e) + g^3)*c^2*d^2)*b^4)*B^2*x^2 + ((6*g^3*log(e) - 5*g^3)*a^3*b*d^4 + (5*c*d^3*g^3 - 12*(
2*f*g^2*log(e) - f*g^2)*d^4)*a^2*b^2 + (36*d^4*f^2*g*log(e) - 24*c*d^3*f*g^2 + 5*c^2*d^2*g^3)*a*b^3 + (12*d^4*
f^3*log(e)^2 - 36*c*d^3*f^2*g*log(e) - (6*g^3*log(e) + 5*g^3)*c^3*d + 12*(2*f*g^2*log(e) + f*g^2)*c^2*d^2)*b^4
)*B^2*x + 3*(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 + 3*(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 + (6*B^2*b^4*d^4*g^3*x^4*log(e) + 2*(a*b^3*d^4*g^3 + (
12*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 + (12*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 + (4*
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*g^3*log(e) - 11*g^3)*a^4*d^4 +
2*(c*d^3*g^3 - 6*(2*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 - 12*(f^2*g*log(e) - f
^2*g)*d^4)*a^2*b^2 - 6*(4*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) - (6*B^2*b^4*d^4*g^3*x^4*log(e) + 2*(a*b^3*d^4*g^3 + (12*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 + (12*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 + (4*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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (f+g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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