\(\int (a g+b g x)^4 (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [97]

Optimal result

Integrand size = 32, antiderivative size = 365 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {B (b c-a d)^2 g^4 (a+b x)^3 \left (4 A+B+4 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (12 A+7 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b d^3}+\frac {B (b c-a d)^4 g^4 (a+b x) \left (12 A+13 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^4}+\frac {B (b c-a d)^5 g^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (12 A+25 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^5}+\frac {2 B^2 (b c-a d)^5 g^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \]

[Out]

-1/10*B*(-a*d+b*c)*g^4*(b*x+a)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/d+1/5*g^4*(b*x+a)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))
^2/b+1/30*B*(-a*d+b*c)^2*g^4*(b*x+a)^3*(4*A+B+4*B*ln(e*(b*x+a)/(d*x+c)))/b/d^2-1/60*B*(-a*d+b*c)^3*g^4*(b*x+a)
^2*(12*A+7*B+12*B*ln(e*(b*x+a)/(d*x+c)))/b/d^3+1/30*B*(-a*d+b*c)^4*g^4*(b*x+a)*(12*A+13*B+12*B*ln(e*(b*x+a)/(d
*x+c)))/b/d^4+1/30*B*(-a*d+b*c)^5*g^4*ln((-a*d+b*c)/b/(d*x+c))*(12*A+25*B+12*B*ln(e*(b*x+a)/(d*x+c)))/b/d^5+2/
5*B^2*(-a*d+b*c)^5*g^4*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^5

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2550, 2381, 2384, 2354, 2438} \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {B g^4 (b c-a d)^5 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+25 B\right )}{30 b d^5}+\frac {B g^4 (a+b x) (b c-a d)^4 \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+13 B\right )}{30 b d^4}-\frac {B g^4 (a+b x)^2 (b c-a d)^3 \left (12 B \log \left (\frac {e (a+b x)}{c+d x}\right )+12 A+7 B\right )}{60 b d^3}+\frac {B g^4 (a+b x)^3 (b c-a d)^2 \left (4 B \log \left (\frac {e (a+b x)}{c+d x}\right )+4 A+B\right )}{30 b d^2}-\frac {B g^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 b}+\frac {2 B^2 g^4 (b c-a d)^5 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \]

[In]

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

[Out]

-1/10*(B*(b*c - a*d)*g^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b*d) + (g^4*(a + b*x)^5*(A + B*Log
[(e*(a + b*x))/(c + d*x)])^2)/(5*b) + (B*(b*c - a*d)^2*g^4*(a + b*x)^3*(4*A + B + 4*B*Log[(e*(a + b*x))/(c + d
*x)]))/(30*b*d^2) - (B*(b*c - a*d)^3*g^4*(a + b*x)^2*(12*A + 7*B + 12*B*Log[(e*(a + b*x))/(c + d*x)]))/(60*b*d
^3) + (B*(b*c - a*d)^4*g^4*(a + b*x)*(12*A + 13*B + 12*B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b*d^4) + (B*(b*c -
a*d)^5*g^4*Log[(b*c - a*d)/(b*(c + d*x))]*(12*A + 25*B + 12*B*Log[(e*(a + b*x))/(c + d*x)]))/(30*b*d^5) + (2*
B^2*(b*c - a*d)^5*g^4*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5)

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2381

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Dist[b*n*(p/(d*(q + 1))), Int[(
f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m
+ q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2438

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 2550

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/b)^m, Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2))
, x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0]
&& NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

Rubi steps \begin{align*} \text {integral}& = \left ((b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {x^4 (A+B \log (e x))^2}{(b-d x)^6} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}-\frac {\left (2 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {x^4 (A+B \log (e x))}{(b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 b} \\ & = -\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {\left (B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {x^3 (4 A+B+4 B \log (e x))}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{10 b d} \\ & = -\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {B (b c-a d)^2 g^4 (a+b x)^3 \left (4 A+B+4 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {\left (B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {x^2 (4 B+3 (4 A+B)+12 B \log (e x))}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{30 b d^2} \\ & = -\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {B (b c-a d)^2 g^4 (a+b x)^3 \left (4 A+B+4 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (12 A+7 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b d^3}+\frac {\left (B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {x (12 B+2 (4 B+3 (4 A+B))+24 B \log (e x))}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{60 b d^3} \\ & = -\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {B (b c-a d)^2 g^4 (a+b x)^3 \left (4 A+B+4 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (12 A+7 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b d^3}+\frac {B (b c-a d)^4 g^4 (a+b x) \left (12 A+13 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^4}-\frac {\left (B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {36 B+2 (4 B+3 (4 A+B))+24 B \log (e x)}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{60 b d^4} \\ & = -\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {B (b c-a d)^2 g^4 (a+b x)^3 \left (4 A+B+4 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (12 A+7 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b d^3}+\frac {B (b c-a d)^4 g^4 (a+b x) \left (12 A+13 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^4}+\frac {B (b c-a d)^5 g^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (12 A+25 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^5}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 b d^5} \\ & = -\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b}+\frac {B (b c-a d)^2 g^4 (a+b x)^3 \left (4 A+B+4 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^2}-\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (12 A+7 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{60 b d^3}+\frac {B (b c-a d)^4 g^4 (a+b x) \left (12 A+13 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^4}+\frac {B (b c-a d)^5 g^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (12 A+25 B+12 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b d^5}+\frac {2 B^2 (b c-a d)^5 g^4 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.40 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B (b c-a d) \left (24 A b d (b c-a d)^3 x+24 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-12 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+8 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-24 B (b c-a d)^4 \log (c+d x)-24 (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^3 (b d x+(-b c+a d) \log (c+d x))+12 B (b c-a d)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{12 d^5}\right )}{5 b} \]

[In]

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

[Out]

(g^4*((a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(b*c - a*d)*(24*A*b*d*(b*c - a*d)^3*x + 24*B*d*(
b*c - a*d)^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] - 12*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x)
)/(c + d*x)]) + 8*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 6*d^4*(a + b*x)^4*(A + B*
Log[(e*(a + b*x))/(c + d*x)]) - 24*B*(b*c - a*d)^4*Log[c + d*x] - 24*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c
+ d*x)])*Log[c + d*x] + 4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*
x]) + B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a
*d)^3*Log[c + d*x]) + 12*B*(b*c - a*d)^3*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 12*B*(b*c - a*d)^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)])))/(12*d^5)
))/(5*b)

Maple [F]

\[\int \left (b g x +a g \right )^{4} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}d x\]

[In]

int((b*g*x+a*g)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

[Out]

int((b*g*x+a*g)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

Fricas [F]

\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \]

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="fricas")

[Out]

integral(A^2*b^4*g^4*x^4 + 4*A^2*a*b^3*g^4*x^3 + 6*A^2*a^2*b^2*g^4*x^2 + 4*A^2*a^3*b*g^4*x + A^2*a^4*g^4 + (B^
2*b^4*g^4*x^4 + 4*B^2*a*b^3*g^4*x^3 + 6*B^2*a^2*b^2*g^4*x^2 + 4*B^2*a^3*b*g^4*x + B^2*a^4*g^4)*log((b*e*x + a*
e)/(d*x + c))^2 + 2*(A*B*b^4*g^4*x^4 + 4*A*B*a*b^3*g^4*x^3 + 6*A*B*a^2*b^2*g^4*x^2 + 4*A*B*a^3*b*g^4*x + A*B*a
^4*g^4)*log((b*e*x + a*e)/(d*x + c)), x)

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Timed out} \]

[In]

integrate((b*g*x+a*g)**4*(A+B*ln(e*(b*x+a)/(d*x+c)))**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2389 vs. \(2 (350) = 700\).

Time = 0.32 (sec) , antiderivative size = 2389, normalized size of antiderivative = 6.55 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="maxima")

[Out]

1/5*A^2*b^4*g^4*x^5 + A^2*a*b^3*g^4*x^4 + 2*A^2*a^2*b^2*g^4*x^3 + 2*A^2*a^3*b*g^4*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*a^4*g^4 + 4*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x
+ c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*a^3*b*g^4 + 2*(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*a^2*b^2*g^4 + 1/3*(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*a*b^3*g^4 + 1/30*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 12*a^5*log
(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 +
6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*A*B*b^4*g^4 + A^2*a^4*g^4*x - 1/30*((12*
g^4*log(e) + 25*g^4)*b^4*c^5 - (60*g^4*log(e) + 113*g^4)*a*b^3*c^4*d + 4*(30*g^4*log(e) + 49*g^4)*a^2*b^2*c^3*
d^2 - 12*(10*g^4*log(e) + 13*g^4)*a^3*b*c^2*d^3 + 12*(5*g^4*log(e) + 4*g^4)*a^4*c*d^4)*B^2*log(d*x + c)/d^5 -
2/5*(b^5*c^5*g^4 - 5*a*b^4*c^4*d*g^4 + 10*a^2*b^3*c^3*d^2*g^4 - 10*a^3*b^2*c^2*d^3*g^4 + 5*a^4*b*c*d^4*g^4 - a
^5*d^5*g^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*d^5)
+ 1/60*(12*B^2*b^5*d^5*g^4*x^5*log(e)^2 - 6*(b^5*c*d^4*g^4*log(e) - (10*g^4*log(e)^2 + g^4*log(e))*a*b^4*d^5)*
B^2*x^4 + 2*((4*g^4*log(e) + g^4)*b^5*c^2*d^3 - 2*(10*g^4*log(e) + g^4)*a*b^4*c*d^4 + (60*g^4*log(e)^2 + 16*g^
4*log(e) + g^4)*a^2*b^3*d^5)*B^2*x^3 - ((12*g^4*log(e) + 7*g^4)*b^5*c^3*d^2 - 3*(20*g^4*log(e) + 9*g^4)*a*b^4*
c^2*d^3 + 3*(40*g^4*log(e) + 11*g^4)*a^2*b^3*c*d^4 - (120*g^4*log(e)^2 + 72*g^4*log(e) + 13*g^4)*a^3*b^2*d^5)*
B^2*x^2 + 2*((12*g^4*log(e) + 13*g^4)*b^5*c^4*d - (60*g^4*log(e) + 59*g^4)*a*b^4*c^3*d^2 + 6*(20*g^4*log(e) +
17*g^4)*a^2*b^3*c^2*d^3 - (120*g^4*log(e) + 79*g^4)*a^3*b^2*c*d^4 + (30*g^4*log(e)^2 + 48*g^4*log(e) + 23*g^4)
*a^4*b*d^5)*B^2*x + 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*a*b^4*d^5*g^4*x^4 + 10*B^2*a^2*b^3*d^5*g^4*x^3 + 10*B^2*a^
3*b^2*d^5*g^4*x^2 + 5*B^2*a^4*b*d^5*g^4*x + B^2*a^5*d^5*g^4)*log(b*x + a)^2 + 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*
a*b^4*d^5*g^4*x^4 + 10*B^2*a^2*b^3*d^5*g^4*x^3 + 10*B^2*a^3*b^2*d^5*g^4*x^2 + 5*B^2*a^4*b*d^5*g^4*x + (b^5*c^5
*g^4 - 5*a*b^4*c^4*d*g^4 + 10*a^2*b^3*c^3*d^2*g^4 - 10*a^3*b^2*c^2*d^3*g^4 + 5*a^4*b*c*d^4*g^4)*B^2)*log(d*x +
c)^2 + 2*(12*B^2*b^5*d^5*g^4*x^5*log(e) - 3*(b^5*c*d^4*g^4 - (20*g^4*log(e) + g^4)*a*b^4*d^5)*B^2*x^4 + 4*(b^
5*c^2*d^3*g^4 - 5*a*b^4*c*d^4*g^4 + 2*(15*g^4*log(e) + 2*g^4)*a^2*b^3*d^5)*B^2*x^3 - 6*(b^5*c^3*d^2*g^4 - 5*a*
b^4*c^2*d^3*g^4 + 10*a^2*b^3*c*d^4*g^4 - 2*(10*g^4*log(e) + 3*g^4)*a^3*b^2*d^5)*B^2*x^2 + 12*(b^5*c^4*d*g^4 -
5*a*b^4*c^3*d^2*g^4 + 10*a^2*b^3*c^2*d^3*g^4 - 10*a^3*b^2*c*d^4*g^4 + (5*g^4*log(e) + 4*g^4)*a^4*b*d^5)*B^2*x
+ (12*a*b^4*c^4*d*g^4 - 54*a^2*b^3*c^3*d^2*g^4 + 94*a^3*b^2*c^2*d^3*g^4 - 77*a^4*b*c*d^4*g^4 + (12*g^4*log(e)
+ 25*g^4)*a^5*d^5)*B^2)*log(b*x + a) - 2*(12*B^2*b^5*d^5*g^4*x^5*log(e) - 3*(b^5*c*d^4*g^4 - (20*g^4*log(e) +
g^4)*a*b^4*d^5)*B^2*x^4 + 4*(b^5*c^2*d^3*g^4 - 5*a*b^4*c*d^4*g^4 + 2*(15*g^4*log(e) + 2*g^4)*a^2*b^3*d^5)*B^2*
x^3 - 6*(b^5*c^3*d^2*g^4 - 5*a*b^4*c^2*d^3*g^4 + 10*a^2*b^3*c*d^4*g^4 - 2*(10*g^4*log(e) + 3*g^4)*a^3*b^2*d^5)
*B^2*x^2 + 12*(b^5*c^4*d*g^4 - 5*a*b^4*c^3*d^2*g^4 + 10*a^2*b^3*c^2*d^3*g^4 - 10*a^3*b^2*c*d^4*g^4 + (5*g^4*lo
g(e) + 4*g^4)*a^4*b*d^5)*B^2*x + 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*a*b^4*d^5*g^4*x^4 + 10*B^2*a^2*b^3*d^5*g^4*x^
3 + 10*B^2*a^3*b^2*d^5*g^4*x^2 + 5*B^2*a^4*b*d^5*g^4*x + B^2*a^5*d^5*g^4)*log(b*x + a))*log(d*x + c))/(b*d^5)

Giac [F]

\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \]

[In]

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="giac")

[Out]

integrate((b*g*x + a*g)^4*(B*log((b*x + a)*e/(d*x + c)) + A)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^4\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \]

[In]

int((a*g + b*g*x)^4*(A + B*log((e*(a + b*x))/(c + d*x)))^2,x)

[Out]

int((a*g + b*g*x)^4*(A + B*log((e*(a + b*x))/(c + d*x)))^2, x)