\(\int (a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [98]

Optimal result

Integrand size = 32, antiderivative size = 309 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=-\frac {B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b d}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b}+\frac {B (b c-a d)^2 g^3 (a+b x)^2 \left (3 A+B+3 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b d^2}-\frac {B (b c-a d)^3 g^3 (a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b d^3}-\frac {B (b c-a d)^4 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+11 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b d^4}-\frac {B^2 (b c-a d)^4 g^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{2 b d^4} \] Output:

-1/6*B*(-a*d+b*c)*g^3*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/d+1/4*g^3*(b 
*x+a)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/b+1/12*B*(-a*d+b*c)^2*g^3*(b*x+a)^2* 
(3*A+B+3*B*ln(e*(b*x+a)/(d*x+c)))/b/d^2-1/12*B*(-a*d+b*c)^3*g^3*(b*x+a)*(6 
*A+5*B+6*B*ln(e*(b*x+a)/(d*x+c)))/b/d^3-1/12*B*(-a*d+b*c)^4*g^3*ln((-a*d+b 
*c)/b/(d*x+c))*(6*A+11*B+6*B*ln(e*(b*x+a)/(d*x+c)))/b/d^4-1/2*B^2*(-a*d+b* 
c)^4*g^3*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^4
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.27 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {g^3 \left ((a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-\frac {B (b c-a d) \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 (b d x+(-b c+a d) \log (c+d x))+3 B (b c-a d)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{3 d^4}\right )}{4 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2950, 2781, 2784, 2784, 2784, 2754, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2950

\(\displaystyle g^3 (b c-a d)^4 \int \frac {(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2781

\(\displaystyle g^3 (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \int \frac {(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle g^3 (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\int \frac {(a+b x)^2 \left (3 A+B+3 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 d}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle g^3 (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}}{3 d}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle g^3 (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {6 A+11 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{2 d}}{3 d}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle g^3 (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {6 B \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+11 B\right )}{d}}{d}}{2 d}}{3 d}\right )}{2 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle g^3 (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+11 B\right )}{d}-\frac {6 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}}{3 d}\right )}{2 b}\right )\)

Input:

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

Output:

(b*c - a*d)^4*g^3*(((a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4 
*b*(c + d*x)^4*(b - (d*(a + b*x))/(c + d*x))^4) - (B*(((a + b*x)^3*(A + B* 
Log[(e*(a + b*x))/(c + d*x)]))/(3*d*(c + d*x)^3*(b - (d*(a + b*x))/(c + d* 
x))^3) - (((a + b*x)^2*(3*A + B + 3*B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d* 
(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^2) - (((a + b*x)*(6*A + 5*B + 6* 
B*Log[(e*(a + b*x))/(c + d*x)]))/(d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x) 
)) - (-(((6*A + 11*B + 6*B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (d*(a + b 
*x))/(b*(c + d*x))])/d) - (6*B*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d) 
/d)/(2*d))/(3*d)))/(2*b))
 

Defintions of rubi rules used

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[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 2781

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] + Simp[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 2784

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] - Simp[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 2838

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 2950

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(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] && E 
qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [F]

\[\int \left (b g x +a g \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1732 vs. \(2 (296) = 592\).

Time = 0.14 (sec) , antiderivative size = 1732, normalized size of antiderivative = 5.61 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*A^2*b^3*g^3*x^4 + A^2*a*b^2*g^3*x^3 + 3/2*A^2*a^2*b*g^3*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^3*g^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*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*A*B*a^2*b*g^3 + (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*b^2*g^3 + 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*b^3*g^3 + A^2*a^3*g^3*x + 1/12*((6*g^3*log(e) + 11*g^3)*b^3*c^4 - 2*( 
12*g^3*log(e) + 19*g^3)*a*b^2*c^3*d + 9*(4*g^3*log(e) + 5*g^3)*a^2*b*c^2*d 
^2 - 6*(4*g^3*log(e) + 3*g^3)*a^3*c*d^3)*B^2*log(d*x + c)/d^4 + 1/2*(b^4*c 
^4*g^3 - 4*a*b^3*c^3*d*g^3 + 6*a^2*b^2*c^2*d^2*g^3 - 4*a^3*b*c*d^3*g^3 + a 
^4*d^4*g^3)*(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^4) + 1/12*(3*B^2*b^4*d^4*g^3*x^4*log(e)^2 
 - 2*(b^4*c*d^3*g^3*log(e) - (6*g^3*log(e)^2 + g^3*log(e))*a*b^3*d^4)*B^2* 
x^3 + ((3*g^3*log(e) + g^3)*b^4*c^2*d^2 - 2*(6*g^3*log(e) + g^3)*a*b^3*c*d 
^3 + (18*g^3*log(e)^2 + 9*g^3*log(e) + g^3)*a^2*b^2*d^4)*B^2*x^2 - ((6*g^3 
*log(e) + 5*g^3)*b^4*c^3*d - (24*g^3*log(e) + 17*g^3)*a*b^3*c^2*d^2 + (36* 
g^3*log(e) + 19*g^3)*a^2*b^2*c*d^3 - (12*g^3*log(e)^2 + 18*g^3*log(e) +...
 

Giac [F]

\[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(g**3*(6*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x 
**2),x)*a**4*b**2*d**5 - 24*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a* 
d*x + b*c*x + b*d*x**2),x)*a**3*b**3*c*d**4 + 36*int((log((a*e + b*e*x)/(c 
 + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b**4*c**2*d**3 - 24*i 
nt((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a* 
b**5*c**3*d**2 + 6*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c 
*x + b*d*x**2),x)*b**6*c**4*d + 6*log(c + d*x)*a**5*d**4 - 24*log(c + d*x) 
*a**4*b*c*d**3 + 11*log(c + d*x)*a**4*b*d**4 + 36*log(c + d*x)*a**3*b**2*c 
**2*d**2 - 44*log(c + d*x)*a**3*b**2*c*d**3 - 24*log(c + d*x)*a**2*b**3*c* 
*3*d + 66*log(c + d*x)*a**2*b**3*c**2*d**2 + 6*log(c + d*x)*a*b**4*c**4 - 
44*log(c + d*x)*a*b**4*c**3*d + 11*log(c + d*x)*b**5*c**4 + 9*log((a*e + b 
*e*x)/(c + d*x))**2*a**3*b**2*c*d**3 + 12*log((a*e + b*e*x)/(c + d*x))**2* 
a**3*b**2*d**4*x - 9*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**3*c**2*d**2 + 
 18*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**3*d**4*x**2 + 3*log((a*e + b*e 
*x)/(c + d*x))**2*a*b**4*c**3*d + 12*log((a*e + b*e*x)/(c + d*x))**2*a*b** 
4*d**4*x**3 + 3*log((a*e + b*e*x)/(c + d*x))**2*b**5*d**4*x**4 + 6*log((a* 
e + b*e*x)/(c + d*x))*a**5*d**4 + 24*log((a*e + b*e*x)/(c + d*x))*a**4*b*d 
**4*x + 11*log((a*e + b*e*x)/(c + d*x))*a**4*b*d**4 - 26*log((a*e + b*e*x) 
/(c + d*x))*a**3*b**2*c*d**3 + 36*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*d 
**4*x**2 + 18*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*d**4*x + 21*log((a...