\(\int (a g+b g x)^4 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [1]

Optimal result

Integrand size = 33, antiderivative size = 188 \[ \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {B (b c-a d)^4 g^4 n x}{5 d^4}-\frac {B (b c-a d)^3 g^4 n (a+b x)^2}{10 b d^3}+\frac {B (b c-a d)^2 g^4 n (a+b x)^3}{15 b d^2}-\frac {B (b c-a d) g^4 n (a+b x)^4}{20 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b}-\frac {B (b c-a d)^5 g^4 n \log (c+d x)}{5 b d^5} \] Output:

1/5*B*(-a*d+b*c)^4*g^4*n*x/d^4-1/10*B*(-a*d+b*c)^3*g^4*n*(b*x+a)^2/b/d^3+1 
/15*B*(-a*d+b*c)^2*g^4*n*(b*x+a)^3/b/d^2-1/20*B*(-a*d+b*c)*g^4*n*(b*x+a)^4 
/b/d+1/5*g^4*(b*x+a)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b-1/5*B*(-a*d+b*c)^ 
5*g^4*n*ln(d*x+c)/b/d^5
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.78 \[ \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {B (b c-a d) n \left (-12 b d (b c-a d)^3 x+6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (-b c+a d) (a+b x)^3+3 d^4 (a+b x)^4+12 (b c-a d)^4 \log (c+d x)\right )}{12 d^5}\right )}{5 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2947, 27, 49, 2009}

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.

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

\(\Big \downarrow \) 2947

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2947

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + 
 B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 1))), x] - Simp[B*n*((b*c - a*d) 
/(g*(m + 1)))   Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; Free 
Q[{a, b, c, d, e, f, g, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] 
&& NeQ[m, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1003\) vs. \(2(176)=352\).

Time = 11.68 (sec) , antiderivative size = 1004, normalized size of antiderivative = 5.34

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1004\)

Input:

int((b*g*x+a*g)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x,method=_RETURNVERBOSE)
 

Output:

1/60*(60*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a*b^4*d^5*g^4*n+36*B*a^4*b*c*d^4* 
g^4*n^2+60*B*a^3*b^2*c^2*d^3*g^4*n^2-90*B*a^2*b^3*c^3*d^2*g^4*n^2+54*B*a*b 
^4*c^4*d*g^4*n^2-180*A*a^4*b*c*d^4*g^4*n+12*B*x^5*ln(e*((b*x+a)/(d*x+c))^n 
)*b^5*d^5*g^4*n+3*B*x^4*a*b^4*d^5*g^4*n^2-3*B*x^4*b^5*c*d^4*g^4*n^2+60*A*x 
^4*a*b^4*d^5*g^4*n+16*B*x^3*a^2*b^3*d^5*g^4*n^2+12*A*x^5*b^5*d^5*g^4*n+12* 
B*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c^5*g^4*n+12*B*ln(b*x+a)*a^5*d^5*g^4*n^2-1 
2*B*ln(b*x+a)*b^5*c^5*g^4*n^2+4*B*x^3*b^5*c^2*d^3*g^4*n^2+120*A*x^3*a^2*b^ 
3*d^5*g^4*n+36*B*x^2*a^3*b^2*d^5*g^4*n^2-6*B*x^2*b^5*c^3*d^2*g^4*n^2+120*A 
*x^2*a^3*b^2*d^5*g^4*n+48*B*x*a^4*b*d^5*g^4*n^2+12*B*x*b^5*c^4*d*g^4*n^2+6 
0*A*x*a^4*b*d^5*g^4*n+120*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*d^5*g^4* 
n-20*B*x^3*a*b^4*c*d^4*g^4*n^2+120*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2 
*d^5*g^4*n-60*B*x^2*a^2*b^3*c*d^4*g^4*n^2+30*B*x^2*a*b^4*c^2*d^3*g^4*n^2+6 
0*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b*d^5*g^4*n-120*B*x*a^3*b^2*c*d^4*g^4* 
n^2+120*B*x*a^2*b^3*c^2*d^3*g^4*n^2-48*B*a^5*d^5*g^4*n^2-12*B*b^5*c^5*g^4* 
n^2-60*A*a^5*d^5*g^4*n-60*B*x*a*b^4*c^3*d^2*g^4*n^2+60*B*ln(e*((b*x+a)/(d* 
x+c))^n)*a^4*b*c*d^4*g^4*n-120*B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^2*d^3 
*g^4*n+120*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*c^3*d^2*g^4*n-60*B*ln(e*((b 
*x+a)/(d*x+c))^n)*a*b^4*c^4*d*g^4*n-60*B*ln(b*x+a)*a^4*b*c*d^4*g^4*n^2+120 
*B*ln(b*x+a)*a^3*b^2*c^2*d^3*g^4*n^2-120*B*ln(b*x+a)*a^2*b^3*c^3*d^2*g^4*n 
^2+60*B*ln(b*x+a)*a*b^4*c^4*d*g^4*n^2)/d^5/n/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (176) = 352\).

Time = 0.15 (sec) , antiderivative size = 569, normalized size of antiderivative = 3.03 \[ \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {12 \, A b^{5} d^{5} g^{4} x^{5} + 12 \, B a^{5} d^{5} g^{4} n \log \left (b x + a\right ) - 12 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} g^{4} n \log \left (d x + c\right ) + 3 \, {\left (20 \, A a b^{4} d^{5} g^{4} - {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{4} n\right )} x^{4} + 4 \, {\left (30 \, A a^{2} b^{3} d^{5} g^{4} + {\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + 4 \, B a^{2} b^{3} d^{5}\right )} g^{4} n\right )} x^{3} + 6 \, {\left (20 \, A a^{3} b^{2} d^{5} g^{4} - {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} - 6 \, B a^{3} b^{2} d^{5}\right )} g^{4} n\right )} x^{2} + 12 \, {\left (5 \, A a^{4} b d^{5} g^{4} + {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 10 \, B a^{3} b^{2} c d^{4} + 4 \, B a^{4} b d^{5}\right )} g^{4} n\right )} x + 12 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B a b^{4} d^{5} g^{4} x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} x^{2} + 5 \, B a^{4} b d^{5} g^{4} x\right )} \log \left (e\right ) + 12 \, {\left (B b^{5} d^{5} g^{4} n x^{5} + 5 \, B a b^{4} d^{5} g^{4} n x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} n x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} n x^{2} + 5 \, B a^{4} b d^{5} g^{4} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{60 \, b d^{5}} \] Input:

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

Output:

1/60*(12*A*b^5*d^5*g^4*x^5 + 12*B*a^5*d^5*g^4*n*log(b*x + a) - 12*(B*b^5*c 
^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2 - 10*B*a^3*b^2*c^2*d^3 + 5*B*a 
^4*b*c*d^4)*g^4*n*log(d*x + c) + 3*(20*A*a*b^4*d^5*g^4 - (B*b^5*c*d^4 - B* 
a*b^4*d^5)*g^4*n)*x^4 + 4*(30*A*a^2*b^3*d^5*g^4 + (B*b^5*c^2*d^3 - 5*B*a*b 
^4*c*d^4 + 4*B*a^2*b^3*d^5)*g^4*n)*x^3 + 6*(20*A*a^3*b^2*d^5*g^4 - (B*b^5* 
c^3*d^2 - 5*B*a*b^4*c^2*d^3 + 10*B*a^2*b^3*c*d^4 - 6*B*a^3*b^2*d^5)*g^4*n) 
*x^2 + 12*(5*A*a^4*b*d^5*g^4 + (B*b^5*c^4*d - 5*B*a*b^4*c^3*d^2 + 10*B*a^2 
*b^3*c^2*d^3 - 10*B*a^3*b^2*c*d^4 + 4*B*a^4*b*d^5)*g^4*n)*x + 12*(B*b^5*d^ 
5*g^4*x^5 + 5*B*a*b^4*d^5*g^4*x^4 + 10*B*a^2*b^3*d^5*g^4*x^3 + 10*B*a^3*b^ 
2*d^5*g^4*x^2 + 5*B*a^4*b*d^5*g^4*x)*log(e) + 12*(B*b^5*d^5*g^4*n*x^5 + 5* 
B*a*b^4*d^5*g^4*n*x^4 + 10*B*a^2*b^3*d^5*g^4*n*x^3 + 10*B*a^3*b^2*d^5*g^4* 
n*x^2 + 5*B*a^4*b*d^5*g^4*n*x)*log((b*x + a)/(d*x + c)))/(b*d^5)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 676 vs. \(2 (176) = 352\).

Time = 0.05 (sec) , antiderivative size = 676, normalized size of antiderivative = 3.60 \[ \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {1}{5} \, B b^{4} g^{4} x^{5} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{5} \, A b^{4} g^{4} x^{5} + B a b^{3} g^{4} x^{4} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A a b^{3} g^{4} x^{4} + 2 \, B a^{2} b^{2} g^{4} x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + 2 \, A a^{2} b^{2} g^{4} x^{3} + 2 \, B a^{3} b g^{4} x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + 2 \, A a^{3} b g^{4} x^{2} + \frac {1}{60} \, B b^{4} g^{4} n {\left (\frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} - \frac {1}{6} \, B a b^{3} g^{4} n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + B a^{2} b^{2} g^{4} n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - 2 \, B a^{3} b g^{4} n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + B a^{4} g^{4} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B a^{4} g^{4} x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A a^{4} g^{4} x \] Input:

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

Output:

1/5*B*b^4*g^4*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A*b^4*g^4*x 
^5 + B*a*b^3*g^4*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a*b^3*g^4* 
x^4 + 2*B*a^2*b^2*g^4*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*a^2 
*b^2*g^4*x^3 + 2*B*a^3*b*g^4*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 
2*A*a^3*b*g^4*x^2 + 1/60*B*b^4*g^4*n*(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)) - 1/6*B*a*b^3*g^4*n*(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)) + B*a^2*b^2*g^4*n*(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)) - 2*B*a^3*b*g^4*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d 
^2 + (b*c - a*d)*x/(b*d)) + B*a^4*g^4*n*(a*log(b*x + a)/b - c*log(d*x + c) 
/d) + B*a^4*g^4*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a^4*g^4*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4462 vs. \(2 (176) = 352\).

Time = 0.87 (sec) , antiderivative size = 4462, normalized size of antiderivative = 23.73 \[ \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/60*(12*(B*b^10*c^6*g^4*n - 6*B*a*b^9*c^5*d*g^4*n - 5*(b*x + a)*B*b^9*c^6 
*d*g^4*n/(d*x + c) + 15*B*a^2*b^8*c^4*d^2*g^4*n + 30*(b*x + a)*B*a*b^8*c^5 
*d^2*g^4*n/(d*x + c) + 10*(b*x + a)^2*B*b^8*c^6*d^2*g^4*n/(d*x + c)^2 - 20 
*B*a^3*b^7*c^3*d^3*g^4*n - 75*(b*x + a)*B*a^2*b^7*c^4*d^3*g^4*n/(d*x + c) 
- 60*(b*x + a)^2*B*a*b^7*c^5*d^3*g^4*n/(d*x + c)^2 - 10*(b*x + a)^3*B*b^7* 
c^6*d^3*g^4*n/(d*x + c)^3 + 15*B*a^4*b^6*c^2*d^4*g^4*n + 100*(b*x + a)*B*a 
^3*b^6*c^3*d^4*g^4*n/(d*x + c) + 150*(b*x + a)^2*B*a^2*b^6*c^4*d^4*g^4*n/( 
d*x + c)^2 + 60*(b*x + a)^3*B*a*b^6*c^5*d^4*g^4*n/(d*x + c)^3 + 5*(b*x + a 
)^4*B*b^6*c^6*d^4*g^4*n/(d*x + c)^4 - 6*B*a^5*b^5*c*d^5*g^4*n - 75*(b*x + 
a)*B*a^4*b^5*c^2*d^5*g^4*n/(d*x + c) - 200*(b*x + a)^2*B*a^3*b^5*c^3*d^5*g 
^4*n/(d*x + c)^2 - 150*(b*x + a)^3*B*a^2*b^5*c^4*d^5*g^4*n/(d*x + c)^3 - 3 
0*(b*x + a)^4*B*a*b^5*c^5*d^5*g^4*n/(d*x + c)^4 + B*a^6*b^4*d^6*g^4*n + 30 
*(b*x + a)*B*a^5*b^4*c*d^6*g^4*n/(d*x + c) + 150*(b*x + a)^2*B*a^4*b^4*c^2 
*d^6*g^4*n/(d*x + c)^2 + 200*(b*x + a)^3*B*a^3*b^4*c^3*d^6*g^4*n/(d*x + c) 
^3 + 75*(b*x + a)^4*B*a^2*b^4*c^4*d^6*g^4*n/(d*x + c)^4 - 5*(b*x + a)*B*a^ 
6*b^3*d^7*g^4*n/(d*x + c) - 60*(b*x + a)^2*B*a^5*b^3*c*d^7*g^4*n/(d*x + c) 
^2 - 150*(b*x + a)^3*B*a^4*b^3*c^2*d^7*g^4*n/(d*x + c)^3 - 100*(b*x + a)^4 
*B*a^3*b^3*c^3*d^7*g^4*n/(d*x + c)^4 + 10*(b*x + a)^2*B*a^6*b^2*d^8*g^4*n/ 
(d*x + c)^2 + 60*(b*x + a)^3*B*a^5*b^2*c*d^8*g^4*n/(d*x + c)^3 + 75*(b*x + 
 a)^4*B*a^4*b^2*c^2*d^8*g^4*n/(d*x + c)^4 - 10*(b*x + a)^3*B*a^6*b*d^9*...
 

Mupad [B] (verification not implemented)

Time = 26.16 (sec) , antiderivative size = 1046, normalized size of antiderivative = 5.56 \[ \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

x^2*(((5*a*d + 5*b*c)*((((b^3*g^4*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n) 
)/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c))/(5*b*d) - (a 
*b^2*g^4*(10*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/d + (A*a*b^3*c*g^4)/d)) 
/(10*b*d) - (a*c*((b^3*g^4*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/(5*d) 
 - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d)))/(2*b*d) + (a^2*b*g^4*(5*A*a*d + 5*A 
*b*c + B*a*d*n - B*b*c*n))/d) - x^3*((((b^3*g^4*(25*A*a*d + 5*A*b*c + B*a* 
d*n - B*b*c*n))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c) 
)/(15*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/(3*d) + 
(A*a*b^3*c*g^4)/(3*d)) + log(e*((a + b*x)/(c + d*x))^n)*((B*b^4*g^4*x^5)/5 
 + B*a^4*g^4*x + 2*B*a^3*b*g^4*x^2 + B*a*b^3*g^4*x^4 + 2*B*a^2*b^2*g^4*x^3 
) + x*((a^3*g^4*(5*A*a*d + 10*A*b*c + 2*B*a*d*n - 2*B*b*c*n))/d - ((5*a*d 
+ 5*b*c)*(((5*a*d + 5*b*c)*((((b^3*g^4*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b 
*c*n))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c))/(5*b*d) 
 - (a*b^2*g^4*(10*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/d + (A*a*b^3*c*g^4 
)/d))/(5*b*d) - (a*c*((b^3*g^4*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/( 
5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d)))/(b*d) + (2*a^2*b*g^4*(5*A*a*d + 
 5*A*b*c + B*a*d*n - B*b*c*n))/d))/(5*b*d) + (a*c*((((b^3*g^4*(25*A*a*d + 
5*A*b*c + B*a*d*n - B*b*c*n))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*( 
5*a*d + 5*b*c))/(5*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c + B*a*d*n - B*b*c 
*n))/d + (A*a*b^3*c*g^4)/d))/(b*d)) + x^4*((b^3*g^4*(25*A*a*d + 5*A*b*c...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.96 \[ \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^{4} \left (60 a^{5} d^{5} x +12 \,\mathrm {log}\left (d x +c \right ) a^{5} d^{5} n -12 \,\mathrm {log}\left (d x +c \right ) b^{5} c^{5} n +12 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{5} d^{5} x^{5}+120 a^{4} b \,d^{5} x^{2}+120 a^{3} b^{2} d^{5} x^{3}+60 a^{2} b^{3} d^{5} x^{4}+12 a \,b^{4} d^{5} x^{5}+60 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{4} b \,d^{5} x +120 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{3} b^{2} d^{5} x^{2}+120 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{2} b^{3} d^{5} x^{3}+60 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a \,b^{4} d^{5} x^{4}+48 a^{4} b \,d^{5} n x +36 a^{3} b^{2} d^{5} n \,x^{2}+16 a^{2} b^{3} d^{5} n \,x^{3}+12 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{5} d^{5}-60 \,\mathrm {log}\left (d x +c \right ) a^{4} b c \,d^{4} n +120 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c^{2} d^{3} n -120 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c^{3} d^{2} n +60 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c^{4} d n -120 a^{3} b^{2} c \,d^{4} n x +120 a^{2} b^{3} c^{2} d^{3} n x -60 a^{2} b^{3} c \,d^{4} n \,x^{2}-60 a \,b^{4} c^{3} d^{2} n x +30 a \,b^{4} c^{2} d^{3} n \,x^{2}-20 a \,b^{4} c \,d^{4} n \,x^{3}+3 a \,b^{4} d^{5} n \,x^{4}+12 b^{5} c^{4} d n x -6 b^{5} c^{3} d^{2} n \,x^{2}+4 b^{5} c^{2} d^{3} n \,x^{3}-3 b^{5} c \,d^{4} n \,x^{4}\right )}{60 d^{5}} \] Input:

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

Output:

(g**4*(12*log(c + d*x)*a**5*d**5*n - 60*log(c + d*x)*a**4*b*c*d**4*n + 120 
*log(c + d*x)*a**3*b**2*c**2*d**3*n - 120*log(c + d*x)*a**2*b**3*c**3*d**2 
*n + 60*log(c + d*x)*a*b**4*c**4*d*n - 12*log(c + d*x)*b**5*c**5*n + 12*lo 
g(((a + b*x)**n*e)/(c + d*x)**n)*a**5*d**5 + 60*log(((a + b*x)**n*e)/(c + 
d*x)**n)*a**4*b*d**5*x + 120*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**2* 
d**5*x**2 + 120*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**3*d**5*x**3 + 6 
0*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**4*d**5*x**4 + 12*log(((a + b*x)* 
*n*e)/(c + d*x)**n)*b**5*d**5*x**5 + 60*a**5*d**5*x + 48*a**4*b*d**5*n*x + 
 120*a**4*b*d**5*x**2 - 120*a**3*b**2*c*d**4*n*x + 36*a**3*b**2*d**5*n*x** 
2 + 120*a**3*b**2*d**5*x**3 + 120*a**2*b**3*c**2*d**3*n*x - 60*a**2*b**3*c 
*d**4*n*x**2 + 16*a**2*b**3*d**5*n*x**3 + 60*a**2*b**3*d**5*x**4 - 60*a*b* 
*4*c**3*d**2*n*x + 30*a*b**4*c**2*d**3*n*x**2 - 20*a*b**4*c*d**4*n*x**3 + 
3*a*b**4*d**5*n*x**4 + 12*a*b**4*d**5*x**5 + 12*b**5*c**4*d*n*x - 6*b**5*c 
**3*d**2*n*x**2 + 4*b**5*c**2*d**3*n*x**3 - 3*b**5*c*d**4*n*x**4))/(60*d** 
5)