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

3.1.1.1 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
 

3.1.1.2 Mathematica [A] (verified)

Time = 0.08 (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)
 

3.1.1.3 Rubi [A] (verified)

Time = 0.31 (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)
 

3.1.1.3.1 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]
 

3.1.1.4 Maple [B] (verified)

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

Time = 17.53 (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*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b*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*b^5*c^4*d*g^4*n^2+60*A*x*a^4 
*b*d^5*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+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^2-120*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^2-60*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+60*B*x^4*ln(e*((b*x+a)/(d*x+c))^n 
)*a*b^4*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^2-48*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+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^2-12*B*ln(b*x+a)*b^5*c^5*g^4*n^2)/d^5/n/b
 

3.1.1.5 Fricas [B] (verification not implemented)

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

Time = 0.34 (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)
 

3.1.1.6 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
 

3.1.1.7 Maxima [B] (verification not implemented)

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

Time = 0.21 (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
 

3.1.1.8 Giac [B] (verification not implemented)

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

Time = 1.06 (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*...
 

3.1.1.9 Mupad [B] (verification not implemented)

Time = 1.38 (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=x^2\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{10\,b\,d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{2\,b\,d}+\frac {a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}\right )-x^3\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{15\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{3\,d}+\frac {A\,a\,b^3\,c\,g^4}{3\,d}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,a^4\,g^4\,x+2\,B\,a^3\,b\,g^4\,x^2+2\,B\,a^2\,b^2\,g^4\,x^3+B\,a\,b^3\,g^4\,x^4+\frac {B\,b^4\,g^4\,x^5}{5}\right )+x\,\left (\frac {a^3\,g^4\,\left (5\,A\,a\,d+10\,A\,b\,c+2\,B\,a\,d\,n-2\,B\,b\,c\,n\right )}{d}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{5\,b\,d}-\frac {a\,c\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{b\,d}+\frac {2\,a^2\,b\,g^4\,\left (5\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}\right )}{5\,b\,d}+\frac {a\,c\,\left (\frac {\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,g^4\,\left (10\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b^3\,c\,g^4}{d}\right )}{b\,d}\right )+x^4\,\left (\frac {b^3\,g^4\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{20\,d}-\frac {A\,b^3\,g^4\,\left (5\,a\,d+5\,b\,c\right )}{20\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (5\,B\,n\,a^4\,c\,d^4\,g^4-10\,B\,n\,a^3\,b\,c^2\,d^3\,g^4+10\,B\,n\,a^2\,b^2\,c^3\,d^2\,g^4-5\,B\,n\,a\,b^3\,c^4\,d\,g^4+B\,n\,b^4\,c^5\,g^4\right )}{5\,d^5}+\frac {A\,b^4\,g^4\,x^5}{5}+\frac {B\,a^5\,g^4\,n\,\ln \left (a+b\,x\right )}{5\,b} \]

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...