\(\int (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [130]

Optimal result

Integrand size = 33, antiderivative size = 156 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d)^3 i^3 n x}{4 b^3}-\frac {B (b c-a d)^2 i^3 n (c+d x)^2}{8 b^2 d}-\frac {B (b c-a d) i^3 n (c+d x)^3}{12 b d}-\frac {B (b c-a d)^4 i^3 n \log (a+b x)}{4 b^4 d}+\frac {i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d} \] Output:

-1/4*B*(-a*d+b*c)^3*i^3*n*x/b^3-1/8*B*(-a*d+b*c)^2*i^3*n*(d*x+c)^2/b^2/d-1 
/12*B*(-a*d+b*c)*i^3*n*(d*x+c)^3/b/d-1/4*B*(-a*d+b*c)^4*i^3*n*ln(b*x+a)/b^ 
4/d+1/4*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.79 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {i^3 \left (-\frac {B (b c-a d) n \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )}{6 b^4}+(c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{4 d} \] Input:

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

Output:

(i^3*(-1/6*(B*(b*c - a*d)*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c 
+ d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]))/b^4 + (c + d 
*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(4*d)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85, 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.

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

\(\Big \downarrow \) 2947

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/4*(B*(b*c - a*d)*i^3*n*((d*(b*c - a*d)^2*x)/b^3 + ((b*c - a*d)*(c + d*x 
)^2)/(2*b^2) + (c + d*x)^3/(3*b) + ((b*c - a*d)^3*Log[a + b*x])/b^4))/d + 
(i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d)
 

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. \(651\) vs. \(2(146)=292\).

Time = 4.89 (sec) , antiderivative size = 652, normalized size of antiderivative = 4.18

method	result	size
parallelrisch	\(\frac {24 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c \,d^{3} i^{3} n +6 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} d^{4} i^{3} n +2 B \,x^{3} a \,b^{3} d^{4} i^{3} n^{2}-2 B \,x^{3} b^{4} c \,d^{3} i^{3} n^{2}+24 A \,x^{3} b^{4} c \,d^{3} i^{3} n -3 B \,x^{2} a^{2} b^{2} d^{4} i^{3} n^{2}-9 B \,x^{2} b^{4} c^{2} d^{2} i^{3} n^{2}+36 A \,x^{2} b^{4} c^{2} d^{2} i^{3} n +6 B x \,a^{3} b \,d^{4} i^{3} n^{2}-18 B x \,b^{4} c^{3} d \,i^{3} n^{2}+24 A x \,b^{4} c^{3} d \,i^{3} n -36 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} i^{3} n^{2}+24 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d \,i^{3} n^{2}-6 B \,a^{4} d^{4} i^{3} n^{2}+18 B \,b^{4} c^{4} i^{3} n^{2}-24 A \,b^{4} c^{4} i^{3} n +21 B \,a^{3} b c \,d^{3} i^{3} n^{2}-24 B \,a^{2} b^{2} c^{2} d^{2} i^{3} n^{2}-9 B a \,b^{3} c^{3} d \,i^{3} n^{2}-60 A a \,b^{3} c^{3} d \,i^{3} n +6 A \,x^{4} b^{4} d^{4} i^{3} n +6 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{4} i^{3} n -6 B \ln \left (b x +a \right ) a^{4} d^{4} i^{3} n^{2}-6 B \ln \left (b x +a \right ) b^{4} c^{4} i^{3} n^{2}+36 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{2} d^{2} i^{3} n +12 B \,x^{2} a \,b^{3} c \,d^{3} i^{3} n^{2}+24 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{3} d \,i^{3} n -24 B x \,a^{2} b^{2} c \,d^{3} i^{3} n^{2}+36 B x a \,b^{3} c^{2} d^{2} i^{3} n^{2}+24 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} i^{3} n^{2}}{24 b^{4} d n}\)	\(652\)

Input:

int((d*i*x+c*i)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x,method=_RETURNVERBOSE)
 

Output:

1/24*(24*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c*d^3*i^3*n+6*B*x^4*ln(e*((b* 
x+a)/(d*x+c))^n)*b^4*d^4*i^3*n+2*B*x^3*a*b^3*d^4*i^3*n^2-2*B*x^3*b^4*c*d^3 
*i^3*n^2+24*A*x^3*b^4*c*d^3*i^3*n-3*B*x^2*a^2*b^2*d^4*i^3*n^2-9*B*x^2*b^4* 
c^2*d^2*i^3*n^2+36*A*x^2*b^4*c^2*d^2*i^3*n+6*B*x*a^3*b*d^4*i^3*n^2-18*B*x* 
b^4*c^3*d*i^3*n^2+24*A*x*b^4*c^3*d*i^3*n-36*B*ln(b*x+a)*a^2*b^2*c^2*d^2*i^ 
3*n^2+24*B*ln(b*x+a)*a*b^3*c^3*d*i^3*n^2-6*B*a^4*d^4*i^3*n^2+18*B*b^4*c^4* 
i^3*n^2-24*A*b^4*c^4*i^3*n+21*B*a^3*b*c*d^3*i^3*n^2-24*B*a^2*b^2*c^2*d^2*i 
^3*n^2-9*B*a*b^3*c^3*d*i^3*n^2-60*A*a*b^3*c^3*d*i^3*n+6*A*x^4*b^4*d^4*i^3* 
n+6*B*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^4*i^3*n-6*B*ln(b*x+a)*a^4*d^4*i^3*n^ 
2-6*B*ln(b*x+a)*b^4*c^4*i^3*n^2+36*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^2 
*d^2*i^3*n+12*B*x^2*a*b^3*c*d^3*i^3*n^2+24*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b 
^4*c^3*d*i^3*n-24*B*x*a^2*b^2*c*d^3*i^3*n^2+36*B*x*a*b^3*c^2*d^2*i^3*n^2+2 
4*B*ln(b*x+a)*a^3*b*c*d^3*i^3*n^2)/b^4/d/n
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (146) = 292\).

Time = 0.11 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.75 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {6 \, A b^{4} d^{4} i^{3} x^{4} - 6 \, B b^{4} c^{4} i^{3} n \log \left (d x + c\right ) + 6 \, {\left (4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} i^{3} n \log \left (b x + a\right ) + 2 \, {\left (12 \, A b^{4} c d^{3} i^{3} - {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i^{3} n\right )} x^{3} + 3 \, {\left (12 \, A b^{4} c^{2} d^{2} i^{3} - {\left (3 \, B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} i^{3} n\right )} x^{2} + 6 \, {\left (4 \, A b^{4} c^{3} d i^{3} - {\left (3 \, B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 4 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} i^{3} n\right )} x + 6 \, {\left (B b^{4} d^{4} i^{3} x^{4} + 4 \, B b^{4} c d^{3} i^{3} x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} x^{2} + 4 \, B b^{4} c^{3} d i^{3} x\right )} \log \left (e\right ) + 6 \, {\left (B b^{4} d^{4} i^{3} n x^{4} + 4 \, B b^{4} c d^{3} i^{3} n x^{3} + 6 \, B b^{4} c^{2} d^{2} i^{3} n x^{2} + 4 \, B b^{4} c^{3} d i^{3} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{24 \, b^{4} d} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fri 
cas")
 

Output:

1/24*(6*A*b^4*d^4*i^3*x^4 - 6*B*b^4*c^4*i^3*n*log(d*x + c) + 6*(4*B*a*b^3* 
c^3*d - 6*B*a^2*b^2*c^2*d^2 + 4*B*a^3*b*c*d^3 - B*a^4*d^4)*i^3*n*log(b*x + 
 a) + 2*(12*A*b^4*c*d^3*i^3 - (B*b^4*c*d^3 - B*a*b^3*d^4)*i^3*n)*x^3 + 3*( 
12*A*b^4*c^2*d^2*i^3 - (3*B*b^4*c^2*d^2 - 4*B*a*b^3*c*d^3 + B*a^2*b^2*d^4) 
*i^3*n)*x^2 + 6*(4*A*b^4*c^3*d*i^3 - (3*B*b^4*c^3*d - 6*B*a*b^3*c^2*d^2 + 
4*B*a^2*b^2*c*d^3 - B*a^3*b*d^4)*i^3*n)*x + 6*(B*b^4*d^4*i^3*x^4 + 4*B*b^4 
*c*d^3*i^3*x^3 + 6*B*b^4*c^2*d^2*i^3*x^2 + 4*B*b^4*c^3*d*i^3*x)*log(e) + 6 
*(B*b^4*d^4*i^3*n*x^4 + 4*B*b^4*c*d^3*i^3*n*x^3 + 6*B*b^4*c^2*d^2*i^3*n*x^ 
2 + 4*B*b^4*c^3*d*i^3*n*x)*log((b*x + a)/(d*x + c)))/(b^4*d)
 

Sympy [F(-1)]

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

integrate((d*i*x+c*i)**3*(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. 479 vs. \(2 (146) = 292\).

Time = 0.06 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.07 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {1}{4} \, B d^{3} i^{3} x^{4} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{4} \, A d^{3} i^{3} x^{4} + B c d^{2} i^{3} x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c d^{2} i^{3} x^{3} + \frac {3}{2} \, B c^{2} d i^{3} x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {3}{2} \, A c^{2} d i^{3} x^{2} - \frac {1}{24} \, B d^{3} i^{3} n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + \frac {1}{2} \, B c d^{2} i^{3} n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - \frac {3}{2} \, B c^{2} d i^{3} n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + B c^{3} i^{3} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B c^{3} i^{3} x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c^{3} i^{3} x \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="max 
ima")
 

Output:

1/4*B*d^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*d^3*i^3*x 
^4 + B*c*d^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*c*d^2*i^3* 
x^3 + 3/2*B*c^2*d*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A*c 
^2*d*i^3*x^2 - 1/24*B*d^3*i^3*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)) + 1/2*B*c*d^2*i^3*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)) - 3/2*B*c^2*d*i^3*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*c^3*i^3*n*(a*log(b*x + a)/b - c*lo 
g(d*x + c)/d) + B*c^3*i^3*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*c^3 
*i^3*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1402 vs. \(2 (146) = 292\).

Time = 0.60 (sec) , antiderivative size = 1402, normalized size of antiderivative = 8.99 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="gia 
c")
 

Output:

1/24*(6*(B*b^5*c^5*i^3*n - 5*B*a*b^4*c^4*d*i^3*n + 10*B*a^2*b^3*c^3*d^2*i^ 
3*n - 10*B*a^3*b^2*c^2*d^3*i^3*n + 5*B*a^4*b*c*d^4*i^3*n - B*a^5*d^5*i^3*n 
)*log((b*x + a)/(d*x + c))/(b^4*d - 4*(b*x + a)*b^3*d^2/(d*x + c) + 6*(b*x 
 + a)^2*b^2*d^3/(d*x + c)^2 - 4*(b*x + a)^3*b*d^4/(d*x + c)^3 + (b*x + a)^ 
4*d^5/(d*x + c)^4) - (11*B*b^8*c^5*i^3*n - 55*B*a*b^7*c^4*d*i^3*n - 26*(b* 
x + a)*B*b^7*c^5*d*i^3*n/(d*x + c) + 110*B*a^2*b^6*c^3*d^2*i^3*n + 130*(b* 
x + a)*B*a*b^6*c^4*d^2*i^3*n/(d*x + c) + 21*(b*x + a)^2*B*b^6*c^5*d^2*i^3* 
n/(d*x + c)^2 - 110*B*a^3*b^5*c^2*d^3*i^3*n - 260*(b*x + a)*B*a^2*b^5*c^3* 
d^3*i^3*n/(d*x + c) - 105*(b*x + a)^2*B*a*b^5*c^4*d^3*i^3*n/(d*x + c)^2 - 
6*(b*x + a)^3*B*b^5*c^5*d^3*i^3*n/(d*x + c)^3 + 55*B*a^4*b^4*c*d^4*i^3*n + 
 260*(b*x + a)*B*a^3*b^4*c^2*d^4*i^3*n/(d*x + c) + 210*(b*x + a)^2*B*a^2*b 
^4*c^3*d^4*i^3*n/(d*x + c)^2 + 30*(b*x + a)^3*B*a*b^4*c^4*d^4*i^3*n/(d*x + 
 c)^3 - 11*B*a^5*b^3*d^5*i^3*n - 130*(b*x + a)*B*a^4*b^3*c*d^5*i^3*n/(d*x 
+ c) - 210*(b*x + a)^2*B*a^3*b^3*c^2*d^5*i^3*n/(d*x + c)^2 - 60*(b*x + a)^ 
3*B*a^2*b^3*c^3*d^5*i^3*n/(d*x + c)^3 + 26*(b*x + a)*B*a^5*b^2*d^6*i^3*n/( 
d*x + c) + 105*(b*x + a)^2*B*a^4*b^2*c*d^6*i^3*n/(d*x + c)^2 + 60*(b*x + a 
)^3*B*a^3*b^2*c^2*d^6*i^3*n/(d*x + c)^3 - 21*(b*x + a)^2*B*a^5*b*d^7*i^3*n 
/(d*x + c)^2 - 30*(b*x + a)^3*B*a^4*b*c*d^7*i^3*n/(d*x + c)^3 + 6*(b*x + a 
)^3*B*a^5*d^8*i^3*n/(d*x + c)^3 - 6*B*b^8*c^5*i^3*log(e) + 30*B*a*b^7*c^4* 
d*i^3*log(e) - 60*B*a^2*b^6*c^3*d^2*i^3*log(e) + 60*B*a^3*b^5*c^2*d^3*i...
 

Mupad [B] (verification not implemented)

Time = 26.27 (sec) , antiderivative size = 588, normalized size of antiderivative = 3.77 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=x^3\,\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{12\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{12\,b}\right )-x^2\,\left (\frac {\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {c\,d\,i^3\,\left (4\,A\,a\,d+6\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{2\,b}+\frac {A\,a\,c\,d^2\,i^3}{2\,b}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,c^3\,i^3\,x+\frac {3\,B\,c^2\,d\,i^3\,x^2}{2}+B\,c\,d^2\,i^3\,x^3+\frac {B\,d^3\,i^3\,x^4}{4}\right )+x\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {c\,d\,i^3\,\left (4\,A\,a\,d+6\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{b}+\frac {A\,a\,c\,d^2\,i^3}{b}\right )}{4\,b\,d}+\frac {c^2\,i^3\,\left (12\,A\,a\,d+8\,A\,b\,c+3\,B\,a\,d\,n-3\,B\,b\,c\,n\right )}{2\,b}-\frac {a\,c\,\left (\frac {d^2\,i^3\,\left (4\,A\,a\,d+16\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4\,b}-\frac {A\,d^2\,i^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b}\right )}{b\,d}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,n\,a^4\,d^3\,i^3-4\,B\,n\,a^3\,b\,c\,d^2\,i^3+6\,B\,n\,a^2\,b^2\,c^2\,d\,i^3-4\,B\,n\,a\,b^3\,c^3\,i^3\right )}{4\,b^4}+\frac {A\,d^3\,i^3\,x^4}{4}-\frac {B\,c^4\,i^3\,n\,\ln \left (c+d\,x\right )}{4\,d} \] Input:

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

Output:

x^3*((d^2*i^3*(4*A*a*d + 16*A*b*c + B*a*d*n - B*b*c*n))/(12*b) - (A*d^2*i^ 
3*(4*a*d + 4*b*c))/(12*b)) - x^2*((((d^2*i^3*(4*A*a*d + 16*A*b*c + B*a*d*n 
 - B*b*c*n))/(4*b) - (A*d^2*i^3*(4*a*d + 4*b*c))/(4*b))*(4*a*d + 4*b*c))/( 
8*b*d) - (c*d*i^3*(4*A*a*d + 6*A*b*c + B*a*d*n - B*b*c*n))/(2*b) + (A*a*c* 
d^2*i^3)/(2*b)) + log(e*((a + b*x)/(c + d*x))^n)*((B*d^3*i^3*x^4)/4 + B*c^ 
3*i^3*x + (3*B*c^2*d*i^3*x^2)/2 + B*c*d^2*i^3*x^3) + x*(((4*a*d + 4*b*c)*( 
(((d^2*i^3*(4*A*a*d + 16*A*b*c + B*a*d*n - B*b*c*n))/(4*b) - (A*d^2*i^3*(4 
*a*d + 4*b*c))/(4*b))*(4*a*d + 4*b*c))/(4*b*d) - (c*d*i^3*(4*A*a*d + 6*A*b 
*c + B*a*d*n - B*b*c*n))/b + (A*a*c*d^2*i^3)/b))/(4*b*d) + (c^2*i^3*(12*A* 
a*d + 8*A*b*c + 3*B*a*d*n - 3*B*b*c*n))/(2*b) - (a*c*((d^2*i^3*(4*A*a*d + 
16*A*b*c + B*a*d*n - B*b*c*n))/(4*b) - (A*d^2*i^3*(4*a*d + 4*b*c))/(4*b))) 
/(b*d)) - (log(a + b*x)*(B*a^4*d^3*i^3*n - 4*B*a*b^3*c^3*i^3*n - 4*B*a^3*b 
*c*d^2*i^3*n + 6*B*a^2*b^2*c^2*d*i^3*n))/(4*b^4) + (A*d^3*i^3*x^4)/4 - (B* 
c^4*i^3*n*log(c + d*x))/(4*d)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.25 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {i \left (-24 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{3} b c \,d^{3}+36 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{2} b^{2} c^{2} d^{2}-24 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a \,b^{3} c^{3} d -24 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} c^{3} d x -36 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} c^{2} d^{2} x^{2}+6 \,\mathrm {log}\left (d x +c \right ) a^{4} d^{4} n +6 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{4} n -6 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} d^{4} x^{4}-6 a \,b^{3} d^{4} x^{4}+6 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{4} d^{4}-24 \,\mathrm {log}\left (d x +c \right ) a^{3} b c \,d^{3} n +36 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c^{2} d^{2} n -24 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c^{3} d n +24 a^{2} b^{2} c \,d^{3} n x -36 a \,b^{3} c^{2} d^{2} n x -12 a \,b^{3} c \,d^{3} n \,x^{2}-6 a^{3} b \,d^{4} n x +3 a^{2} b^{2} d^{4} n \,x^{2}-2 a \,b^{3} d^{4} n \,x^{3}+18 b^{4} c^{3} d n x +9 b^{4} c^{2} d^{2} n \,x^{2}+2 b^{4} c \,d^{3} n \,x^{3}-24 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} c \,d^{3} x^{3}-24 a \,b^{3} c^{3} d x -36 a \,b^{3} c^{2} d^{2} x^{2}-24 a \,b^{3} c \,d^{3} x^{3}\right )}{24 b^{3} d} \] Input:

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

Output:

(i*(6*log(c + d*x)*a**4*d**4*n - 24*log(c + d*x)*a**3*b*c*d**3*n + 36*log( 
c + d*x)*a**2*b**2*c**2*d**2*n - 24*log(c + d*x)*a*b**3*c**3*d*n + 6*log(c 
 + d*x)*b**4*c**4*n + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*d**4 - 24* 
log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b*c*d**3 + 36*log(((a + b*x)**n*e) 
/(c + d*x)**n)*a**2*b**2*c**2*d**2 - 24*log(((a + b*x)**n*e)/(c + d*x)**n) 
*a*b**3*c**3*d - 24*log(((a + b*x)**n*e)/(c + d*x)**n)*b**4*c**3*d*x - 36* 
log(((a + b*x)**n*e)/(c + d*x)**n)*b**4*c**2*d**2*x**2 - 24*log(((a + b*x) 
**n*e)/(c + d*x)**n)*b**4*c*d**3*x**3 - 6*log(((a + b*x)**n*e)/(c + d*x)** 
n)*b**4*d**4*x**4 - 6*a**3*b*d**4*n*x + 24*a**2*b**2*c*d**3*n*x + 3*a**2*b 
**2*d**4*n*x**2 - 24*a*b**3*c**3*d*x - 36*a*b**3*c**2*d**2*n*x - 36*a*b**3 
*c**2*d**2*x**2 - 12*a*b**3*c*d**3*n*x**2 - 24*a*b**3*c*d**3*x**3 - 2*a*b* 
*3*d**4*n*x**3 - 6*a*b**3*d**4*x**4 + 18*b**4*c**3*d*n*x + 9*b**4*c**2*d** 
2*n*x**2 + 2*b**4*c*d**3*n*x**3))/(24*b**3*d)