\(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^3} \, dx\) [26]

Optimal result

Integrand size = 40, antiderivative size = 345 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {2 B d (b c-a d) i^3 (c+d x)}{b^3 g^3 (a+b x)}-\frac {B (b c-a d) i^3 (c+d x)^2}{4 b^2 g^3 (a+b x)^2}+\frac {d^3 i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^3}-\frac {2 d (b c-a d) i^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}-\frac {(b c-a d) i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^3 (a+b x)^2}-\frac {B d^2 (b c-a d) i^3 \log (c+d x)}{b^4 g^3}-\frac {3 d^2 (b c-a d) i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^3}+\frac {3 B d^2 (b c-a d) i^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^3} \] Output:

-2*B*d*(-a*d+b*c)*i^3*(d*x+c)/b^3/g^3/(b*x+a)-1/4*B*(-a*d+b*c)*i^3*(d*x+c) 
^2/b^2/g^3/(b*x+a)^2+d^3*i^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^4/g^3-2 
*d*(-a*d+b*c)*i^3*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^3/g^3/(b*x+a)-1/2* 
(-a*d+b*c)*i^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/b^2/g^3/(b*x+a)^2-B*d 
^2*(-a*d+b*c)*i^3*ln(d*x+c)/b^4/g^3-3*d^2*(-a*d+b*c)*i^3*(A+B*ln(e*(b*x+a) 
/(d*x+c)))*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/g^3+3*B*d^2*(-a*d+b*c)*i^3*polylo 
g(2,b*(d*x+c)/d/(b*x+a))/b^4/g^3
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.91 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=\frac {i^3 \left (4 A b d^3 x-\frac {B (b c-a d)^3}{(a+b x)^2}-\frac {10 B d (b c-a d)^2}{a+b x}+10 B d^2 (-b c+a d) \log (a+b x)+4 B d^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-\frac {2 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}-\frac {12 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+12 d^2 (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 B d^2 (b c-a d) \log (c+d x)+6 B d^2 (-b c+a d) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{4 b^4 g^3} \] Input:

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

Output:

(i^3*(4*A*b*d^3*x - (B*(b*c - a*d)^3)/(a + b*x)^2 - (10*B*d*(b*c - a*d)^2) 
/(a + b*x) + 10*B*d^2*(-(b*c) + a*d)*Log[a + b*x] + 4*B*d^3*(a + b*x)*Log[ 
(e*(a + b*x))/(c + d*x)] - (2*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + 
d*x)]))/(a + b*x)^2 - (12*d*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d* 
x)]))/(a + b*x) + 12*d^2*(b*c - a*d)*Log[a + b*x]*(A + B*Log[(e*(a + b*x)) 
/(c + d*x)]) + 6*B*d^2*(b*c - a*d)*Log[c + d*x] + 6*B*d^2*(-(b*c) + a*d)*( 
Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog 
[2, (d*(a + b*x))/(-(b*c) + a*d)])))/(4*b^4*g^3)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2962, 2793, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2793

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[a + b*Log[c*x^n], 
 (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, 
 f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && Integer 
Q[r]))
 

rule 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(731\) vs. \(2(341)=682\).

Time = 3.22 (sec) , antiderivative size = 732, normalized size of antiderivative = 2.12

method	result	size
parts	\(\frac {i^{3} A \left (\frac {x \,d^{3}}{b^{3}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{2 b^{4} \left (b x +a \right )^{2}}-\frac {3 d^{2} \left (d a -b c \right ) \ln \left (b x +a \right )}{b^{4}}-\frac {3 d \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}{b^{4} \left (b x +a \right )}\right )}{g^{3}}-\frac {i^{3} B \left (d a -b c \right )^{4} e^{4} \left (\frac {2 \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right ) d^{6}}{\left (d a -b c \right )^{3} b^{3} e^{3}}+\frac {\left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right ) d^{8}}{\left (d a -b c \right )^{3} b^{3} e^{3}}+\frac {3 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d^{7}}{2 \left (d a -b c \right )^{3} b^{4} e^{4}}+\frac {\left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right ) d^{5}}{\left (d a -b c \right )^{3} b^{2} e^{2}}-\frac {3 \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{8}}{\left (d a -b c \right )^{3} b^{4} e^{4}}\right )}{g^{3} d^{5}}\)	\(732\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {i^{3} d^{2} e^{3} A \left (\frac {d^{2}}{b^{3} e^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {3 d^{2} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{4} e^{4}}-\frac {1}{2 b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {3 d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b^{4} e^{4}}-\frac {2 d}{b^{3} e^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}\right )}{g^{3}}+\frac {i^{3} d^{2} e^{3} B \left (-\frac {3 \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{3}}{b^{4} e^{4}}+\frac {3 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d^{2}}{2 b^{4} e^{4}}+\frac {\left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right ) d^{3}}{b^{3} e^{3}}+\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}}{b^{2} e^{2}}+\frac {2 \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right ) d}{b^{3} e^{3}}\right )}{g^{3}}\right )}{d^{2}}\)	\(779\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {i^{3} d^{2} e^{3} A \left (\frac {d^{2}}{b^{3} e^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {3 d^{2} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{4} e^{4}}-\frac {1}{2 b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {3 d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b^{4} e^{4}}-\frac {2 d}{b^{3} e^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}\right )}{g^{3}}+\frac {i^{3} d^{2} e^{3} B \left (-\frac {3 \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{3}}{b^{4} e^{4}}+\frac {3 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d^{2}}{2 b^{4} e^{4}}+\frac {\left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right ) d^{3}}{b^{3} e^{3}}+\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}}{b^{2} e^{2}}+\frac {2 \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right ) d}{b^{3} e^{3}}\right )}{g^{3}}\right )}{d^{2}}\)	\(779\)
risch	\(\text {Expression too large to display}\)	\(4194\)

Input:

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

Output:

i^3*A/g^3*(x*d^3/b^3-1/2/b^4*(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+b^3*c^3 
)/(b*x+a)^2-3/b^4*d^2*(a*d-b*c)*ln(b*x+a)-3/b^4*d*(a^2*d^2-2*a*b*c*d+b^2*c 
^2)/(b*x+a))-i^3*B/g^3/d^5*(a*d-b*c)^4*e^4*(2*(-1/(b*e/d+(a*d-b*c)*e/d/(d* 
x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c)))/(a* 
d-b*c)^3*d^6/b^3/e^3+(1/b/e*ln((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/d-ln(b 
*e/d+(a*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/b/e/((b*e/d+(a*d 
-b*c)*e/d/(d*x+c))*d-b*e))/(a*d-b*c)^3*d^8/b^3/e^3+3/2*ln(b*e/d+(a*d-b*c)* 
e/d/(d*x+c))^2/(a*d-b*c)^3*d^7/b^4/e^4+(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c)) 
^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)/(a 
*d-b*c)^3*d^5/b^2/e^2-3*(dilog(-((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/b/e) 
/d+ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b* 
e)/b/e)/d)/(a*d-b*c)^3*d^8/b^4/e^4)
 

Fricas [F]

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

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algo 
rithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2302 vs. \(2 (340) = 680\).

Time = 0.17 (sec) , antiderivative size = 2302, normalized size of antiderivative = 6.67 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3,x, algo 
rithm="maxima")
 

Output:

-3/4*B*c^2*d*i^3*(2*(2*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^4* 
g^3*x^2 + 2*a*b^3*g^3*x + a^2*b^2*g^3) + (3*a*b*c - a^2*d + 2*(2*b^2*c - a 
*b*d)*x)/((b^5*c - a*b^4*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)*g^3*x + (a^2 
*b^3*c - a^3*b^2*d)*g^3) + 2*(2*b*c*d - a*d^2)*log(b*x + a)/((b^4*c^2 - 2* 
a*b^3*c*d + a^2*b^2*d^2)*g^3) - 2*(2*b*c*d - a*d^2)*log(d*x + c)/((b^4*c^2 
 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3)) - 1/2*A*d^3*i^3*((6*a^2*b*x + 5*a^3)/( 
b^6*g^3*x^2 + 2*a*b^5*g^3*x + a^2*b^4*g^3) - 2*x/(b^3*g^3) + 6*a*log(b*x + 
 a)/(b^4*g^3)) + 3/2*A*c*d^2*i^3*((4*a*b*x + 3*a^2)/(b^5*g^3*x^2 + 2*a*b^4 
*g^3*x + a^2*b^3*g^3) + 2*log(b*x + a)/(b^3*g^3)) + 1/4*B*c^3*i^3*((2*b*d* 
x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^2*d)*g^3* 
x + (a^2*b^2*c - a^3*b*d)*g^3) - 2*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b 
^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3) + 2*d^2*log(b*x + a)/((b^3*c^2 - 2 
*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 - 2*a*b^2*c*d 
+ a^2*b*d^2)*g^3)) - 3/2*(2*b*x + a)*A*c^2*d*i^3/(b^4*g^3*x^2 + 2*a*b^3*g^ 
3*x + a^2*b^2*g^3) - 1/2*A*c^3*i^3/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^ 
3) - 1/2*(2*b^3*c^3*d^2*i^3 + 8*a*b^2*c^2*d^3*i^3 - 13*a^2*b*c*d^4*i^3 + 5 
*a^3*d^5*i^3)*B*log(d*x + c)/(b^6*c^2*g^3 - 2*a*b^5*c*d*g^3 + a^2*b^4*d^2* 
g^3) + 1/4*(4*(b^5*c^2*d^3*i^3*log(e) - 2*a*b^4*c*d^4*i^3*log(e) + a^2*b^3 
*d^5*i^3*log(e))*B*x^3 + 8*(a*b^4*c^2*d^3*i^3*log(e) - 2*a^2*b^3*c*d^4*i^3 
*log(e) + a^3*b^2*d^5*i^3*log(e))*B*x^2 + 2*(12*(i^3*log(e) + i^3)*a*b^...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(i*( - 4*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**3 + 3*a**2*b*x + 3*a* 
b**2*x**2 + b**3*x**3),x)*a**6*b**5*d**5 + 8*int((log((a*e + b*e*x)/(c + d 
*x))*x**3)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a**5*b**6*c* 
d**4 - 8*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**3 + 3*a**2*b*x + 3*a* 
b**2*x**2 + b**3*x**3),x)*a**5*b**6*d**5*x - 4*int((log((a*e + b*e*x)/(c + 
 d*x))*x**3)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a**4*b**7* 
c**2*d**3 + 16*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**3 + 3*a**2*b*x 
+ 3*a*b**2*x**2 + b**3*x**3),x)*a**4*b**7*c*d**4*x - 4*int((log((a*e + b*e 
*x)/(c + d*x))*x**3)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a* 
*4*b**7*d**5*x**2 - 8*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**3 + 3*a* 
*2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a**3*b**8*c**2*d**3*x + 8*int((log( 
(a*e + b*e*x)/(c + d*x))*x**3)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x 
**3),x)*a**3*b**8*c*d**4*x**2 - 4*int((log((a*e + b*e*x)/(c + d*x))*x**3)/ 
(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a**2*b**9*c**2*d**3*x** 
2 - 12*int((log((a*e + b*e*x)/(c + d*x))*x**2)/(a**3 + 3*a**2*b*x + 3*a*b* 
*2*x**2 + b**3*x**3),x)*a**6*b**5*c*d**4 + 24*int((log((a*e + b*e*x)/(c + 
d*x))*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a**5*b**6*c 
**2*d**3 - 24*int((log((a*e + b*e*x)/(c + d*x))*x**2)/(a**3 + 3*a**2*b*x + 
 3*a*b**2*x**2 + b**3*x**3),x)*a**5*b**6*c*d**4*x - 12*int((log((a*e + b*e 
*x)/(c + d*x))*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)...