\(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^2} \, dx\) [25]

Optimal result

Integrand size = 40, antiderivative size = 373 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=-\frac {B d^2 (b c-a d) i^3 x}{2 b^3 g^2}-\frac {B (b c-a d)^2 i^3 (c+d x)}{b^3 g^2 (a+b x)}-\frac {B d (b c-a d)^2 i^3 \log \left (\frac {a+b x}{c+d x}\right )}{2 b^4 g^2}+\frac {2 d^2 (b c-a d) i^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^4 g^2}-\frac {(b c-a d)^2 i^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {d i^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b^2 g^2}-\frac {5 B d (b c-a d)^2 i^3 \log (c+d x)}{2 b^4 g^2}-\frac {3 d (b c-a d)^2 i^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2}+\frac {3 B d (b c-a d)^2 i^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\frac {i^3 \left (2 A b d^2 (3 b c-2 a d) x-b B d^2 (b c-a d) x-\frac {2 B (b c-a d)^3}{a+b x}-a^2 B d^3 \log (a+b x)-2 B d (b c-a d)^2 \log (a+b x)+2 B d^2 (3 b c-2 a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+b^2 d^3 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {2 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+6 d (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+b^2 B c^2 d \log (c+d x)+2 B d (b c-a d)^2 \log (c+d x)-2 B d (-b c+a d) (-3 b c+2 a d) \log (c+d x)-3 B d (b c-a d)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{2 b^4 g^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.87, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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. \(862\) vs. \(2(365)=730\).

Time = 3.30 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.31

method	result	size
parts	\(\frac {i^{3} A \left (\frac {d^{2} \left (\frac {1}{2} b d \,x^{2}-2 x d a +3 b c x \right )}{b^{3}}+\frac {3 d \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \ln \left (b x +a \right )}{b^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{b^{4} \left (b x +a \right )}\right )}{g^{2}}-\frac {i^{3} B \left (d a -b c \right )^{4} e^{4} \left (\frac {\left (-\frac {1}{2 b e d \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 )}-\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 )}{2 b^{2} e^{2} 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 ) \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -2 b e \right )}{2 b^{2} e^{2} \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 )^{2}}\right ) d^{7}}{\left (d a -b c \right )^{2} b^{2} e^{2}}-\frac {\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^{5}}{\left (d a -b c \right )^{2} b^{3} e^{3}}-\frac {2 \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^{7}}{\left (d a -b c \right )^{2} 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^{6}}{2 \left (d a -b c \right )^{2} b^{4} e^{4}}+\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^{7}}{\left (d a -b c \right )^{2} b^{4} e^{4}}\right )}{g^{2} d^{5}}\)	\(863\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (-\frac {A \,d^{2} e^{3} i^{3} \left (d a -b c \right ) \left (-\frac {3 d \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 {2 d}{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 {d}{2 b^{2} e^{2} \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 )^{2}}-\frac {1}{b^{3} e^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {3 d \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}}\right )}{g^{2}}-\frac {B \,d^{2} e^{3} i^{3} \left (d a -b c \right ) \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^{2}}{b^{4} e^{4}}+\frac {-\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 )}}}{b^{3} e^{3}}+\frac {2 \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^{2}}{b^{3} e^{3}}-\frac {\left (\frac {1}{2 b e d \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 {\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 )}{2 b^{2} e^{2} 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 ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \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 )^{2}}\right ) d^{2}}{b^{2} e^{2}}+\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 b^{4} e^{4}}\right )}{g^{2}}\right )}{d^{2}}\)	\(937\)
default	\(-\frac {e \left (d a -b c \right ) \left (-\frac {A \,d^{2} e^{3} i^{3} \left (d a -b c \right ) \left (-\frac {3 d \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 {2 d}{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 {d}{2 b^{2} e^{2} \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 )^{2}}-\frac {1}{b^{3} e^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {3 d \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}}\right )}{g^{2}}-\frac {B \,d^{2} e^{3} i^{3} \left (d a -b c \right ) \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^{2}}{b^{4} e^{4}}+\frac {-\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 )}}}{b^{3} e^{3}}+\frac {2 \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^{2}}{b^{3} e^{3}}-\frac {\left (\frac {1}{2 b e d \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 {\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 )}{2 b^{2} e^{2} 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 ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \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 )^{2}}\right ) d^{2}}{b^{2} e^{2}}+\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 b^{4} e^{4}}\right )}{g^{2}}\right )}{d^{2}}\)	\(937\)
risch	\(\text {Expression too large to display}\)	\(4732\)

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1501 vs. \(2 (364) = 728\).

Time = 0.12 (sec) , antiderivative size = 1501, normalized size of antiderivative = 4.02 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algo 
rithm="maxima")
 

Output:

-3*A*(a^2/(b^4*g^2*x + a*b^3*g^2) - x/(b^2*g^2) + 2*a*log(b*x + a)/(b^3*g^ 
2))*c*d^2*i^3 + 1/2*(2*a^3/(b^5*g^2*x + a*b^4*g^2) + 6*a^2*log(b*x + a)/(b 
^4*g^2) + (b*x^2 - 4*a*x)/(b^3*g^2))*A*d^3*i^3 + 3*A*c^2*d*i^3*(a/(b^3*g^2 
*x + a*b^2*g^2) + log(b*x + a)/(b^2*g^2)) - B*c^3*i^3*(log(b*e*x/(d*x + c) 
 + a*e/(d*x + c))/(b^2*g^2*x + a*b*g^2) + 1/(b^2*g^2*x + a*b*g^2) + d*log( 
b*x + a)/((b^2*c - a*b*d)*g^2) - d*log(d*x + c)/((b^2*c - a*b*d)*g^2)) - A 
*c^3*i^3/(b^2*g^2*x + a*b*g^2) - 1/2*(5*b^3*c^3*d*i^3 - 3*a*b^2*c^2*d^2*i^ 
3 - 2*a^2*b*c*d^3*i^3 + 2*a^3*d^4*i^3)*B*log(d*x + c)/(b^5*c*g^2 - a*b^4*d 
*g^2) + 1/2*((b^4*c*d^3*i^3*log(e) - a*b^3*d^4*i^3*log(e))*B*x^3 + ((6*i^3 
*log(e) - i^3)*b^4*c^2*d^2 - (9*i^3*log(e) - 2*i^3)*a*b^3*c*d^3 + (3*i^3*l 
og(e) - i^3)*a^2*b^2*d^4)*B*x^2 + ((6*i^3*log(e) - i^3)*a*b^3*c^2*d^2 - 2* 
(5*i^3*log(e) - i^3)*a^2*b^2*c*d^3 + (4*i^3*log(e) - i^3)*a^3*b*d^4)*B*x + 
 3*((b^4*c^3*d*i^3 - 3*a*b^3*c^2*d^2*i^3 + 3*a^2*b^2*c*d^3*i^3 - a^3*b*d^4 
*i^3)*B*x + (a*b^3*c^3*d*i^3 - 3*a^2*b^2*c^2*d^2*i^3 + 3*a^3*b*c*d^3*i^3 - 
 a^4*d^4*i^3)*B)*log(b*x + a)^2 + 2*(3*(i^3*log(e) + i^3)*a*b^3*c^3*d - 6* 
(i^3*log(e) + i^3)*a^2*b^2*c^2*d^2 + 4*(i^3*log(e) + i^3)*a^3*b*c*d^3 - (i 
^3*log(e) + i^3)*a^4*d^4)*B + ((b^4*c*d^3*i^3 - a*b^3*d^4*i^3)*B*x^3 + 3*( 
2*b^4*c^2*d^2*i^3 - 3*a*b^3*c*d^3*i^3 + a^2*b^2*d^4*i^3)*B*x^2 + (6*b^4*c^ 
3*d*i^3*log(e) - 18*(i^3*log(e) - i^3)*a*b^3*c^2*d^2 + 9*(2*i^3*log(e) - 3 
*i^3)*a^2*b^2*c*d^3 - (6*i^3*log(e) - 11*i^3)*a^3*b*d^4)*B*x - (18*a^2*...
 

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(i*( - 2*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**2 + 2*a*b*x + b**2*x* 
*2),x)*a**3*b**5*d**4 + 2*int((log((a*e + b*e*x)/(c + d*x))*x**3)/(a**2 + 
2*a*b*x + b**2*x**2),x)*a**2*b**6*c*d**3 - 2*int((log((a*e + b*e*x)/(c + d 
*x))*x**3)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**6*d**4*x + 2*int((log(( 
a*e + b*e*x)/(c + d*x))*x**3)/(a**2 + 2*a*b*x + b**2*x**2),x)*a*b**7*c*d** 
3*x - 6*int((log((a*e + b*e*x)/(c + d*x))*x**2)/(a**2 + 2*a*b*x + b**2*x** 
2),x)*a**3*b**5*c*d**3 + 6*int((log((a*e + b*e*x)/(c + d*x))*x**2)/(a**2 + 
 2*a*b*x + b**2*x**2),x)*a**2*b**6*c**2*d**2 - 6*int((log((a*e + b*e*x)/(c 
 + d*x))*x**2)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**6*c*d**3*x + 6*int( 
(log((a*e + b*e*x)/(c + d*x))*x**2)/(a**2 + 2*a*b*x + b**2*x**2),x)*a*b**7 
*c**2*d**2*x - 6*int((log((a*e + b*e*x)/(c + d*x))*x)/(a**2 + 2*a*b*x + b* 
*2*x**2),x)*a**3*b**5*c**2*d**2 + 6*int((log((a*e + b*e*x)/(c + d*x))*x)/( 
a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**6*c**3*d - 6*int((log((a*e + b*e*x) 
/(c + d*x))*x)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**6*c**2*d**2*x + 6*i 
nt((log((a*e + b*e*x)/(c + d*x))*x)/(a**2 + 2*a*b*x + b**2*x**2),x)*a*b**7 
*c**3*d*x - 6*log(a + b*x)*a**6*d**4 + 18*log(a + b*x)*a**5*b*c*d**3 - 6*l 
og(a + b*x)*a**5*b*d**4*x - 18*log(a + b*x)*a**4*b**2*c**2*d**2 + 18*log(a 
 + b*x)*a**4*b**2*c*d**3*x + 6*log(a + b*x)*a**3*b**3*c**3*d - 18*log(a + 
b*x)*a**3*b**3*c**2*d**2*x + 6*log(a + b*x)*a**2*b**4*c**3*d*x - 2*log(a + 
 b*x)*a*b**5*c**4 - 2*log(a + b*x)*b**6*c**4*x + 2*log(c + d*x)*a*b**5*...