\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{c i+d i x} \, dx\) [34]

Optimal result

Integrand size = 30, antiderivative size = 81 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c i+d i x} \, dx=-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}-\frac {B \operatorname {PolyLog}\left (2,1-\frac {b c-a d}{b (c+d x)}\right )}{d i} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.17 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c i+d i x} \, dx=\frac {\log (i (c+d x)) \left (2 A-2 B \log \left (\frac {d (a+b x)}{-b c+a d}\right )+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )+B \log (i (c+d x))\right )-2 B \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{2 d i} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2944, 2858, 27, 25, 2778, 2005, 2752}

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.

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

\(\Big \downarrow \) 2944

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

\(\Big \downarrow \) 2858

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2778

\(\displaystyle \frac {B (b c-a d) \int \frac {(c+d x) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{b c-a d-b (c+d x)}d\frac {1}{c+d x}}{d i}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i}\)

\(\Big \downarrow \) 2005

\(\displaystyle \frac {B (b c-a d) \int \frac {\log \left (\frac {b c-a d}{b (c+d x)}\right )}{\frac {b c-a d}{c+d x}-b}d\frac {1}{c+d x}}{d i}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d i}\)

\(\Big \downarrow \) 2752

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2778

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), 
x_Symbol] :> Simp[1/n   Subst[Int[(a + b*Log[c*x])/(x*(d + e*x^(r/n))), x], 
 x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2944

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

Maple [A] (verified)

Time = 2.85 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.80

method	result	size
parts	\(\frac {A \ln \left (d x +c \right )}{i d}+\frac {B \left (-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{i}\)	\(146\)
risch	\(\frac {A \ln \left (d x +c \right )}{i d}-\frac {B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{i d}-\frac {B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{i d}\)	\(148\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {d A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{i e \left (d a -b c \right )}-\frac {d^{2} B \left (-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{i e \left (d a -b c \right )}\right )}{d^{2}}\)	\(217\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {d A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{i e \left (d a -b c \right )}-\frac {d^{2} B \left (-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{i e \left (d a -b c \right )}\right )}{d^{2}}\)	\(217\)

Input:

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

Output:

A/i*ln(d*x+c)/d+B/i*(-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)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/2*B*(log(d*x + c)^2/(d*i) - 2*integrate((log(b*x + a) + log(e))/(d*i*x 
+ c*i), x)) + A*log(d*i*x + c*i)/(d*i)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (80) = 160\).

Time = 36.84 (sec) , antiderivative size = 617, normalized size of antiderivative = 7.62 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c i+d i x} \, dx=\frac {1}{2} \, {\left (\frac {{\left (B b^{3} c^{3} e^{3} - 3 \, B a b^{2} c^{2} d e^{3} + 3 \, B a^{2} b c d^{2} e^{3} - B a^{3} d^{3} e^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} d e^{2} i - \frac {2 \, {\left (b e x + a e\right )} b d^{2} e i}{d x + c} + \frac {{\left (b e x + a e\right )}^{2} d^{3} i}{{\left (d x + c\right )}^{2}}} + \frac {A b^{4} c^{3} e^{3} - B b^{4} c^{3} e^{3} - 3 \, A a b^{3} c^{2} d e^{3} + 3 \, B a b^{3} c^{2} d e^{3} + 3 \, A a^{2} b^{2} c d^{2} e^{3} - 3 \, B a^{2} b^{2} c d^{2} e^{3} - A a^{3} b d^{3} e^{3} + B a^{3} b d^{3} e^{3} + \frac {{\left (b e x + a e\right )} B b^{3} c^{3} d e^{2}}{d x + c} - \frac {3 \, {\left (b e x + a e\right )} B a b^{2} c^{2} d^{2} e^{2}}{d x + c} + \frac {3 \, {\left (b e x + a e\right )} B a^{2} b c d^{3} e^{2}}{d x + c} - \frac {{\left (b e x + a e\right )} B a^{3} d^{4} e^{2}}{d x + c}}{b^{3} d e^{2} i - \frac {2 \, {\left (b e x + a e\right )} b^{2} d^{2} e i}{d x + c} + \frac {{\left (b e x + a e\right )}^{2} b d^{3} i}{{\left (d x + c\right )}^{2}}} + \frac {{\left (B b^{3} c^{3} e - 3 \, B a b^{2} c^{2} d e + 3 \, B a^{2} b c d^{2} e - B a^{3} d^{3} e\right )} \log \left (-b e + \frac {{\left (b e x + a e\right )} d}{d x + c}\right )}{b^{2} d i} - \frac {{\left (B b^{3} c^{3} e - 3 \, B a b^{2} c^{2} d e + 3 \, B a^{2} b c d^{2} e - B a^{3} d^{3} e\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} d i}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - i*(int(log((a*e + b*e*x)/(c + d*x))/(c + d*x),x)*b*d + log(c + d*x)*a) 
)/d