\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{c i+d i x} \, dx\) [87]

Optimal result

Integrand size = 32, antiderivative size = 127 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c i+d i x} \, dx=-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d i}-\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d i}+\frac {2 B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d i} \] Output:

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

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.98 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c i+d i x} \, dx=\frac {A^2 \log (c+d x)+2 A B \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )-2 A B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )-B^2 \log ^2\left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+A B \log ^2\left (\frac {b c-a d}{b c+b d x}\right )-2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )-2 A B \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )+2 B^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d i} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2952, 2754, 2821, 7143}

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.

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2952

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/d)^m   Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, ( 
a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[ 
n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f 
 - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(337\) vs. \(2(127)=254\).

Time = 3.11 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.66

method	result	size
parts	\(\frac {A^{2} \ln \left (d x +c \right )}{i d}-\frac {B^{2} \left (\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \operatorname {polylog}\left (2, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )-2 \operatorname {polylog}\left (3, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )\right )}{i d}+\frac {2 A B \left (-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{i}\)	\(338\)
risch	\(\frac {A^{2} \ln \left (d x +c \right )}{i d}-\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{d i}-\frac {2 B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \operatorname {polylog}\left (2, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{d i}+\frac {2 B^{2} \operatorname {polylog}\left (3, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{d i}-\frac {2 A B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{i d}-\frac {2 A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{i d}\)	\(356\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {d \,A^{2} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{i e \left (d a -b c \right )}+\frac {d \,B^{2} \left (\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \operatorname {polylog}\left (2, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )-2 \operatorname {polylog}\left (3, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )\right )}{i e \left (d a -b c \right )}-\frac {2 d^{2} A B \left (-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{i e \left (d a -b c \right )}\right )}{d^{2}}\)	\(418\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {d \,A^{2} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{i e \left (d a -b c \right )}+\frac {d \,B^{2} \left (\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (1-\frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )+2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \operatorname {polylog}\left (2, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )-2 \operatorname {polylog}\left (3, \frac {d \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )\right )}{i e \left (d a -b c \right )}-\frac {2 d^{2} A B \left (-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{i e \left (d a -b c \right )}\right )}{d^{2}}\)	\(418\)

Input:

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

Output:

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

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

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i),x, algorithm="fricas" 
)
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(i*( - int(log((a*e + b*e*x)/(c + d*x))**2/(c + d*x),x)*b**2*d - 2*int(log 
((a*e + b*e*x)/(c + d*x))/(c + d*x),x)*a*b*d - log(c + d*x)*a**2))/d