\(\int \frac {(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(c i+d i x)^2} \, dx\) [40]

Optimal result

Integrand size = 40, antiderivative size = 260 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=-\frac {2 B (b c-a d) g^2 (a+b x)}{d^2 i^2 (c+d x)}+\frac {(2 A+B) (b c-a d) g^2 (a+b x)}{d^2 i^2 (c+d x)}+\frac {2 B (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^2 (c+d x)}+\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i^2 (c+d x)}+\frac {b (b c-a d) g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^2}+\frac {2 b B (b c-a d) g^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.92 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\frac {g^2 \left (A b^2 d x+\frac {B (b c-a d)^2}{c+d x}+b B (b c-a d) \log (a+b x)+b B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-\frac {(b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}-2 b B (b c-a d) \log (c+d x)-2 b (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+b B (b c-a d) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3 i^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2962, 2784, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2784

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)/(e*(q + 1))), x] - Simp[f/(e*(q + 1))   Int[(f*x)^(m - 1)*(d + e*x)^(q + 
1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, 
x] && ILtQ[q, -1] && GtQ[m, 0]
 

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. \(531\) vs. \(2(260)=520\).

Time = 3.06 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.05

method	result	size
parts	\(\frac {g^{2} A \left (\frac {x \,b^{2}}{d^{2}}+\frac {2 b \left (d a -b c \right ) \ln \left (d x +c \right )}{d^{3}}-\frac {a^{2} d^{2}-2 a c d b +c^{2} b^{2}}{d^{3} \left (d x +c \right )}\right )}{i^{2}}-\frac {g^{2} B \left (\frac {\left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right ) \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}{d^{2}}+\frac {\left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right ) b^{2} e^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}{d^{2}}+\frac {2 \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) b e \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}{d^{2}}\right )}{i^{2} \left (d a -b c \right ) e}\)	\(532\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {g^{2} d^{2} A \left (\frac {\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}{d^{2}}+\frac {2 b e \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d^{3}}+\frac {b^{2} e^{2}}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{e^{2} i^{2}}+\frac {g^{2} d^{2} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}}{d^{2}}+\frac {2 \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) b e}{d^{2}}+\frac {\left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right ) b^{2} e^{2}}{d^{2}}\right )}{e^{2} i^{2}}\right )}{d^{2}}\)	\(538\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {g^{2} d^{2} A \left (\frac {\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}{d^{2}}+\frac {2 b e \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{d^{3}}+\frac {b^{2} e^{2}}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{e^{2} i^{2}}+\frac {g^{2} d^{2} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}}{d^{2}}+\frac {2 \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) b e}{d^{2}}+\frac {\left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right ) b^{2} e^{2}}{d^{2}}\right )}{e^{2} i^{2}}\right )}{d^{2}}\)	\(538\)
risch	\(\text {Expression too large to display}\)	\(2042\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 886 vs. \(2 (259) = 518\).

Time = 0.10 (sec) , antiderivative size = 886, normalized size of antiderivative = 3.41 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1774 vs. \(2 (259) = 518\).

Time = 59.34 (sec) , antiderivative size = 1774, normalized size of antiderivative = 6.82 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/6*(2*(B*b^6*c^4*e^4*g^2 - 4*B*a*b^5*c^3*d*e^4*g^2 + 6*B*a^2*b^4*c^2*d^2* 
e^4*g^2 - 4*B*a^3*b^3*c*d^3*e^4*g^2 + B*a^4*b^2*d^4*e^4*g^2 - 3*(b*e*x + a 
*e)*B*b^5*c^4*d*e^3*g^2/(d*x + c) + 12*(b*e*x + a*e)*B*a*b^4*c^3*d^2*e^3*g 
^2/(d*x + c) - 18*(b*e*x + a*e)*B*a^2*b^3*c^2*d^3*e^3*g^2/(d*x + c) + 12*( 
b*e*x + a*e)*B*a^3*b^2*c*d^4*e^3*g^2/(d*x + c) - 3*(b*e*x + a*e)*B*a^4*b*d 
^5*e^3*g^2/(d*x + c) + 3*(b*e*x + a*e)^2*B*b^4*c^4*d^2*e^2*g^2/(d*x + c)^2 
 - 12*(b*e*x + a*e)^2*B*a*b^3*c^3*d^3*e^2*g^2/(d*x + c)^2 + 18*(b*e*x + a* 
e)^2*B*a^2*b^2*c^2*d^4*e^2*g^2/(d*x + c)^2 - 12*(b*e*x + a*e)^2*B*a^3*b*c* 
d^5*e^2*g^2/(d*x + c)^2 + 3*(b*e*x + a*e)^2*B*a^4*d^6*e^2*g^2/(d*x + c)^2) 
*log((b*e*x + a*e)/(d*x + c))/(b^3*d^3*e^3*i^2 - 3*(b*e*x + a*e)*b^2*d^4*e 
^2*i^2/(d*x + c) + 3*(b*e*x + a*e)^2*b*d^5*e*i^2/(d*x + c)^2 - (b*e*x + a* 
e)^3*d^6*i^2/(d*x + c)^3) + (2*A*b^6*c^4*e^4*g^2 + 3*B*b^6*c^4*e^4*g^2 - 8 
*A*a*b^5*c^3*d*e^4*g^2 - 12*B*a*b^5*c^3*d*e^4*g^2 + 12*A*a^2*b^4*c^2*d^2*e 
^4*g^2 + 18*B*a^2*b^4*c^2*d^2*e^4*g^2 - 8*A*a^3*b^3*c*d^3*e^4*g^2 - 12*B*a 
^3*b^3*c*d^3*e^4*g^2 + 2*A*a^4*b^2*d^4*e^4*g^2 + 3*B*a^4*b^2*d^4*e^4*g^2 - 
 6*(b*e*x + a*e)*A*b^5*c^4*d*e^3*g^2/(d*x + c) - 7*(b*e*x + a*e)*B*b^5*c^4 
*d*e^3*g^2/(d*x + c) + 24*(b*e*x + a*e)*A*a*b^4*c^3*d^2*e^3*g^2/(d*x + c) 
+ 28*(b*e*x + a*e)*B*a*b^4*c^3*d^2*e^3*g^2/(d*x + c) - 36*(b*e*x + a*e)*A* 
a^2*b^3*c^2*d^3*e^3*g^2/(d*x + c) - 42*(b*e*x + a*e)*B*a^2*b^3*c^2*d^3*e^3 
*g^2/(d*x + c) + 24*(b*e*x + a*e)*A*a^3*b^2*c*d^4*e^3*g^2/(d*x + c) + 2...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**2*( - int((log((a*e + b*e*x)/(c + d*x))*x**2)/(c**2 + 2*c*d*x + d**2*x 
**2),x)*a*b**3*c**2*d**4 - int((log((a*e + b*e*x)/(c + d*x))*x**2)/(c**2 + 
 2*c*d*x + d**2*x**2),x)*a*b**3*c*d**5*x + int((log((a*e + b*e*x)/(c + d*x 
))*x**2)/(c**2 + 2*c*d*x + d**2*x**2),x)*b**4*c**3*d**3 + int((log((a*e + 
b*e*x)/(c + d*x))*x**2)/(c**2 + 2*c*d*x + d**2*x**2),x)*b**4*c**2*d**4*x - 
 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a* 
*2*b**2*c**2*d**4 - 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(c**2 + 2*c*d*x 
 + d**2*x**2),x)*a**2*b**2*c*d**5*x + 2*int((log((a*e + b*e*x)/(c + d*x))* 
x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a*b**3*c**3*d**3 + 2*int((log((a*e + b* 
e*x)/(c + d*x))*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*a*b**3*c**2*d**4*x + lo 
g(a + b*x)*a**3*b*c*d**3 + log(a + b*x)*a**3*b*d**4*x - 2*log(c + d*x)*a** 
3*b*c**2*d**2 - 2*log(c + d*x)*a**3*b*c*d**3*x - log(c + d*x)*a**3*b*c*d** 
3 - log(c + d*x)*a**3*b*d**4*x + 4*log(c + d*x)*a**2*b**2*c**3*d + 4*log(c 
 + d*x)*a**2*b**2*c**2*d**2*x - 2*log(c + d*x)*a*b**3*c**4 - 2*log(c + d*x 
)*a*b**3*c**3*d*x - log((a*e + b*e*x)/(c + d*x))*a**3*b*d**4*x + log((a*e 
+ b*e*x)/(c + d*x))*a**2*b**2*c*d**3*x - a**4*d**4*x + 3*a**3*b*c*d**3*x + 
 a**3*b*d**4*x - 4*a**2*b**2*c**2*d**2*x - a**2*b**2*c*d**3*x**2 - a**2*b* 
*2*c*d**3*x + 2*a*b**3*c**3*d*x + a*b**3*c**2*d**2*x**2))/(c*d**3*(a*c*d + 
 a*d**2*x - b*c**2 - b*c*d*x))