\(\int \frac {(a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(c i+d i x)^2} \, dx\) [39]

Optimal result

Integrand size = 40, antiderivative size = 341 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\frac {3 B (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}-\frac {(6 A+5 B) (b c-a d)^2 g^3 (a+b x)}{2 d^3 i^2 (c+d x)}-\frac {3 B (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d i^2 (c+d x)}-\frac {(b c-a d) g^3 (a+b x)^2 \left (3 A+B+3 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^2 i^2 (c+d x)}-\frac {b (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d^4 i^2}-\frac {3 b B (b c-a d)^2 g^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.05 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\frac {g^3 \left (-2 A b^2 d (2 b c-3 a d) x-2 b B d (2 b c-3 a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+b^3 d^2 x^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+\frac {2 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}+2 b B (2 b c-3 a d) (b c-a d) \log (c+d x)+6 b (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 B (b c-a d)^2 \left (\frac {b c-a d}{c+d x}+b \log (a+b x)-b \log (c+d x)\right )+b B \left (-a^2 d^2 \log (a+b x)+b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )-3 b B (b c-a d)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^4 i^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2962, 2784, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2784

\(\displaystyle \frac {g^3 (b c-a d)^2 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {(a+b x) \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{d}}{2 d}\right )}{i^2}\)

\(\Big \downarrow \) 2793

\(\displaystyle \frac {g^3 (b c-a d)^2 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \left (-\frac {6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )}{d}-\frac {b \left (6 A+5 B+6 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d \left (\frac {d (a+b x)}{c+d x}-b\right )}\right )d\frac {a+b x}{c+d x}}{d}}{2 d}\right )}{i^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {g^3 (b c-a d)^2 \left (\frac {(a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x)^2 \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {b \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{d^2}-\frac {(6 A+5 B) (a+b x)}{d (c+d x)}-\frac {6 b B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2}-\frac {6 B (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d (c+d x)}+\frac {6 B (a+b x)}{d (c+d x)}}{d}}{2 d}\right )}{i^2}\)

Input:

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

Output:

((b*c - a*d)^2*g^3*(((a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2* 
d*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^2) - (((a + b*x)^2*(3*A + B + 
3*B*Log[(e*(a + b*x))/(c + d*x)]))/(d*(c + d*x)^2*(b - (d*(a + b*x))/(c + 
d*x))) - ((6*B*(a + b*x))/(d*(c + d*x)) - ((6*A + 5*B)*(a + b*x))/(d*(c + 
d*x)) - (6*B*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d*(c + d*x)) - (b*(6 
*A + 5*B + 6*B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (d*(a + b*x))/(b*(c + 
 d*x))])/d^2 - (6*b*B*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d^2)/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. \(845\) vs. \(2(333)=666\).

Time = 3.29 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.48

method	result	size
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (-\frac {A \,d^{2} g^{3} \left (d a -b c \right ) \left (-\frac {\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}{d^{3}}-\frac {3 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^{4}}-\frac {3 b^{2} e^{2}}{d^{4} \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 {b^{3} e^{3}}{2 d^{4} \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 )}{e^{2} i^{2}}-\frac {B \,d^{2} g^{3} \left (d a -b c \right ) \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^{3}}-\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 ) b e}{d^{3}}-\frac {3 \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^{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 ) b^{3} e^{3}}{d^{3}}\right )}{e^{2} i^{2}}\right )}{d^{2}}\)	\(846\)
default	\(-\frac {e \left (d a -b c \right ) \left (-\frac {A \,d^{2} g^{3} \left (d a -b c \right ) \left (-\frac {\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}{d^{3}}-\frac {3 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^{4}}-\frac {3 b^{2} e^{2}}{d^{4} \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 {b^{3} e^{3}}{2 d^{4} \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 )}{e^{2} i^{2}}-\frac {B \,d^{2} g^{3} \left (d a -b c \right ) \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^{3}}-\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 ) b e}{d^{3}}-\frac {3 \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^{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 ) b^{3} e^{3}}{d^{3}}\right )}{e^{2} i^{2}}\right )}{d^{2}}\)	\(846\)
parts	\(\frac {g^{3} A \left (\frac {b^{2} \left (\frac {1}{2} b d \,x^{2}+3 x d a -2 b c x \right )}{d^{3}}+\frac {3 b \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \ln \left (d x +c \right )}{d^{4}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{d^{4} \left (d x +c \right )}\right )}{i^{2}}-\frac {g^{3} 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^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{3}}+\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 ) b^{3} e^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{3}}+\frac {3 \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^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{3}}+\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 ) b e \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{3}}\right )}{i^{2} \left (d a -b c \right ) e}\)	\(896\)
risch	\(\text {Expression too large to display}\)	\(3868\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

integrate((b*g*x+a*g)**3*(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. 1341 vs. \(2 (332) = 664\).

Time = 0.11 (sec) , antiderivative size = 1341, normalized size of antiderivative = 3.93 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(2*c^3/(d^5*i^2*x + c*d^4*i^2) + 6*c^2*log(d*x + c)/(d^4*i^2) + (d*x^2 
 - 4*c*x)/(d^3*i^2))*A*b^3*g^3 - 3*A*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^3 + 3*A*a^2*b*g^3*(c/(d^3*i^2* 
x + c*d^2*i^2) + log(d*x + c)/(d^2*i^2)) - B*a^3*g^3*(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^3*g^3/(d^2*i^2*x + c*d*i^2) - 1/2*(6*a^3*b*d^3*g^3*log(e) - (6*g^3*log(e 
) + 7*g^3)*b^4*c^3 + (18*g^3*log(e) + 17*g^3)*a*b^3*c^2*d - 6*(3*g^3*log(e 
) + 2*g^3)*a^2*b^2*c*d^2)*B*log(d*x + c)/(b*c*d^4*i^2 - a*d^5*i^2) + 1/2*( 
(b^4*c*d^3*g^3*log(e) - a*b^3*d^4*g^3*log(e))*B*x^3 - ((3*g^3*log(e) + g^3 
)*b^4*c^2*d^2 - (9*g^3*log(e) + 2*g^3)*a*b^3*c*d^3 + (6*g^3*log(e) + g^3)* 
a^2*b^2*d^4)*B*x^2 - ((4*g^3*log(e) + g^3)*b^4*c^3*d - 2*(5*g^3*log(e) + g 
^3)*a*b^3*c^2*d^2 + (6*g^3*log(e) + g^3)*a^2*b^2*c*d^3)*B*x - 3*((b^4*c^3* 
d*g^3 - 3*a*b^3*c^2*d^2*g^3 + 3*a^2*b^2*c*d^3*g^3 - a^3*b*d^4*g^3)*B*x + ( 
b^4*c^4*g^3 - 3*a*b^3*c^3*d*g^3 + 3*a^2*b^2*c^2*d^2*g^3 - a^3*b*c*d^3*g^3) 
*B)*log(d*x + c)^2 + 2*((g^3*log(e) - g^3)*b^4*c^4 - 4*(g^3*log(e) - g^3)* 
a*b^3*c^3*d + 6*(g^3*log(e) - g^3)*a^2*b^2*c^2*d^2 - 3*(g^3*log(e) - g^3)* 
a^3*b*c*d^3)*B + ((b^4*c*d^3*g^3 - a*b^3*d^4*g^3)*B*x^3 - 3*(b^4*c^2*d^2*g 
^3 - 3*a*b^3*c*d^3*g^3 + 2*a^2*b^2*d^4*g^3)*B*x^2 - (6*b^4*c^3*d*g^3 - 12* 
a*b^3*c^2*d^2*g^3 + 3*a^2*b^2*c*d^3*g^3 + 5*a^3*b*d^4*g^3)*B*x - (6*a*b...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2814 vs. \(2 (332) = 664\).

Time = 69.45 (sec) , antiderivative size = 2814, normalized size of antiderivative = 8.25 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/24*(6*(B*b^8*c^5*e^5*g^3 - 5*B*a*b^7*c^4*d*e^5*g^3 + 10*B*a^2*b^6*c^3*d 
^2*e^5*g^3 - 10*B*a^3*b^5*c^2*d^3*e^5*g^3 + 5*B*a^4*b^4*c*d^4*e^5*g^3 - B* 
a^5*b^3*d^5*e^5*g^3 - 4*(b*e*x + a*e)*B*b^7*c^5*d*e^4*g^3/(d*x + c) + 20*( 
b*e*x + a*e)*B*a*b^6*c^4*d^2*e^4*g^3/(d*x + c) - 40*(b*e*x + a*e)*B*a^2*b^ 
5*c^3*d^3*e^4*g^3/(d*x + c) + 40*(b*e*x + a*e)*B*a^3*b^4*c^2*d^4*e^4*g^3/( 
d*x + c) - 20*(b*e*x + a*e)*B*a^4*b^3*c*d^5*e^4*g^3/(d*x + c) + 4*(b*e*x + 
 a*e)*B*a^5*b^2*d^6*e^4*g^3/(d*x + c) + 6*(b*e*x + a*e)^2*B*b^6*c^5*d^2*e^ 
3*g^3/(d*x + c)^2 - 30*(b*e*x + a*e)^2*B*a*b^5*c^4*d^3*e^3*g^3/(d*x + c)^2 
 + 60*(b*e*x + a*e)^2*B*a^2*b^4*c^3*d^4*e^3*g^3/(d*x + c)^2 - 60*(b*e*x + 
a*e)^2*B*a^3*b^3*c^2*d^5*e^3*g^3/(d*x + c)^2 + 30*(b*e*x + a*e)^2*B*a^4*b^ 
2*c*d^6*e^3*g^3/(d*x + c)^2 - 6*(b*e*x + a*e)^2*B*a^5*b*d^7*e^3*g^3/(d*x + 
 c)^2 - 4*(b*e*x + a*e)^3*B*b^5*c^5*d^3*e^2*g^3/(d*x + c)^3 + 20*(b*e*x + 
a*e)^3*B*a*b^4*c^4*d^4*e^2*g^3/(d*x + c)^3 - 40*(b*e*x + a*e)^3*B*a^2*b^3* 
c^3*d^5*e^2*g^3/(d*x + c)^3 + 40*(b*e*x + a*e)^3*B*a^3*b^2*c^2*d^6*e^2*g^3 
/(d*x + c)^3 - 20*(b*e*x + a*e)^3*B*a^4*b*c*d^7*e^2*g^3/(d*x + c)^3 + 4*(b 
*e*x + a*e)^3*B*a^5*d^8*e^2*g^3/(d*x + c)^3)*log((b*e*x + a*e)/(d*x + c))/ 
(b^4*d^4*e^4*i^2 - 4*(b*e*x + a*e)*b^3*d^5*e^3*i^2/(d*x + c) + 6*(b*e*x + 
a*e)^2*b^2*d^6*e^2*i^2/(d*x + c)^2 - 4*(b*e*x + a*e)^3*b*d^7*e*i^2/(d*x + 
c)^3 + (b*e*x + a*e)^4*d^8*i^2/(d*x + c)^4) + (6*A*b^8*c^5*e^5*g^3 + 11*B* 
b^8*c^5*e^5*g^3 - 30*A*a*b^7*c^4*d*e^5*g^3 - 55*B*a*b^7*c^4*d*e^5*g^3 +...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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