\(\int \frac {(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(c i+d i x)^2} \, dx\) [93]

Optimal result

Integrand size = 42, antiderivative size = 469 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=-\frac {2 A B (b c-a d) g^2 (a+b x)}{d^2 i^2 (c+d x)}+\frac {2 B^2 (b c-a d) g^2 (a+b x)}{d^2 i^2 (c+d x)}-\frac {2 B^2 (b c-a d) g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^2 (c+d x)}+\frac {2 b B (b c-a d) g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^2}+\frac {b g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2 i^2}+\frac {(b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2 i^2 (c+d x)}+\frac {2 b (b c-a d) g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2}+\frac {2 b B^2 (b c-a d) g^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}+\frac {4 b B (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2}-\frac {4 b B^2 (b c-a d) g^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2622\) vs. \(2(469)=938\).

Time = 6.49 (sec) , antiderivative size = 2622, normalized size of antiderivative = 5.59 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(g^2*(3*A^2*b^2*d*x - (3*A^2*(b*c - a*d)^2)/(c + d*x) + 6*A^2*b*(-(b*c) + 
a*d)*Log[c + d*x] + (6*a^2*A*B*d^2*(b*c - a*d + b*(c + d*x)*Log[a/b + x] + 
 (-(b*c) + a*d)*Log[(e*(a + b*x))/(c + d*x)] - b*c*Log[(b*(c + d*x))/(b*c 
- a*d)] - b*d*x*Log[(b*(c + d*x))/(b*c - a*d)]))/((b*c - a*d)*(c + d*x)) + 
 6*a*A*b*B*d*(-Log[c/d + x]^2 + 2*Log[c/d + x]*Log[c + d*x] + 2*(-(c/(c + 
d*x)) + (b*c*Log[a + b*x])/(-(b*c) + a*d) + (b*c*Log[c + d*x])/(b*c - a*d) 
 - Log[a/b + x]*Log[c + d*x] + Log[(e*(a + b*x))/(c + d*x)]*(c/(c + d*x) + 
 Log[c + d*x]) + Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)]) + 2*PolyLog[ 
2, (d*(a + b*x))/(-(b*c) + a*d)]) + 6*A*b^2*B*(d*(a/b + x)*(-1 + Log[a/b + 
 x]) - (c^2*Log[a/b + x])/(c + d*x) - (c + d*x)*(-1 + Log[c/d + x]) + c*Lo 
g[c/d + x]^2 + (c^2*(1 + Log[c/d + x]))/(c + d*x) + (b*c^2*(Log[a + b*x] - 
 Log[c + d*x]))/(b*c - a*d) + (-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + 
b*x))/(c + d*x)])*(d*x - c^2/(c + d*x) - 2*c*Log[c + d*x]) - 2*c*(Log[a/b 
+ x]*Log[(b*(c + d*x))/(b*c - a*d)] + PolyLog[2, (d*(a + b*x))/(-(b*c) + a 
*d)])) - (3*a^2*B^2*d^2*(2*b*c - 2*a*d + 2*b*(c + d*x)*Log[a + b*x] - 2*(b 
*c - a*d)*Log[(e*(a + b*x))/(c + d*x)] - 2*b*(c + d*x)*Log[a + b*x]*Log[(e 
*(a + b*x))/(c + d*x)] + (b*c - a*d)*Log[(e*(a + b*x))/(c + d*x)]^2 - 2*b* 
(c + d*x)*Log[c + d*x] - 2*b*(c + d*x)*Log[(e*(a + b*x))/(c + d*x)]*Log[(b 
*c - a*d)/(b*c + b*d*x)] + b*(c + d*x)*(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...
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 2795, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((b*c - a*d)*g^2*((-2*A*B*(a + b*x))/(d^2*(c + d*x)) + (2*B^2*(a + b*x))/( 
d^2*(c + d*x)) - (2*B^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^2*(c + 
d*x)) + ((a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^2*(c + d*x)) 
 + (b*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^2*(c + d*x)*(b 
- (d*(a + b*x))/(c + d*x))) + (2*b*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])* 
Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^3 + (2*b*(A + B*Log[(e*(a + b*x))/ 
(c + d*x)])^2*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^3 + (2*b*B^2*PolyLog 
[2, (d*(a + b*x))/(b*(c + d*x))])/d^3 + (4*b*B*(A + B*Log[(e*(a + b*x))/(c 
 + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d^3 - (4*b*B^2*PolyLog[ 
3, (d*(a + b*x))/(b*(c + d*x))])/d^3))/i^2
 

Defintions of rubi rules used

rule 2009

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

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[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 [F]

\[\int \frac {\left (b g x +a g \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (d i x +c i \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

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