\(\int \frac {(a g+b g x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(c i+d i x)^3} \, dx\) [152]

Optimal result

Integrand size = 43, antiderivative size = 263 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c i+d i x)^3} \, dx=\frac {B g^2 n (a+b x)^2}{4 d i^3 (c+d x)^2}-\frac {A b g^2 (a+b x)}{d^2 i^3 (c+d x)}+\frac {b B g^2 n (a+b x)}{d^2 i^3 (c+d x)}-\frac {b B g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^2 i^3 (c+d x)}-\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d i^3 (c+d x)^2}-\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^3 i^3}-\frac {b^2 B g^2 n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.98 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c i+d i x)^3} \, dx=\frac {g^2 \left (\frac {B (b c-a d)^2 n}{(c+d x)^2}-\frac {6 b B (b c-a d) n}{c+d x}-6 b^2 B n \log (a+b x)-\frac {2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2}+\frac {8 b (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}+6 b^2 B n \log (c+d x)+4 b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-2 b^2 B n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{4 d^3 i^3} \] Input:

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

Output:

(g^2*((B*(b*c - a*d)^2*n)/(c + d*x)^2 - (6*b*B*(b*c - a*d)*n)/(c + d*x) - 
6*b^2*B*n*Log[a + b*x] - (2*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x 
))^n]))/(c + d*x)^2 + (8*b*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^ 
n]))/(c + d*x) + 6*b^2*B*n*Log[c + d*x] + 4*b^2*(A + B*Log[e*((a + b*x)/(c 
 + d*x))^n])*Log[c + d*x] - 2*b^2*B*n*((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)])) 
)/(4*d^3*i^3)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2961, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2793

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

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 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [F]

\[\int \frac {\left (b g x +a g \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (d i x +c i \right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

integrate((b*g*x+a*g)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(d*i*x+c*i)**3,x 
)
 

Output:

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

Maxima [F]

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

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

Output:

1/2*B*a*b*g^2*n*((b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/((b*c*d^4 - a*d 
^5)*i^3*x^2 + 2*(b*c^2*d^3 - a*c*d^4)*i^3*x + (b*c^3*d^2 - a*c^2*d^3)*i^3) 
 + 2*(b^2*c - 2*a*b*d)*log(b*x + a)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4) 
*i^3) - 2*(b^2*c - 2*a*b*d)*log(d*x + c)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2 
*d^4)*i^3)) + 1/4*B*a^2*g^2*n*((2*b*d*x + 3*b*c - a*d)/((b*c*d^3 - a*d^4)* 
i^3*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*i^3*x + (b*c^3*d - a*c^2*d^2)*i^3) + 2*b 
^2*log(b*x + a)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3) - 2*b^2*log(d*x 
+ c)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3)) + 1/2*A*b^2*g^2*((4*c*d*x 
+ 3*c^2)/(d^5*i^3*x^2 + 2*c*d^4*i^3*x + c^2*d^3*i^3) + 2*log(d*x + c)/(d^3 
*i^3)) - 1/2*B*b^2*g^2*((2*(d^2*n*x^2 + 2*c*d*n*x + c^2*n)*log(b*x + a)*lo 
g(d*x + c) - (d^2*n*x^2 + 2*c*d*n*x + c^2*n)*log(d*x + c)^2 - (4*c*d*x + 3 
*c^2 + 2*(d^2*x^2 + 2*c*d*x + c^2)*log(d*x + c))*log((b*x + a)^n) + (4*c*d 
*x + 3*c^2 + 2*(d^2*x^2 + 2*c*d*x + c^2)*log(d*x + c))*log((d*x + c)^n))/( 
d^5*i^3*x^2 + 2*c*d^4*i^3*x + c^2*d^3*i^3) - 2*integrate(1/2*(2*b*d^3*x^3* 
log(e) + 2*a*d^3*x^2*log(e) - 3*b*c^3*n + 3*a*c^2*d*n - 4*(b*c^2*d*n - a*c 
*d^2*n)*x + 2*(b*d^3*n*x^3 + a*c^2*d*n + (2*b*c*d^2*n + a*d^3*n)*x^2 + (b* 
c^2*d*n + 2*a*c*d^2*n)*x)*log(b*x + a))/(b*d^6*i^3*x^4 + a*c^3*d^3*i^3 + ( 
3*b*c*d^5*i^3 + a*d^6*i^3)*x^3 + 3*(b*c^2*d^4*i^3 + a*c*d^5*i^3)*x^2 + (b* 
c^3*d^3*i^3 + 3*a*c^2*d^4*i^3)*x), x)) - (2*d*x + c)*B*a*b*g^2*log(e*(b*x/ 
(d*x + c) + a/(d*x + c))^n)/(d^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^3)...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**2*i*(4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(c**3 + 3*c**2*d* 
x + 3*c*d**2*x**2 + d**3*x**3),x)*a**2*b**3*c**4*d**5 + 8*int((log(((a + b 
*x)**n*e)/(c + d*x)**n)*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x* 
*3),x)*a**2*b**3*c**3*d**6*x + 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x 
**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a**2*b**3*c**2*d** 
7*x**2 - 8*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(c**3 + 3*c**2*d* 
x + 3*c*d**2*x**2 + d**3*x**3),x)*a*b**4*c**5*d**4 - 16*int((log(((a + b*x 
)**n*e)/(c + d*x)**n)*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3 
),x)*a*b**4*c**4*d**5*x - 8*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/ 
(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a*b**4*c**3*d**6*x**2 + 
 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(c**3 + 3*c**2*d*x + 3*c* 
d**2*x**2 + d**3*x**3),x)*b**5*c**6*d**3 + 8*int((log(((a + b*x)**n*e)/(c 
+ d*x)**n)*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b**5*c 
**5*d**4*x + 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(c**3 + 3*c** 
2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b**5*c**4*d**5*x**2 - 2*log(a + b*x) 
*a**4*b*c**2*d**4*n - 4*log(a + b*x)*a**4*b*c*d**5*n*x - 2*log(a + b*x)*a* 
*4*b*d**6*n*x**2 + 2*log(c + d*x)*a**4*b*c**2*d**4*n + 4*log(c + d*x)*a**4 
*b*c*d**5*n*x + 2*log(c + d*x)*a**4*b*d**6*n*x**2 + 4*log(c + d*x)*a**3*b* 
*2*c**4*d**2 + 8*log(c + d*x)*a**3*b**2*c**3*d**3*x + 4*log(c + d*x)*a**3* 
b**2*c**2*d**4*x**2 - 8*log(c + d*x)*a**2*b**3*c**5*d - 16*log(c + d*x)...