\(\int (f+g x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [69]

Optimal result

Integrand size = 30, antiderivative size = 290 \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {B (b c-a d) g n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 d}-\frac {(b f-a g)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g}+\frac {(f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 g}+\frac {B (b c-a d) (2 b d f-b c g-a d g) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{b^2 d^2}+\frac {B^2 (b c-a d)^2 g n^2 \log (c+d x)}{b^2 d^2}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.25 \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {(f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {B n \left (2 A b d (b c-a d) g^2 x+2 B d (b c-a d) g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 d^2 (b f-a g)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 g^2 n \log (c+d x)-2 b^2 (d f-c g)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-B d^2 (b f-a g)^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+b^2 B (d f-c g)^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b^2 d^2}}{2 g} \] Input:

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

Output:

((f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (B*n*(2*A*b*d*(b*c 
 - a*d)*g^2*x + 2*B*d*(b*c - a*d)*g^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x) 
)^n] + 2*d^2*(b*f - a*g)^2*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x)) 
^n]) - 2*B*(b*c - a*d)^2*g^2*n*Log[c + d*x] - 2*b^2*(d*f - c*g)^2*(A + B*L 
og[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - B*d^2*(b*f - a*g)^2*n*(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)]) + b^2*B*(d*f - c*g)^2*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)])))/(b^2*d^2))/(2*g)
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.41, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2953, 2798, 2804, 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.

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

\(\Big \downarrow \) 2953

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

\(\Big \downarrow \) 2798

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

\(\Big \downarrow \) 2804

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2798

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

rule 2804

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ 
u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / 
; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
 

rule 2953

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)   Sub 
st[Int[(b*f - a*g - (d*f - c*g)*x)^m*((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] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (285) = 570\).

Time = 0.57 (sec) , antiderivative size = 899, normalized size of antiderivative = 3.10 \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

A*B*g*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*A^2*g*x^2 - A*B*g*n 
*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) + 2*A 
*B*f*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + 2*A*B*f*x*log(e*(b*x/(d*x + 
 c) + a/(d*x + c))^n) + A^2*f*x - (a*c*d*g*n^2 + (2*c*d*f*n*log(e) - (g*n^ 
2 + g*n*log(e))*c^2)*b)*B^2*log(d*x + c)/(b*d^2) + (2*a*b*d^2*f*n^2 - a^2* 
d^2*g*n^2 - (2*c*d*f*n^2 - c^2*g*n^2)*b^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^2/(b^2*d^2) + 1/2 
*(B^2*b^2*d^2*g*x^2*log(e)^2 + 2*(2*c*d*f*n^2 - c^2*g*n^2)*B^2*b^2*log(b*x 
 + a)*log(d*x + c) - (2*c*d*f*n^2 - c^2*g*n^2)*B^2*b^2*log(d*x + c)^2 - (2 
*a*b*d^2*f*n^2 - a^2*d^2*g*n^2)*B^2*log(b*x + a)^2 + 2*(a*b*d^2*g*n*log(e) 
 - (c*d*g*n*log(e) - d^2*f*log(e)^2)*b^2)*B^2*x + 2*((g*n^2 - g*n*log(e))* 
a^2*d^2 - (c*d*g*n^2 - 2*d^2*f*n*log(e))*a*b)*B^2*log(b*x + a) + (B^2*b^2* 
d^2*g*x^2 + 2*B^2*b^2*d^2*f*x)*log((b*x + a)^n)^2 + (B^2*b^2*d^2*g*x^2 + 2 
*B^2*b^2*d^2*f*x)*log((d*x + c)^n)^2 + 2*(B^2*b^2*d^2*g*x^2*log(e) - (2*c* 
d*f*n - c^2*g*n)*B^2*b^2*log(d*x + c) + (a*b*d^2*g*n - (c*d*g*n - 2*d^2*f* 
log(e))*b^2)*B^2*x + (2*a*b*d^2*f*n - a^2*d^2*g*n)*B^2*log(b*x + a))*log(( 
b*x + a)^n) - 2*(B^2*b^2*d^2*g*x^2*log(e) - (2*c*d*f*n - c^2*g*n)*B^2*b^2* 
log(d*x + c) + (a*b*d^2*g*n - (c*d*g*n - 2*d^2*f*log(e))*b^2)*B^2*x + (2*a 
*b*d^2*f*n - a^2*d^2*g*n)*B^2*log(b*x + a) + (B^2*b^2*d^2*g*x^2 + 2*B^2*b^ 
2*d^2*f*x)*log((b*x + a)^n))*log((d*x + c)^n))/(b^2*d^2)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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