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

Optimal result

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

[Out]

-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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2553, 2398, 2404, 2338, 2351, 31, 2354, 2438} \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B n (b c-a d) (-a d g-b c g+2 b d f) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d^2}-\frac {(b f-a g)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^2 g}-\frac {B g n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d}+\frac {(f+g x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g}+\frac {B^2 n^2 (b c-a d) (-a d g-b c g+2 b d f) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}+\frac {B^2 g n^2 (b c-a d)^2 \log (c+d x)}{b^2 d^2} \]

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2351

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

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2398

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] - Dist[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 2404

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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2553

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[b*c - a*d, Subst[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] && Inte
gerQ[m] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = (b c-a d) \text {Subst}\left (\int \frac {(b f-a g-(d f-c g) x) \left (A+B \log \left (e x^n\right )\right )^2}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \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 n) \text {Subst}\left (\int \frac {(b f-a g+(-d f+c g) x)^2 \left (A+B \log \left (e x^n\right )\right )}{x (b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{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 n) \text {Subst}\left (\int \left (\frac {(b f-a g)^2 \left (A+B \log \left (e x^n\right )\right )}{b^2 x}+\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e x^n\right )\right )}{b d (b-d x)^2}+\frac {(b c-a d) g (2 b d f-b c g-a d g) \left (A+B \log \left (e x^n\right )\right )}{b^2 d (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{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 {\left (B (b c-a d)^2 g n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{b d}-\frac {\left (B (b f-a g)^2 n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b^2 g}-\frac {(B (b c-a d) (2 b d f-b c g-a d g) n) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b^2 d} \\ & = -\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 {\left (B^2 (b c-a d)^2 g n^2\right ) \text {Subst}\left (\int \frac {1}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b^2 d}-\frac {\left (B^2 (b c-a d) (2 b d f-b c g-a d g) n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b^2 d^2} \\ & = -\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 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.25 \[ \int (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {(f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {B n \left (2 A b d (b c-a d) g^2 x+2 B d (b c-a d) g^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 d^2 (b f-a g)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 g^2 n \log (c+d x)-2 b^2 (d f-c g)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-B d^2 (b f-a g)^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+b^2 B (d f-c g)^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b^2 d^2}}{2 g} \]

[In]

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

[Out]

((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*Log[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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

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

Time = 0.68 (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=A B g x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{2} \, A^{2} g x^{2} - A B g n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + 2 \, A B f n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + 2 \, A B f x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A^{2} f x - \frac {{\left (a c d g n^{2} + {\left (2 \, c d f n \log \left (e\right ) - {\left (g n^{2} + g n \log \left (e\right )\right )} c^{2}\right )} b\right )} B^{2} \log \left (d x + c\right )}{b d^{2}} + \frac {{\left (2 \, a b d^{2} f n^{2} - a^{2} d^{2} g n^{2} - {\left (2 \, c d f n^{2} - c^{2} g n^{2}\right )} b^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B^{2}}{b^{2} d^{2}} + \frac {B^{2} b^{2} d^{2} g x^{2} \log \left (e\right )^{2} + 2 \, {\left (2 \, c d f n^{2} - c^{2} g n^{2}\right )} B^{2} b^{2} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (2 \, c d f n^{2} - c^{2} g n^{2}\right )} B^{2} b^{2} \log \left (d x + c\right )^{2} - {\left (2 \, a b d^{2} f n^{2} - a^{2} d^{2} g n^{2}\right )} B^{2} \log \left (b x + a\right )^{2} + 2 \, {\left (a b d^{2} g n \log \left (e\right ) - {\left (c d g n \log \left (e\right ) - d^{2} f \log \left (e\right )^{2}\right )} b^{2}\right )} B^{2} x + 2 \, {\left ({\left (g n^{2} - g n \log \left (e\right )\right )} a^{2} d^{2} - {\left (c d g n^{2} - 2 \, d^{2} f n \log \left (e\right )\right )} a b\right )} B^{2} \log \left (b x + a\right ) + {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} b^{2} d^{2} f x\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} b^{2} d^{2} f x\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + 2 \, {\left (B^{2} b^{2} d^{2} g x^{2} \log \left (e\right ) - {\left (2 \, c d f n - c^{2} g n\right )} B^{2} b^{2} \log \left (d x + c\right ) + {\left (a b d^{2} g n - {\left (c d g n - 2 \, d^{2} f \log \left (e\right )\right )} b^{2}\right )} B^{2} x + {\left (2 \, a b d^{2} f n - a^{2} d^{2} g n\right )} B^{2} \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B^{2} b^{2} d^{2} g x^{2} \log \left (e\right ) - {\left (2 \, c d f n - c^{2} g n\right )} B^{2} b^{2} \log \left (d x + c\right ) + {\left (a b d^{2} g n - {\left (c d g n - 2 \, d^{2} f \log \left (e\right )\right )} b^{2}\right )} B^{2} x + {\left (2 \, a b d^{2} f n - a^{2} d^{2} g n\right )} B^{2} \log \left (b x + a\right ) + {\left (B^{2} b^{2} d^{2} g x^{2} + 2 \, B^{2} b^{2} d^{2} f x\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b^{2} d^{2}} \]

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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