\(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{a g+b g x} \, dx\) [121]

Optimal result

Integrand size = 43, antiderivative size = 289 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=-\frac {B d (b c-a d) i^2 n x}{2 b^2 g}+\frac {d (b c-a d) i^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {i^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g}-\frac {B (b c-a d)^2 i^2 n \log \left (\frac {a+b x}{c+d x}\right )}{2 b^3 g}-\frac {3 B (b c-a d)^2 i^2 n \log (c+d x)}{2 b^3 g}-\frac {(b c-a d)^2 i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {B (b c-a d)^2 i^2 n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2561, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\frac {d i^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}-\frac {i^2 (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac {i^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b g}+\frac {B i^2 n (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}-\frac {B i^2 n (b c-a d)^2 \log \left (\frac {a+b x}{c+d x}\right )}{2 b^3 g}-\frac {3 B i^2 n (b c-a d)^2 \log (c+d x)}{2 b^3 g}-\frac {B d i^2 n x (b c-a d)}{2 b^2 g} \]

[In]

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

[Out]

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

Rule 31

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

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 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(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] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.91 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=\frac {i^2 \left (2 A b d (b c-a d) x-B (b c-a d) n (b d x+(b c-a d) \log (a+b x))+2 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 (b c-a d)^2 \log (g (a+b x)) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 n \log (c+d x)+B (b c-a d)^2 n \left (-\log (g (a+b x)) \left (\log (g (a+b x))-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{2 b^3 g} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (280) = 560\).

Time = 0.49 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.01 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx=2 \, A c d i^{2} {\left (\frac {x}{b g} - \frac {a \log \left (b x + a\right )}{b^{2} g}\right )} + \frac {1}{2} \, A d^{2} i^{2} {\left (\frac {2 \, a^{2} \log \left (b x + a\right )}{b^{3} g} + \frac {b x^{2} - 2 \, a x}{b^{2} g}\right )} + \frac {A c^{2} i^{2} \log \left (b g x + a g\right )}{b g} - \frac {{\left (3 \, b c^{2} i^{2} n - 2 \, a c d i^{2} n\right )} B \log \left (d x + c\right )}{2 \, b^{2} g} + \frac {{\left (b^{2} c^{2} i^{2} n - 2 \, a b c d i^{2} n + a^{2} d^{2} i^{2} n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{b^{3} g} + \frac {B b^{2} d^{2} i^{2} x^{2} \log \left (e\right ) - {\left (b^{2} c^{2} i^{2} n - 2 \, a b c d i^{2} n + a^{2} d^{2} i^{2} n\right )} B \log \left (b x + a\right )^{2} - {\left ({\left (i^{2} n - 4 \, i^{2} \log \left (e\right )\right )} b^{2} c d - {\left (i^{2} n - 2 \, i^{2} \log \left (e\right )\right )} a b d^{2}\right )} B x + {\left (2 \, b^{2} c^{2} i^{2} \log \left (e\right ) + 4 \, {\left (i^{2} n - i^{2} \log \left (e\right )\right )} a b c d - {\left (3 \, i^{2} n - 2 \, i^{2} \log \left (e\right )\right )} a^{2} d^{2}\right )} B \log \left (b x + a\right ) + {\left (B b^{2} d^{2} i^{2} x^{2} + 2 \, {\left (2 \, b^{2} c d i^{2} - a b d^{2} i^{2}\right )} B x + 2 \, {\left (b^{2} c^{2} i^{2} - 2 \, a b c d i^{2} + a^{2} d^{2} i^{2}\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (B b^{2} d^{2} i^{2} x^{2} + 2 \, {\left (2 \, b^{2} c d i^{2} - a b d^{2} i^{2}\right )} B x + 2 \, {\left (b^{2} c^{2} i^{2} - 2 \, a b c d i^{2} + a^{2} d^{2} i^{2}\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b^{3} g} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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