\(\int \frac {1}{(a g+b g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2} \, dx\) [28]

Optimal result

Integrand size = 35, antiderivative size = 314 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=-\frac {2 b e^{\frac {2 A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n} (c+d x)^2 \operatorname {ExpIntegralEi}\left (-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 (b c-a d)^2 g^3 n^2 (a+b x)^2}+\frac {d e^{\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} (c+d x) \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 (b c-a d)^2 g^3 n^2 (a+b x)}+\frac {d (c+d x)}{B (b c-a d)^2 g^3 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}-\frac {b (c+d x)^2}{B (b c-a d)^2 g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2549, 2395, 2343, 2347, 2209} \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=-\frac {2 b e^{\frac {2 A}{B n}} (c+d x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n} \operatorname {ExpIntegralEi}\left (-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 g^3 n^2 (a+b x)^2 (b c-a d)^2}+\frac {d e^{\frac {A}{B n}} (c+d x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 g^3 n^2 (a+b x) (b c-a d)^2}-\frac {b (c+d x)^2}{B g^3 n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}+\frac {d (c+d x)}{B g^3 n (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )} \]

[In]

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

[Out]

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2343

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

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2395

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 2549

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/b)^m, Subst[Int[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] && IntegersQ[m,
p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\frac {(c+d x) \left (B (-b c+a d) n-2 b e^{\frac {2 A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n} (c+d x) \operatorname {ExpIntegralEi}\left (-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+d e^{\frac {A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{B^2 (b c-a d)^2 g^3 n^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {1}{\left (b g x +a g \right )^{3} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (312) = 624\).

Time = 0.29 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.40 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=-\frac {{\left (B b c d - B a d^{2}\right )} n x - {\left (A b^{2} d x^{2} + 2 \, A a b d x + A a^{2} d + {\left (B b^{2} d x^{2} + 2 \, B a b d x + B a^{2} d\right )} \log \left (e\right ) + {\left (B b^{2} d n x^{2} + 2 \, B a b d n x + B a^{2} d n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} e^{\left (\frac {B \log \left (e\right ) + A}{B n}\right )} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (-\frac {B \log \left (e\right ) + A}{B n}\right )}}{b x + a}\right ) + 2 \, {\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b + {\left (B b^{3} x^{2} + 2 \, B a b^{2} x + B a^{2} b\right )} \log \left (e\right ) + {\left (B b^{3} n x^{2} + 2 \, B a b^{2} n x + B a^{2} b n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} e^{\left (\frac {2 \, {\left (B \log \left (e\right ) + A\right )}}{B n}\right )} \operatorname {log\_integral}\left (\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} e^{\left (-\frac {2 \, {\left (B \log \left (e\right ) + A\right )}}{B n}\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + {\left (B b c^{2} - B a c d\right )} n}{{\left (A B^{2} b^{4} c^{2} - 2 \, A B^{2} a b^{3} c d + A B^{2} a^{2} b^{2} d^{2}\right )} g^{3} n^{2} x^{2} + 2 \, {\left (A B^{2} a b^{3} c^{2} - 2 \, A B^{2} a^{2} b^{2} c d + A B^{2} a^{3} b d^{2}\right )} g^{3} n^{2} x + {\left (A B^{2} a^{2} b^{2} c^{2} - 2 \, A B^{2} a^{3} b c d + A B^{2} a^{4} d^{2}\right )} g^{3} n^{2} + {\left ({\left (B^{3} b^{4} c^{2} - 2 \, B^{3} a b^{3} c d + B^{3} a^{2} b^{2} d^{2}\right )} g^{3} n^{2} x^{2} + 2 \, {\left (B^{3} a b^{3} c^{2} - 2 \, B^{3} a^{2} b^{2} c d + B^{3} a^{3} b d^{2}\right )} g^{3} n^{2} x + {\left (B^{3} a^{2} b^{2} c^{2} - 2 \, B^{3} a^{3} b c d + B^{3} a^{4} d^{2}\right )} g^{3} n^{2}\right )} \log \left (e\right ) + {\left ({\left (B^{3} b^{4} c^{2} - 2 \, B^{3} a b^{3} c d + B^{3} a^{2} b^{2} d^{2}\right )} g^{3} n^{3} x^{2} + 2 \, {\left (B^{3} a b^{3} c^{2} - 2 \, B^{3} a^{2} b^{2} c d + B^{3} a^{3} b d^{2}\right )} g^{3} n^{3} x + {\left (B^{3} a^{2} b^{2} c^{2} - 2 \, B^{3} a^{3} b c d + B^{3} a^{4} d^{2}\right )} g^{3} n^{3}\right )} \log \left (\frac {b x + a}{d x + c}\right )} \]

[In]

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}} \,d x } \]

[In]

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}} \,d x } \]

[In]

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\int \frac {1}{{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]

[In]

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

[Out]

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