3.2.49 \(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^3 (c i+d i x)^2} \, dx\) [149]

Optimal. Leaf size=380 \[ \frac {B d^3 n (a+b x)}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {3 b^2 B d n (c+d x)}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 B n (c+d x)^2}{4 (b c-a d)^4 g^3 i^2 (a+b x)^2}-\frac {d^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {3 b^2 d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^3 i^2 (a+b x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^3 i^2}-\frac {3 b B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^3 i^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {2561, 45, 2372, 12, 14, 2338} \begin {gather*} -\frac {b^3 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^2 (a+b x)^2 (b c-a d)^4}+\frac {3 b^2 d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (a+b x) (b c-a d)^4}-\frac {d^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (c+d x) (b c-a d)^4}+\frac {3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (b c-a d)^4}-\frac {b^3 B n (c+d x)^2}{4 g^3 i^2 (a+b x)^2 (b c-a d)^4}+\frac {3 b^2 B d n (c+d x)}{g^3 i^2 (a+b x) (b c-a d)^4}+\frac {B d^3 n (a+b x)}{g^3 i^2 (c+d x) (b c-a d)^4}-\frac {3 b B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^3 i^2 (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -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*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(149 c+149 d x)^2 (a g+b g x)^3} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^2 g^3 (a+b x)^3}-\frac {2 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)^2}-\frac {3 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (3 b^2 d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b d^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}-\frac {\left (2 b^2 d\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{22201 (b c-a d)^3 g^3}-\frac {d^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{22201 (b c-a d)^3 g^3}+\frac {b^2 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{22201 (b c-a d)^2 g^3}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}+\frac {\left (3 b B d^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}-\frac {(2 b B d n) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{22201 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{22201 (b c-a d)^3 g^3}+\frac {(b B n) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{44402 (b c-a d)^2 g^3}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{22201 (b c-a d)^4 g^3}+\frac {\left (3 b B d^2 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{22201 (b c-a d)^4 g^3}-\frac {(2 b B d n) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{22201 (b c-a d)^2 g^3}-\frac {\left (B d^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{22201 (b c-a d)^2 g^3}+\frac {(b B n) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{44402 (b c-a d) g^3}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b^2 B d^2 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}+\frac {\left (3 b^2 B d^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}+\frac {\left (3 b B d^3 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}-\frac {(2 b B d n) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{22201 (b c-a d)^2 g^3}-\frac {\left (B d^2 n\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{22201 (b c-a d)^2 g^3}+\frac {(b B n) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{44402 (b c-a d) g^3}\\ &=-\frac {b B n}{88804 (b c-a d)^2 g^3 (a+b x)^2}+\frac {5 b B d n}{44402 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2 n}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b B d^2 n \log (a+b x)}{44402 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log (c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b^2 B d^2 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}\\ &=-\frac {b B n}{88804 (b c-a d)^2 g^3 (a+b x)^2}+\frac {5 b B d n}{44402 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2 n}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b B d^2 n \log (a+b x)}{44402 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log ^2(a+b x)}{44402 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log (c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log ^2(c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{22201 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{22201 (b c-a d)^4 g^3}\\ &=-\frac {b B n}{88804 (b c-a d)^2 g^3 (a+b x)^2}+\frac {5 b B d n}{44402 (b c-a d)^3 g^3 (a+b x)}-\frac {B d^2 n}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b B d^2 n \log (a+b x)}{44402 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log ^2(a+b x)}{44402 (b c-a d)^4 g^3}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac {3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log (c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac {3 b B d^2 n \log ^2(c+d x)}{44402 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}+\frac {3 b B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.49, size = 478, normalized size = 1.26 \begin {gather*} \frac {-\frac {b B (b c-a d)^2 n}{(a+b x)^2}+\frac {8 b^2 B c d n}{a+b x}-\frac {8 a b B d^2 n}{a+b x}+\frac {2 b B d (b c-a d) n}{a+b x}-\frac {4 b B c d^2 n}{c+d x}+\frac {4 a B d^3 n}{c+d x}+6 b B d^2 n \log (a+b x)-\frac {2 b (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}+\frac {8 b d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}+\frac {4 d^2 (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}+12 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 b B d^2 n \log (c+d x)-12 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-6 b B d^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 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+6 b B d^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 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^4 g^3 i^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b*B*(b*c - a*d)^2*n)/(a + b*x)^2) + (8*b^2*B*c*d*n)/(a + b*x) - (8*a*b*B*d^2*n)/(a + b*x) + (2*b*B*d*(b*c
- a*d)*n)/(a + b*x) - (4*b*B*c*d^2*n)/(c + d*x) + (4*a*B*d^3*n)/(c + d*x) + 6*b*B*d^2*n*Log[a + b*x] - (2*b*(b
*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 + (8*b*d*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(
c + d*x))^n]))/(a + b*x) + (4*d^2*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) + 12*b*d^2*Log
[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*b*B*d^2*n*Log[c + d*x] - 12*b*d^2*(A + B*Log[e*((a + b*x)
/(c + d*x))^n])*Log[c + d*x] - 6*b*B*d^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)]) + 6*b*B*d^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)]))/(4*(b*c - a*d)^4*g^3*i^2)

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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b g x +a g \right )^{3} \left (d i x +c i \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1623 vs. \(2 (354) = 708\).
time = 0.42, size = 1623, normalized size = 4.27 \begin {gather*} -\frac {1}{2} \, B {\left (\frac {6 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 2 \, a^{2} d^{2} + 3 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x}{{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} g^{3} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} g^{3} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} g^{3} x + {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3}\right )} g^{3}} + \frac {6 \, b d^{2} \log \left (b x + a\right )}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} g^{3}} - \frac {6 \, b d^{2} \log \left (d x + c\right )}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} g^{3}}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {{\left (b^{3} c^{3} - 12 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )^{2} + 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )^{2} - 3 \, {\left (3 \, b^{3} c^{2} d - 2 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x - 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x - 2 \, {\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} + {\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} + {\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B n}{4 \, {\left (a^{2} b^{4} c^{5} g^{3} - 4 \, a^{3} b^{3} c^{4} d g^{3} + 6 \, a^{4} b^{2} c^{3} d^{2} g^{3} - 4 \, a^{5} b c^{2} d^{3} g^{3} + a^{6} c d^{4} g^{3} + {\left (b^{6} c^{4} d g^{3} - 4 \, a b^{5} c^{3} d^{2} g^{3} + 6 \, a^{2} b^{4} c^{2} d^{3} g^{3} - 4 \, a^{3} b^{3} c d^{4} g^{3} + a^{4} b^{2} d^{5} g^{3}\right )} x^{3} + {\left (b^{6} c^{5} g^{3} - 2 \, a b^{5} c^{4} d g^{3} - 2 \, a^{2} b^{4} c^{3} d^{2} g^{3} + 8 \, a^{3} b^{3} c^{2} d^{3} g^{3} - 7 \, a^{4} b^{2} c d^{4} g^{3} + 2 \, a^{5} b d^{5} g^{3}\right )} x^{2} + {\left (2 \, a b^{5} c^{5} g^{3} - 7 \, a^{2} b^{4} c^{4} d g^{3} + 8 \, a^{3} b^{3} c^{3} d^{2} g^{3} - 2 \, a^{4} b^{2} c^{2} d^{3} g^{3} - 2 \, a^{5} b c d^{4} g^{3} + a^{6} d^{5} g^{3}\right )} x\right )}} - \frac {1}{2} \, A {\left (\frac {6 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 2 \, a^{2} d^{2} + 3 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x}{{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} g^{3} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} g^{3} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} g^{3} x + {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3}\right )} g^{3}} + \frac {6 \, b d^{2} \log \left (b x + a\right )}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} g^{3}} - \frac {6 \, b d^{2} \log \left (d x + c\right )}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} g^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*B*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3*a*b*d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^
2*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)*g^3*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3
- 2*a^4*b*d^4)*g^3*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c^2*d^2 + a^4*b*c*d^3 - a^5*d^4)*g^3*x + (
a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3)*g^3) + 6*b*d^2*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c
^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3) - 6*b*d^2*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*
a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3))*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 1/4*(b^3*c^3 - 12*a*
b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c
*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b*x + a)^2 + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*
c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(d*x + c)^2 - 3*(3*b^3*c^2*d - 2*a*b^2*c*d^2 - a^
2*b*d^3)*x - 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log
(b*x + a) + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x - 2*(
b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b*x + a))*log(d
*x + c))*B*n/(a^2*b^4*c^5*g^3 - 4*a^3*b^3*c^4*d*g^3 + 6*a^4*b^2*c^3*d^2*g^3 - 4*a^5*b*c^2*d^3*g^3 + a^6*c*d^4*
g^3 + (b^6*c^4*d*g^3 - 4*a*b^5*c^3*d^2*g^3 + 6*a^2*b^4*c^2*d^3*g^3 - 4*a^3*b^3*c*d^4*g^3 + a^4*b^2*d^5*g^3)*x^
3 + (b^6*c^5*g^3 - 2*a*b^5*c^4*d*g^3 - 2*a^2*b^4*c^3*d^2*g^3 + 8*a^3*b^3*c^2*d^3*g^3 - 7*a^4*b^2*c*d^4*g^3 + 2
*a^5*b*d^5*g^3)*x^2 + (2*a*b^5*c^5*g^3 - 7*a^2*b^4*c^4*d*g^3 + 8*a^3*b^3*c^3*d^2*g^3 - 2*a^4*b^2*c^2*d^3*g^3 -
2*a^5*b*c*d^4*g^3 + a^6*d^5*g^3)*x) - 1/2*A*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d +
3*a*b*d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^2*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)*g^3*x^3 + (b^5*c^4 - a*b^4*c^3*d
- 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 - 2*a^4*b*d^4)*g^3*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c^2*
d^2 + a^4*b*c*d^3 - a^5*d^4)*g^3*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3)*g^3) + 6*b*
d^2*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3) - 6*b*d^2*log(d
*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (354) = 708\).
time = 0.41, size = 776, normalized size = 2.04 \begin {gather*} \frac {2 \, {\left (A + B\right )} b^{3} c^{3} - 12 \, {\left (A + B\right )} a b^{2} c^{2} d + 6 \, {\left (A + B\right )} a^{2} b c d^{2} + 4 \, {\left (A + B\right )} a^{3} d^{3} - 6 \, {\left (2 \, {\left (A + B\right )} b^{3} c d^{2} - 2 \, {\left (A + B\right )} a b^{2} d^{3} + {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n\right )} x^{2} - 6 \, {\left (B b^{3} d^{3} n x^{3} + B a^{2} b c d^{2} n + {\left (B b^{3} c d^{2} + 2 \, B a b^{2} d^{3}\right )} n x^{2} + {\left (2 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B b^{3} c^{3} - 12 \, B a b^{2} c^{2} d + 15 \, B a^{2} b c d^{2} - 4 \, B a^{3} d^{3}\right )} n - 3 \, {\left (2 \, {\left (A + B\right )} b^{3} c^{2} d + 4 \, {\left (A + B\right )} a b^{2} c d^{2} - 6 \, {\left (A + B\right )} a^{2} b d^{3} + {\left (3 \, B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2} - B a^{2} b d^{3}\right )} n\right )} x - 2 \, {\left (6 \, {\left (A + B\right )} a^{2} b c d^{2} + 3 \, {\left (B b^{3} d^{3} n + 2 \, {\left (A + B\right )} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (3 \, B b^{3} c d^{2} n + 2 \, {\left (A + B\right )} b^{3} c d^{2} + 4 \, {\left (A + B\right )} a b^{2} d^{3}\right )} x^{2} - {\left (B b^{3} c^{3} - 6 \, B a b^{2} c^{2} d + 2 \, B a^{3} d^{3}\right )} n + 3 \, {\left (4 \, {\left (A + B\right )} a b^{2} c d^{2} + 2 \, {\left (A + B\right )} a^{2} b d^{3} + {\left (B b^{3} c^{2} d + 4 \, B a b^{2} c d^{2} - 2 \, B a^{2} b d^{3}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{6} c^{4} d - 4 \, a b^{5} c^{3} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{3} - 4 \, a^{3} b^{3} c d^{4} + a^{4} b^{2} d^{5}\right )} g^{3} x^{3} + {\left (b^{6} c^{5} - 2 \, a b^{5} c^{4} d - 2 \, a^{2} b^{4} c^{3} d^{2} + 8 \, a^{3} b^{3} c^{2} d^{3} - 7 \, a^{4} b^{2} c d^{4} + 2 \, a^{5} b d^{5}\right )} g^{3} x^{2} + {\left (2 \, a b^{5} c^{5} - 7 \, a^{2} b^{4} c^{4} d + 8 \, a^{3} b^{3} c^{3} d^{2} - 2 \, a^{4} b^{2} c^{2} d^{3} - 2 \, a^{5} b c d^{4} + a^{6} d^{5}\right )} g^{3} x + {\left (a^{2} b^{4} c^{5} - 4 \, a^{3} b^{3} c^{4} d + 6 \, a^{4} b^{2} c^{3} d^{2} - 4 \, a^{5} b c^{2} d^{3} + a^{6} c d^{4}\right )} g^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(A + B)*b^3*c^3 - 12*(A + B)*a*b^2*c^2*d + 6*(A + B)*a^2*b*c*d^2 + 4*(A + B)*a^3*d^3 - 6*(2*(A + B)*b^3
*c*d^2 - 2*(A + B)*a*b^2*d^3 + (B*b^3*c*d^2 - B*a*b^2*d^3)*n)*x^2 - 6*(B*b^3*d^3*n*x^3 + B*a^2*b*c*d^2*n + (B*
b^3*c*d^2 + 2*B*a*b^2*d^3)*n*x^2 + (2*B*a*b^2*c*d^2 + B*a^2*b*d^3)*n*x)*log((b*x + a)/(d*x + c))^2 + (B*b^3*c^
3 - 12*B*a*b^2*c^2*d + 15*B*a^2*b*c*d^2 - 4*B*a^3*d^3)*n - 3*(2*(A + B)*b^3*c^2*d + 4*(A + B)*a*b^2*c*d^2 - 6*
(A + B)*a^2*b*d^3 + (3*B*b^3*c^2*d - 2*B*a*b^2*c*d^2 - B*a^2*b*d^3)*n)*x - 2*(6*(A + B)*a^2*b*c*d^2 + 3*(B*b^3
*d^3*n + 2*(A + B)*b^3*d^3)*x^3 + 3*(3*B*b^3*c*d^2*n + 2*(A + B)*b^3*c*d^2 + 4*(A + B)*a*b^2*d^3)*x^2 - (B*b^3
*c^3 - 6*B*a*b^2*c^2*d + 2*B*a^3*d^3)*n + 3*(4*(A + B)*a*b^2*c*d^2 + 2*(A + B)*a^2*b*d^3 + (B*b^3*c^2*d + 4*B*
a*b^2*c*d^2 - 2*B*a^2*b*d^3)*n)*x)*log((b*x + a)/(d*x + c)))/((b^6*c^4*d - 4*a*b^5*c^3*d^2 + 6*a^2*b^4*c^2*d^3
- 4*a^3*b^3*c*d^4 + a^4*b^2*d^5)*g^3*x^3 + (b^6*c^5 - 2*a*b^5*c^4*d - 2*a^2*b^4*c^3*d^2 + 8*a^3*b^3*c^2*d^3 -
7*a^4*b^2*c*d^4 + 2*a^5*b*d^5)*g^3*x^2 + (2*a*b^5*c^5 - 7*a^2*b^4*c^4*d + 8*a^3*b^3*c^3*d^2 - 2*a^4*b^2*c^2*d
^3 - 2*a^5*b*c*d^4 + a^6*d^5)*g^3*x + (a^2*b^4*c^5 - 4*a^3*b^3*c^4*d + 6*a^4*b^2*c^3*d^2 - 4*a^5*b*c^2*d^3 + a
^6*c*d^4)*g^3)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 124.71, size = 222, normalized size = 0.58 \begin {gather*} \frac {1}{4} \, {\left (\frac {2 \, {\left (B b n - \frac {2 \, {\left (b x + a\right )} B d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}} + \frac {B b n - \frac {4 \, {\left (b x + a\right )} B d n}{d x + c} + 2 \, A b + 2 \, B b - \frac {4 \, {\left (b x + a\right )} A d}{d x + c} - \frac {4 \, {\left (b x + a\right )} B d}{d x + c}}{\frac {{\left (b x + a\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 7.38, size = 1016, normalized size = 2.67 \begin {gather*} \frac {3\,B\,b\,d^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g^3\,i^2\,n\,{\left (a\,d-b\,c\right )}^4}-\frac {\frac {4\,A\,a^2\,d^2-2\,A\,b^2\,c^2-4\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+10\,A\,a\,b\,c\,d+11\,B\,a\,b\,c\,d\,n}{2\,\left (a\,d-b\,c\right )}+\frac {3\,x^2\,\left (2\,A\,b^2\,d^2+B\,b^2\,d^2\,n\right )}{a\,d-b\,c}+\frac {3\,x\,\left (6\,A\,a\,b\,d^2+2\,A\,b^2\,c\,d+B\,a\,b\,d^2\,n+3\,B\,b^2\,c\,d\,n\right )}{2\,\left (a\,d-b\,c\right )}}{x\,\left (2\,a^4\,d^3\,g^3\,i^2-6\,a^2\,b^2\,c^2\,d\,g^3\,i^2+4\,a\,b^3\,c^3\,g^3\,i^2\right )+x^2\,\left (4\,a^3\,b\,d^3\,g^3\,i^2-6\,a^2\,b^2\,c\,d^2\,g^3\,i^2+2\,b^4\,c^3\,g^3\,i^2\right )+x^3\,\left (2\,a^2\,b^2\,d^3\,g^3\,i^2-4\,a\,b^3\,c\,d^2\,g^3\,i^2+2\,b^4\,c^2\,d\,g^3\,i^2\right )+2\,a^2\,b^2\,c^3\,g^3\,i^2+2\,a^4\,c\,d^2\,g^3\,i^2-4\,a^3\,b\,c^2\,d\,g^3\,i^2}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {\frac {B\,\left (2\,a\,d+b\,c\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {3\,B\,b\,d\,x}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (d\,a^2\,g^3\,i^2+2\,b\,c\,a\,g^3\,i^2\right )+x^2\,\left (c\,b^2\,g^3\,i^2+2\,a\,d\,b\,g^3\,i^2\right )+a^2\,c\,g^3\,i^2+b^2\,d\,g^3\,i^2\,x^3}+\frac {3\,B\,b\,d^2\,\left (b\,g^3\,i^2\,n\,x^2\,\left (a\,d-b\,c\right )+\frac {a\,c\,g^3\,i^2\,n\,\left (a\,d-b\,c\right )}{d}+\frac {g^3\,i^2\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{d}\right )}{g^3\,i^2\,n\,{\left (a\,d-b\,c\right )}^4\,\left (x\,\left (d\,a^2\,g^3\,i^2+2\,b\,c\,a\,g^3\,i^2\right )+x^2\,\left (c\,b^2\,g^3\,i^2+2\,a\,d\,b\,g^3\,i^2\right )+a^2\,c\,g^3\,i^2+b^2\,d\,g^3\,i^2\,x^3\right )}\right )-\frac {b\,d^2\,\mathrm {atan}\left (\frac {b\,d^2\,\left (2\,A+B\,n\right )\,\left (\frac {a^4\,d^4\,g^3\,i^2-2\,a^3\,b\,c\,d^3\,g^3\,i^2+2\,a\,b^3\,c^3\,d\,g^3\,i^2-b^4\,c^4\,g^3\,i^2}{a^3\,d^3\,g^3\,i^2-3\,a^2\,b\,c\,d^2\,g^3\,i^2+3\,a\,b^2\,c^2\,d\,g^3\,i^2-b^3\,c^3\,g^3\,i^2}+2\,b\,d\,x\right )\,\left (a^3\,d^3\,g^3\,i^2-3\,a^2\,b\,c\,d^2\,g^3\,i^2+3\,a\,b^2\,c^2\,d\,g^3\,i^2-b^3\,c^3\,g^3\,i^2\right )\,3{}\mathrm {i}}{g^3\,i^2\,\left (6\,A\,b\,d^2+3\,B\,b\,d^2\,n\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (2\,A+B\,n\right )\,3{}\mathrm {i}}{g^3\,i^2\,{\left (a\,d-b\,c\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*B*b*d^2*log(e*((a + b*x)/(c + d*x))^n)^2)/(2*g^3*i^2*n*(a*d - b*c)^4) - ((4*A*a^2*d^2 - 2*A*b^2*c^2 - 4*B*a
^2*d^2*n - B*b^2*c^2*n + 10*A*a*b*c*d + 11*B*a*b*c*d*n)/(2*(a*d - b*c)) + (3*x^2*(2*A*b^2*d^2 + B*b^2*d^2*n))/
(a*d - b*c) + (3*x*(6*A*a*b*d^2 + 2*A*b^2*c*d + B*a*b*d^2*n + 3*B*b^2*c*d*n))/(2*(a*d - b*c)))/(x*(2*a^4*d^3*g
^3*i^2 + 4*a*b^3*c^3*g^3*i^2 - 6*a^2*b^2*c^2*d*g^3*i^2) + x^2*(2*b^4*c^3*g^3*i^2 + 4*a^3*b*d^3*g^3*i^2 - 6*a^2
*b^2*c*d^2*g^3*i^2) + x^3*(2*a^2*b^2*d^3*g^3*i^2 + 2*b^4*c^2*d*g^3*i^2 - 4*a*b^3*c*d^2*g^3*i^2) + 2*a^2*b^2*c^
3*g^3*i^2 + 2*a^4*c*d^2*g^3*i^2 - 4*a^3*b*c^2*d*g^3*i^2) - (b*d^2*atan((b*d^2*(2*A + B*n)*((a^4*d^4*g^3*i^2 -
b^4*c^4*g^3*i^2 + 2*a*b^3*c^3*d*g^3*i^2 - 2*a^3*b*c*d^3*g^3*i^2)/(a^3*d^3*g^3*i^2 - b^3*c^3*g^3*i^2 + 3*a*b^2*
c^2*d*g^3*i^2 - 3*a^2*b*c*d^2*g^3*i^2) + 2*b*d*x)*(a^3*d^3*g^3*i^2 - b^3*c^3*g^3*i^2 + 3*a*b^2*c^2*d*g^3*i^2 -
3*a^2*b*c*d^2*g^3*i^2)*3i)/(g^3*i^2*(6*A*b*d^2 + 3*B*b*d^2*n)*(a*d - b*c)^4))*(2*A + B*n)*3i)/(g^3*i^2*(a*d -
b*c)^4) - log(e*((a + b*x)/(c + d*x))^n)*(((B*(2*a*d + b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (3*B*b*d*x
)/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x*(a^2*d*g^3*i^2 + 2*a*b*c*g^3*i^2) + x^2*(b^2*c*g^3*i^2 + 2*a*b*d*g^3
*i^2) + a^2*c*g^3*i^2 + b^2*d*g^3*i^2*x^3) + (3*B*b*d^2*(b*g^3*i^2*n*x^2*(a*d - b*c) + (a*c*g^3*i^2*n*(a*d - b
*c))/d + (g^3*i^2*n*x*(a*d + b*c)*(a*d - b*c))/d))/(g^3*i^2*n*(a*d - b*c)^4*(x*(a^2*d*g^3*i^2 + 2*a*b*c*g^3*i^
2) + x^2*(b^2*c*g^3*i^2 + 2*a*b*d*g^3*i^2) + a^2*c*g^3*i^2 + b^2*d*g^3*i^2*x^3)))

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