3.313 \(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(g+h x)^2} \, dx\)

Optimal. Leaf size=302 \[ \frac {6 B^2 n^2 (b c-a d) \text {Li}_2\left (\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(b g-a h) (d g-c h)}+\frac {3 B n (b c-a d) \log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(b g-a h) (d g-c h)}+\frac {(a+b x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(g+h x) (b g-a h)}-\frac {6 B^3 n^3 (b c-a d) \text {Li}_3\left (\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)} \]

[Out]

(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(-a*h+b*g)/(h*x+g)+3*B*(-a*d+b*c)*n*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n
)))^2*ln(1-(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/(-a*h+b*g)/(-c*h+d*g)+6*B^2*(-a*d+b*c)*n^2*(A+B*ln(e*(b*x+a)
^n/((d*x+c)^n)))*polylog(2,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/(-a*h+b*g)/(-c*h+d*g)-6*B^3*(-a*d+b*c)*n^3*p
olylog(3,(-c*h+d*g)*(b*x+a)/(-a*h+b*g)/(d*x+c))/(-a*h+b*g)/(-c*h+d*g)

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Rubi [B]  time = 0.81, antiderivative size = 650, normalized size of antiderivative = 2.15, number of steps used = 14, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6742, 2490, 36, 31, 2503, 2502, 2315, 2506, 6610} \[ \frac {6 A B^2 n^2 (b c-a d) \text {PolyLog}\left (2,1-\frac {(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}+\frac {6 B^3 n^2 (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {PolyLog}\left (2,1-\frac {(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}-\frac {6 B^3 n^3 (b c-a d) \text {PolyLog}\left (3,1-\frac {(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x) (b g-a h)}-\frac {3 A^2 B n (b c-a d) \log (c+d x)}{(b g-a h) (d g-c h)}+\frac {3 A^2 B n (b c-a d) \log (g+h x)}{(b g-a h) (d g-c h)}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x) (b g-a h)}+\frac {6 A B^2 n (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x) (b g-a h)}+\frac {3 B^3 n (b c-a d) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}-\frac {A^3}{h (g+h x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A^3/(h*(g + h*x))) - (3*A^2*B*(b*c - a*d)*n*Log[c + d*x])/((b*g - a*h)*(d*g - c*h)) + (3*A^2*B*(a + b*x)*Log
[(e*(a + b*x)^n)/(c + d*x)^n])/((b*g - a*h)*(g + h*x)) + (3*A*B^2*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2
)/((b*g - a*h)*(g + h*x)) + (B^3*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3)/((b*g - a*h)*(g + h*x)) + (3*A^
2*B*(b*c - a*d)*n*Log[g + h*x])/((b*g - a*h)*(d*g - c*h)) + (6*A*B^2*(b*c - a*d)*n*Log[(e*(a + b*x)^n)/(c + d*
x)^n]*Log[((b*c - a*d)*(g + h*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h)) + (3*B^3*(b*c - a*d)*n*L
og[(e*(a + b*x)^n)/(c + d*x)^n]^2*Log[((b*c - a*d)*(g + h*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*
h)) + (6*A*B^2*(b*c - a*d)*n^2*PolyLog[2, 1 - ((b*c - a*d)*(g + h*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(
d*g - c*h)) + (6*B^3*(b*c - a*d)*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]*PolyLog[2, 1 - ((b*c - a*d)*(g + h*x))/(
(b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h)) - (6*B^3*(b*c - a*d)*n^3*PolyLog[3, 1 - ((b*c - a*d)*(g + h
*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h))

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2490

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

Rule 2502

Int[Log[((e_.)*((c_.) + (d_.)*(x_)))/((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{g = Coeff[Simplify[1/(u*(a
+ b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Dist[(b - d*e)/(h*(b*c - a*d)), Subst[Int[Log[
e*x]/(1 - e*x), x], x, (c + d*x)/(a + b*x)], x] /; EqQ[g*(b - d*e) - h*(a - c*e), 0]] /; FreeQ[{a, b, c, d, e}
, x] && NeQ[b*c - a*d, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]

Rule 2503

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Simp[(Log[e*(f*(
a + b*x)^p*(c + d*x)^q)^r]^s*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/(b*g - a*h), x] + Dist[(
p*r*s*(b*c - a*d))/(b*g - a*h), Int[(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)*Log[-(((b*c - a*d)*(g + h*x)
)/((d*g - c*h)*(a + b*x)))])/((a + b*x)*(c + d*x)), x], x] /; NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0]] /; FreeQ
[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0] && LinearQ[Simplify[1/
(u*(a + b*x))], x]

Rule 2506

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo
g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log
[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx &=\int \left (\frac {A^3}{(g+h x)^2}+\frac {3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}+\frac {3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}+\frac {B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}\right ) \, dx\\ &=-\frac {A^3}{h (g+h x)}+\left (3 A^2 B\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx+\left (3 A B^2\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx+B^3 \int \frac {\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx\\ &=-\frac {A^3}{h (g+h x)}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}-\frac {\left (3 A^2 B (b c-a d) n\right ) \int \frac {1}{(c+d x) (g+h x)} \, dx}{b g-a h}-\frac {\left (6 A B^2 (b c-a d) n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{b g-a h}-\frac {\left (3 B^3 (b c-a d) n\right ) \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{b g-a h}\\ &=-\frac {A^3}{h (g+h x)}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {6 A B^2 (b c-a d) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac {3 B^3 (b c-a d) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}-\frac {\left (3 A^2 B d (b c-a d) n\right ) \int \frac {1}{c+d x} \, dx}{(b g-a h) (d g-c h)}+\frac {\left (3 A^2 B (b c-a d) h n\right ) \int \frac {1}{g+h x} \, dx}{(b g-a h) (d g-c h)}-\frac {\left (6 A B^2 (b c-a d)^2 n^2\right ) \int \frac {\log \left (-\frac {(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h) (d g-c h)}-\frac {\left (6 B^3 (b c-a d)^2 n^2\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h) (d g-c h)}\\ &=-\frac {A^3}{h (g+h x)}-\frac {3 A^2 B (b c-a d) n \log (c+d x)}{(b g-a h) (d g-c h)}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {3 A^2 B (b c-a d) n \log (g+h x)}{(b g-a h) (d g-c h)}+\frac {6 A B^2 (b c-a d) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac {3 B^3 (b c-a d) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac {6 B^3 (b c-a d) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac {\left (6 A B^2 (b c-a d)^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {(-b c+a d) x}{b g-a h}\right )}{1+\frac {(-b c+a d) x}{b g-a h}} \, dx,x,\frac {g+h x}{c+d x}\right )}{(b g-a h)^2 (d g-c h)}-\frac {\left (6 B^3 (b c-a d)^2 n^3\right ) \int \frac {\text {Li}_2\left (1+\frac {(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h) (d g-c h)}\\ &=-\frac {A^3}{h (g+h x)}-\frac {3 A^2 B (b c-a d) n \log (c+d x)}{(b g-a h) (d g-c h)}+\frac {3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac {3 A^2 B (b c-a d) n \log (g+h x)}{(b g-a h) (d g-c h)}+\frac {6 A B^2 (b c-a d) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac {3 B^3 (b c-a d) n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac {6 A B^2 (b c-a d) n^2 \text {Li}_2\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac {6 B^3 (b c-a d) n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}-\frac {6 B^3 (b c-a d) n^3 \text {Li}_3\left (1-\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}\\ \end {align*}

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Mathematica [F]  time = 3.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(g+h x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 3 \, A B^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 3 \, A^{2} B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{3}}{h^{2} x^{2} + 2 \, g h x + g^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 3.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{3}}{\left (h x +g \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {B^{3} \log \left ({\left (d x + c\right )}^{n}\right )^{3}}{h^{2} x + g h} + \frac {3 \, {\left (\frac {b e n \log \left (b x + a\right )}{b g h - a h^{2}} - \frac {d e n \log \left (d x + c\right )}{d g h - c h^{2}} - \frac {{\left (b c e n - a d e n\right )} \log \left (h x + g\right )}{{\left (d g h - c h^{2}\right )} a - {\left (d g^{2} - c g h\right )} b}\right )} A^{2} B}{e} - \frac {3 \, A^{2} B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{h^{2} x + g h} - \frac {A^{3}}{h^{2} x + g h} + \int \frac {B^{3} c h \log \relax (e)^{3} + 3 \, A B^{2} c h \log \relax (e)^{2} + {\left (B^{3} d h x + B^{3} c h\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{3} + 3 \, {\left (B^{3} c h \log \relax (e) + A B^{2} c h + {\left (B^{3} d h \log \relax (e) + A B^{2} d h\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 3 \, {\left (A B^{2} c h - {\left (d g n - c h \log \relax (e)\right )} B^{3} - {\left ({\left (h n - h \log \relax (e)\right )} B^{3} d - A B^{2} d h\right )} x + {\left (B^{3} d h x + B^{3} c h\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + {\left (B^{3} d h \log \relax (e)^{3} + 3 \, A B^{2} d h \log \relax (e)^{2}\right )} x + 3 \, {\left (B^{3} c h \log \relax (e)^{2} + 2 \, A B^{2} c h \log \relax (e) + {\left (B^{3} d h \log \relax (e)^{2} + 2 \, A B^{2} d h \log \relax (e)\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 3 \, {\left (B^{3} c h \log \relax (e)^{2} + 2 \, A B^{2} c h \log \relax (e) + {\left (B^{3} d h x + B^{3} c h\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + {\left (B^{3} d h \log \relax (e)^{2} + 2 \, A B^{2} d h \log \relax (e)\right )} x + 2 \, {\left (B^{3} c h \log \relax (e) + A B^{2} c h + {\left (B^{3} d h \log \relax (e) + A B^{2} d h\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{d h^{3} x^{3} + c g^{2} h + {\left (2 \, d g h^{2} + c h^{3}\right )} x^{2} + {\left (d g^{2} h + 2 \, c g h^{2}\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3}{{\left (g+h\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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