\(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^2}{(g+h x)^3} \, dx\) [308]

Optimal result

Integrand size = 33, antiderivative size = 393 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\frac {B (b c-a d) h n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b g-a h)^2 (d g-c h) (g+h x)}+\frac {b^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 h (b g-a h)^2}-\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 h (g+h x)^2}+\frac {B^2 (b c-a d)^2 h n^2 \log \left (\frac {g+h x}{c+d x}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {B (b c-a d) (2 b d g-b c h-a d h) n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2573, 2553, 2398, 2404, 2338, 2351, 31, 2354, 2438} \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\frac {b^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 h (b g-a h)^2}+\frac {B h n (a+b x) (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(g+h x) (b g-a h)^2 (d g-c h)}+\frac {B n (b c-a d) (-a d h-b c h+2 b d g) \log \left (1-\frac {(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(b g-a h)^2 (d g-c h)^2}-\frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 h (g+h x)^2}+\frac {B^2 n^2 (b c-a d) (-a d h-b c h+2 b d g) \operatorname {PolyLog}\left (2,\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac {B^2 h n^2 (b c-a d)^2 \log \left (\frac {g+h x}{c+d x}\right )}{(b g-a h)^2 (d g-c h)^2} \]

[In]

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

[Out]

(B*(b*c - a*d)*h*n*(a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*g - a*h)^2*(d*g - c*h)*(g + h*x)) +
(b^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*h*(b*g - a*h)^2) - (A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n
])^2/(2*h*(g + h*x)^2) + (B^2*(b*c - a*d)^2*h*n^2*Log[(g + h*x)/(c + d*x)])/((b*g - a*h)^2*(d*g - c*h)^2) + (B
*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*Log[1 - ((d*g - c*h)*(a + b*
x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^2) + (B^2*(b*c - a*d)*(2*b*d*g - b*c*h - a*d*h)*n^2*P
olyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)^2*(d*g - c*h)^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]

Rule 2573

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^
n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(13182\) vs. \(2(393)=786\).

Time = 5.23 (sec) , antiderivative size = 13182, normalized size of antiderivative = 33.54 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(g+h x)^3} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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