\(\int \frac {(a+b \log (c (d+e x)^n))^2}{x^3 (f+g x^2)^2} \, dx\) [324]

Optimal result

Integrand size = 29, antiderivative size = 970 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )^2} \, dx=\frac {b^2 e^2 n^2 \log (x)}{d^2 f^2}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}+\frac {e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac {b e \left (e f+d \sqrt {-f} \sqrt {g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^3 \left (e^2 f+d^2 g\right )}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}-\frac {b e \left (e f-d \sqrt {-f} \sqrt {g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^3 \left (e^2 f+d^2 g\right )}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{d^2 f^2}+\frac {b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right )}{d^2 f^2}-\frac {b^2 e \left (e \sqrt {-f}+d \sqrt {g}\right ) g n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {b^2 e \left (e f+d \sqrt {-f} \sqrt {g}\right ) g n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^3 \left (e^2 f+d^2 g\right )}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}-\frac {4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{f^3}-\frac {2 b^2 g n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {2 b^2 g n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}+\frac {4 b^2 g n^2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )}{f^3} \]

[Out]

b^2*e^2*n^2*ln(x)/d^2/f^2-b*e*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))/d^2/f^2/x+1/2*e^2*g*(a+b*ln(c*(e*x+d)^n))^2/f^2/
(d^2*g+e^2*f)-1/2*(a+b*ln(c*(e*x+d)^n))^2/f^2/x^2-1/2*g*(a+b*ln(c*(e*x+d)^n))^2/f^2/(g*x^2+f)-2*g*ln(-e*x/d)*(
a+b*ln(c*(e*x+d)^n))^2/f^3-b*e^2*n*(a+b*ln(c*(e*x+d)^n))*ln(1-d/(e*x+d))/d^2/f^2+g*(a+b*ln(c*(e*x+d)^n))^2*ln(
e*((-f)^(1/2)-x*g^(1/2))/(e*(-f)^(1/2)+d*g^(1/2)))/f^3+g*(a+b*ln(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)+x*g^(1/2))/(
e*(-f)^(1/2)-d*g^(1/2)))/f^3+b^2*e^2*n^2*polylog(2,d/(e*x+d))/d^2/f^2-4*b*g*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,
1+e*x/d)/f^3+2*b*g*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))/f^3+2*b*g*n*(a
+b*ln(c*(e*x+d)^n))*polylog(2,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))/f^3+4*b^2*g*n^2*polylog(3,1+e*x/d)/f^3
-2*b^2*g*n^2*polylog(3,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))/f^3-2*b^2*g*n^2*polylog(3,(e*x+d)*g^(1/2)/(e
*(-f)^(1/2)+d*g^(1/2)))/f^3-1/2*b^2*e*g*n^2*polylog(2,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))*(e*(-f)^(1/2)
+d*g^(1/2))/(-f)^(5/2)/(d^2*g+e^2*f)-1/2*b*e*g*n*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)+x*g^(1/2))/(e*(-f)^(1/
2)-d*g^(1/2)))*(e*f-d*(-f)^(1/2)*g^(1/2))/f^3/(d^2*g+e^2*f)-1/2*b*e*g*n*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)
-x*g^(1/2))/(e*(-f)^(1/2)+d*g^(1/2)))*(e*f+d*(-f)^(1/2)*g^(1/2))/f^3/(d^2*g+e^2*f)-1/2*b^2*e*g*n^2*polylog(2,(
e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))*(e*f+d*(-f)^(1/2)*g^(1/2))/f^3/(d^2*g+e^2*f)

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 970, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {2463, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2443, 2481, 2421, 6724, 2460, 2465, 2437, 2338, 2441, 2440} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )^2} \, dx=\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 e^2}{2 f^2 \left (g d^2+e^2 f\right )}+\frac {b^2 n^2 \log (x) e^2}{d^2 f^2}-\frac {b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right ) e^2}{d^2 f^2}+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x}\right ) e^2}{d^2 f^2}-\frac {b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) e}{d^2 f^2 x}-\frac {b \left (\sqrt {-f} \sqrt {g} d+e f\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right ) e}{2 f^3 \left (g d^2+e^2 f\right )}-\frac {b \left (e f-d \sqrt {-f} \sqrt {g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) e}{2 f^3 \left (g d^2+e^2 f\right )}-\frac {b^2 \left (\sqrt {g} d+e \sqrt {-f}\right ) g n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) e}{2 (-f)^{5/2} \left (g d^2+e^2 f\right )}-\frac {b^2 \left (\sqrt {-f} \sqrt {g} d+e f\right ) g n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) e}{2 f^3 \left (g d^2+e^2 f\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (g x^2+f\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right )}{f^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{f^3}-\frac {4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{f^3}-\frac {2 b^2 g n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {2 b^2 g n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{f^3}+\frac {4 b^2 g n^2 \operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right )}{f^3} \]

[In]

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

[Out]

(b^2*e^2*n^2*Log[x])/(d^2*f^2) - (b*e*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(d^2*f^2*x) + (e^2*g*(a + b*Log[
c*(d + e*x)^n])^2)/(2*f^2*(e^2*f + d^2*g)) - (a + b*Log[c*(d + e*x)^n])^2/(2*f^2*x^2) - (g*(a + b*Log[c*(d + e
*x)^n])^2)/(2*f^2*(f + g*x^2)) - (2*g*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n])^2)/f^3 - (b*e*(e*f + d*Sqrt[-
f]*Sqrt[g])*g*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^3*(e
^2*f + d^2*g)) + (g*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3
- (b*e*(e*f - d*Sqrt[-f]*Sqrt[g])*g*n*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] -
d*Sqrt[g])])/(2*f^3*(e^2*f + d^2*g)) + (g*(a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt
[-f] - d*Sqrt[g])])/f^3 - (b*e^2*n*(a + b*Log[c*(d + e*x)^n])*Log[1 - d/(d + e*x)])/(d^2*f^2) + (b^2*e^2*n^2*P
olyLog[2, d/(d + e*x)])/(d^2*f^2) - (b^2*e*(e*Sqrt[-f] + d*Sqrt[g])*g*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*
Sqrt[-f] - d*Sqrt[g]))])/(2*(-f)^(5/2)*(e^2*f + d^2*g)) + (2*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, -((Sq
rt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^3 - (b^2*e*(e*f + d*Sqrt[-f]*Sqrt[g])*g*n^2*PolyLog[2, (Sqrt[g]
*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*f^3*(e^2*f + d^2*g)) + (2*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2
, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3 - (4*b*g*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, 1 + (e*x
)/d])/f^3 - (2*b^2*g*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/f^3 - (2*b^2*g*n^2*PolyL
og[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3 + (4*b^2*g*n^2*PolyLog[3, 1 + (e*x)/d])/f^3

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 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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 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 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2443

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

Rule 2445

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

Rule 2458

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

Rule 2460

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

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2465

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2481

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 6724

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 x^3}-\frac {2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3 x}+\frac {g^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 \left (f+g x^2\right )^2}+\frac {2 g^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3 \left (f+g x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx}{f^2}-\frac {(2 g) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{f^3}+\frac {\left (2 g^2\right ) \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f^3}+\frac {g^2 \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx}{f^2} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}+\frac {\left (2 g^2\right ) \int \left (-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f^3}+\frac {(b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx}{f^2}+\frac {(4 b e g n) \int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{f^3}+\frac {(b e g n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \left (f+g x^2\right )} \, dx}{f^2} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac {g^{3/2} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{f^3}+\frac {g^{3/2} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{f^3}+\frac {(b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{f^2}+\frac {(4 b g n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x\right )}{f^3}+\frac {(b e g n) \int \left (\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (e^2 f+d^2 g\right ) (d+e x)}-\frac {g (-d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (e^2 f+d^2 g\right ) \left (f+g x^2\right )}\right ) \, dx}{f^2} \\ & = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3}+\frac {(b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{d f^2}-\frac {(b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e x\right )}{d f^2}-\frac {(2 b e g n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{f^3}-\frac {(2 b e g n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{f^3}+\frac {\left (b e^3 g n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{d+e x} \, dx}{f^2 \left (e^2 f+d^2 g\right )}-\frac {\left (b e g^2 n\right ) \int \frac {(-d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{f^2 \left (e^2 f+d^2 g\right )}+\frac {\left (4 b^2 g n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{f^3} \\ & = -\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{d^2 f^2}-\frac {4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3}+\frac {4 b^2 g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{f^3}-\frac {(2 b g n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}+d \sqrt {g}}{e}-\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{f^3}-\frac {(2 b g n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}-d \sqrt {g}}{e}+\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{f^3}+\frac {\left (b e^2 g n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{f^2 \left (e^2 f+d^2 g\right )}-\frac {\left (b e g^2 n\right ) \int \left (\frac {\left (-d \sqrt {-f}-\frac {e f}{\sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\left (-d \sqrt {-f}+\frac {e f}{\sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f^2 \left (e^2 f+d^2 g\right )}+\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{d^2 f^2}+\frac {\left (b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+e x\right )}{d^2 f^2} \\ & = \frac {b^2 e^2 n^2 \log (x)}{d^2 f^2}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}+\frac {e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{d^2 f^2}+\frac {b^2 e^2 n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{d^2 f^2}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}-\frac {4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3}+\frac {4 b^2 g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{f^3}-\frac {\left (b e \left (\frac {d}{\sqrt {-f}}+\frac {e}{\sqrt {g}}\right ) g^2 n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 f^2 \left (e^2 f+d^2 g\right )}+\frac {\left (b e \left (\frac {d f}{(-f)^{3/2}}+\frac {e}{\sqrt {g}}\right ) g^2 n\right ) \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac {\left (2 b^2 g n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{f^3}-\frac {\left (2 b^2 g n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{f^3} \\ & = \frac {b^2 e^2 n^2 \log (x)}{d^2 f^2}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}+\frac {e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac {b e \left (e f+d \sqrt {-f} \sqrt {g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^3 \left (e^2 f+d^2 g\right )}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}-\frac {b e \left (e \sqrt {-f}+d \sqrt {g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{d^2 f^2}+\frac {b^2 e^2 n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{d^2 f^2}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}-\frac {4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3}-\frac {2 b^2 g n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {2 b^2 g n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}+\frac {4 b^2 g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{f^3}+\frac {\left (b^2 e^2 \left (e \sqrt {-f}+d \sqrt {g}\right ) g n^2\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{2 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac {\left (b^2 e^2 \left (e f+d \sqrt {-f} \sqrt {g}\right ) g n^2\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{2 f^3 \left (e^2 f+d^2 g\right )} \\ & = \frac {b^2 e^2 n^2 \log (x)}{d^2 f^2}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}+\frac {e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac {b e \left (e f+d \sqrt {-f} \sqrt {g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^3 \left (e^2 f+d^2 g\right )}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}-\frac {b e \left (e \sqrt {-f}+d \sqrt {g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{d^2 f^2}+\frac {b^2 e^2 n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{d^2 f^2}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}-\frac {4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3}-\frac {2 b^2 g n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {2 b^2 g n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}+\frac {4 b^2 g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{f^3}+\frac {\left (b^2 e \left (e \sqrt {-f}+d \sqrt {g}\right ) g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac {\left (b^2 e \left (e f+d \sqrt {-f} \sqrt {g}\right ) g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 f^3 \left (e^2 f+d^2 g\right )} \\ & = \frac {b^2 e^2 n^2 \log (x)}{d^2 f^2}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f^2 x}+\frac {e^2 g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2 \left (f+g x^2\right )}-\frac {2 g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^3}-\frac {b e \left (e f+d \sqrt {-f} \sqrt {g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^3 \left (e^2 f+d^2 g\right )}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}-\frac {b e \left (e \sqrt {-f}+d \sqrt {g}\right ) g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (1-\frac {d}{d+e x}\right )}{d^2 f^2}+\frac {b^2 e^2 n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{d^2 f^2}-\frac {b^2 e \left (e \sqrt {-f}+d \sqrt {g}\right ) g n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2} \left (e^2 f+d^2 g\right )}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {b^2 e \left (e f+d \sqrt {-f} \sqrt {g}\right ) g n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^3 \left (e^2 f+d^2 g\right )}+\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}-\frac {4 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^3}-\frac {2 b^2 g n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^3}-\frac {2 b^2 g n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^3}+\frac {4 b^2 g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{f^3} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.90 (sec) , antiderivative size = 1391, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )^2} \, dx=\frac {-\frac {2 f \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{x^2}-\frac {2 f g \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2}-8 g \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+4 g \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (f+g x^2\right )+2 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-\frac {2 f \left (d e x+e^2 x^2 \log (x)+\left (d^2-e^2 x^2\right ) \log (d+e x)\right )}{d^2 x^2}+\frac {i \sqrt {f} g \left (\sqrt {g} (d+e x) \log (d+e x)+i e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (i \sqrt {f}-\sqrt {g} x\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}+\frac {i \sqrt {f} g \left (-\sqrt {g} (d+e x) \log (d+e x)+e \left (i \sqrt {f}+\sqrt {g} x\right ) \log \left (i \sqrt {f}+\sqrt {g} x\right )\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+4 g \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+4 g \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )-8 g \left (\log \left (-\frac {e x}{d}\right ) \log (d+e x)+\operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )\right )\right )+b^2 n^2 \left (\frac {i \sqrt {f} g \left (-\sqrt {g} (d+e x) \log ^2(d+e x)+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}-\frac {\sqrt {f} g \left (\log (d+e x) \left (-i \sqrt {g} (d+e x) \log (d+e x)+2 e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+2 e \left (\sqrt {f}+i \sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}-\frac {2 f \left (-2 e^2 \log (x)+\frac {\log (d+e x) \left (2 e^2 x^2 \log \left (-\frac {e x}{d}\right )+(d+e x) (2 e x+(d-e x) \log (d+e x))\right )}{x^2}+2 e^2 \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )\right )}{d^2}+4 g \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )-2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )+4 g \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )-2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )-8 g \left (\log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)+2 \log (d+e x) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )\right )\right )}{4 f^3} \]

[In]

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

[Out]

((-2*f*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/x^2 - (2*f*g*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)
^n])^2)/(f + g*x^2) - 8*g*Log[x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 4*g*(a - b*n*Log[d + e*x] +
b*Log[c*(d + e*x)^n])^2*Log[f + g*x^2] + 2*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*((-2*f*(d*e*x +
e^2*x^2*Log[x] + (d^2 - e^2*x^2)*Log[d + e*x]))/(d^2*x^2) + (I*Sqrt[f]*g*(Sqrt[g]*(d + e*x)*Log[d + e*x] + I*e
*(Sqrt[f] + I*Sqrt[g]*x)*Log[I*Sqrt[f] - Sqrt[g]*x]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x)) + (I
*Sqrt[f]*g*(-(Sqrt[g]*(d + e*x)*Log[d + e*x]) + e*(I*Sqrt[f] + Sqrt[g]*x)*Log[I*Sqrt[f] + Sqrt[g]*x]))/((e*Sqr
t[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) + 4*g*(Log[d + e*x]*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] -
I*d*Sqrt[g])] + PolyLog[2, ((-I)*Sqrt[g]*(d + e*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) + 4*g*(Log[d + e*x]*Log[(e*(Sq
rt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])] + PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d*Sqrt[g])]
) - 8*g*(Log[-((e*x)/d)]*Log[d + e*x] + PolyLog[2, 1 + (e*x)/d])) + b^2*n^2*((I*Sqrt[f]*g*(-(Sqrt[g]*(d + e*x)
*Log[d + e*x]^2) + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*S
qrt[g])] + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d*Sqrt[g])]))/((e*Sqrt[
f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) - (Sqrt[f]*g*(Log[d + e*x]*((-I)*Sqrt[g]*(d + e*x)*Log[d + e*x] + 2
*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) + 2*e*(Sqrt[f] + I*Sqrt
[g]*x)*PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])]))/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqr
t[g]*x)) - (2*f*(-2*e^2*Log[x] + (Log[d + e*x]*(2*e^2*x^2*Log[-((e*x)/d)] + (d + e*x)*(2*e*x + (d - e*x)*Log[d
+ e*x])))/x^2 + 2*e^2*PolyLog[2, 1 + (e*x)/d]))/d^2 + 4*g*(Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/((-I)*e
*Sqrt[f] + d*Sqrt[g])] + 2*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyL
og[3, (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])]) + 4*g*(Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/(I*
e*Sqrt[f] + d*Sqrt[g])] + 2*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyLog
[3, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])]) - 8*g*(Log[-((e*x)/d)]*Log[d + e*x]^2 + 2*Log[d + e*x]*Pol
yLog[2, 1 + (e*x)/d] - 2*PolyLog[3, 1 + (e*x)/d])))/(4*f^3)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b^2*log((e*x + d)^n*c)^2 + 2*a*b*log((e*x + d)^n*c) + a^2)/(g^2*x^7 + 2*f*g*x^5 + f^2*x^3), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-1/2*a^2*((2*g*x^2 + f)/(f^2*g*x^4 + f^3*x^2) - 2*g*log(g*x^2 + f)/f^3 + 4*g*log(x)/f^3) + integrate((b^2*log(
(e*x + d)^n)^2 + b^2*log(c)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log((e*x + d)^n))/(g^2*x^7 + 2*f*g*x^5 + f
^2*x^3), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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