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

Optimal. Leaf size=460 \[ \frac{b g n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^3}+\frac{b g n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{f^3}-\frac{2 b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f^3}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac{g \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac{g \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f^2 x^2}+\frac{b d e g^{3/2} n \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 f^{5/2} \left (d^2 g+e^2 f\right )}-\frac{b e^2 g n \log \left (f+g x^2\right )}{4 f^2 \left (d^2 g+e^2 f\right )}+\frac{b e^2 g n \log (d+e x)}{2 f^2 \left (d^2 g+e^2 f\right )}-\frac{b e^2 n \log (x)}{2 d^2 f^2}+\frac{b e^2 n \log (d+e x)}{2 d^2 f^2}-\frac{b e n}{2 d f^2 x} \]

[Out]

-(b*e*n)/(2*d*f^2*x) + (b*d*e*g^(3/2)*n*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(2*f^(5/2)*(e^2*f + d^2*g)) - (b*e^2*n*Lo
g[x])/(2*d^2*f^2) + (b*e^2*n*Log[d + e*x])/(2*d^2*f^2) + (b*e^2*g*n*Log[d + e*x])/(2*f^2*(e^2*f + d^2*g)) - (a
+ b*Log[c*(d + e*x)^n])/(2*f^2*x^2) - (g*(a + b*Log[c*(d + e*x)^n]))/(2*f^2*(f + g*x^2)) - (2*g*Log[-((e*x)/d
)]*(a + b*Log[c*(d + e*x)^n]))/f^3 + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f]
+ d*Sqrt[g])])/f^3 + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/f
^3 - (b*e^2*g*n*Log[f + g*x^2])/(4*f^2*(e^2*f + d^2*g)) + (b*g*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f]
- d*Sqrt[g]))])/f^3 + (b*g*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3 - (2*b*g*n*PolyLog[
2, 1 + (e*x)/d])/f^3

________________________________________________________________________________________

Rubi [A]  time = 0.516832, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 14, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.518, Rules used = {266, 44, 2416, 2395, 2394, 2315, 2413, 706, 31, 635, 205, 260, 2393, 2391} \[ \frac{b g n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{f^3}+\frac{b g n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{f^3}-\frac{2 b g n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f^3}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f^2 \left (f+g x^2\right )}-\frac{2 g \log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac{g \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}+\frac{g \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^3}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 f^2 x^2}+\frac{b d e g^{3/2} n \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 f^{5/2} \left (d^2 g+e^2 f\right )}-\frac{b e^2 g n \log \left (f+g x^2\right )}{4 f^2 \left (d^2 g+e^2 f\right )}+\frac{b e^2 g n \log (d+e x)}{2 f^2 \left (d^2 g+e^2 f\right )}-\frac{b e^2 n \log (x)}{2 d^2 f^2}+\frac{b e^2 n \log (d+e x)}{2 d^2 f^2}-\frac{b e n}{2 d f^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*n)/(2*d*f^2*x) + (b*d*e*g^(3/2)*n*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(2*f^(5/2)*(e^2*f + d^2*g)) - (b*e^2*n*Lo
g[x])/(2*d^2*f^2) + (b*e^2*n*Log[d + e*x])/(2*d^2*f^2) + (b*e^2*g*n*Log[d + e*x])/(2*f^2*(e^2*f + d^2*g)) - (a
+ b*Log[c*(d + e*x)^n])/(2*f^2*x^2) - (g*(a + b*Log[c*(d + e*x)^n]))/(2*f^2*(f + g*x^2)) - (2*g*Log[-((e*x)/d
)]*(a + b*Log[c*(d + e*x)^n]))/f^3 + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f]
+ d*Sqrt[g])])/f^3 + (g*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/f
^3 - (b*e^2*g*n*Log[f + g*x^2])/(4*f^2*(e^2*f + d^2*g)) + (b*g*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f]
- d*Sqrt[g]))])/f^3 + (b*g*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/f^3 - (2*b*g*n*PolyLog[
2, 1 + (e*x)/d])/f^3

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2416

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 2395

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

Rule 2394

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 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 2413

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 706

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 + a*e^2), Int[1/(d + e*x), x],
x] + Dist[1/(c*d^2 + a*e^2), Int[(c*d - c*e*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a
*e^2, 0]

Rule 31

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

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

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

Rule 2393

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 2391

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]

Rubi steps

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

Mathematica [C]  time = 1.32859, size = 596, normalized size = 1.3 \[ \frac{b n \left (4 g \left (\text{PolyLog}\left (2,-\frac{i \sqrt{g} (d+e x)}{e \sqrt{f}-i d \sqrt{g}}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt{f}+i \sqrt{g} x\right )}{e \sqrt{f}-i d \sqrt{g}}\right )\right )+4 g \left (\text{PolyLog}\left (2,\frac{i \sqrt{g} (d+e x)}{e \sqrt{f}+i d \sqrt{g}}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt{f}-i \sqrt{g} x\right )}{e \sqrt{f}+i d \sqrt{g}}\right )\right )-8 g \left (\text{PolyLog}\left (2,\frac{e x}{d}+1\right )+\log \left (-\frac{e x}{d}\right ) \log (d+e x)\right )-\frac{2 f \left (\left (d^2-e^2 x^2\right ) \log (d+e x)+d e x+e^2 x^2 \log (x)\right )}{d^2 x^2}+\frac{i \sqrt{f} g \left (\sqrt{g} (d+e x) \log (d+e x)+i e \left (\sqrt{f}+i \sqrt{g} x\right ) \log \left (-\sqrt{g} x+i \sqrt{f}\right )\right )}{\left (\sqrt{f}+i \sqrt{g} x\right ) \left (e \sqrt{f}-i d \sqrt{g}\right )}+\frac{i \sqrt{f} g \left (-\sqrt{g} (d+e x) \log (d+e x)+e \left (\sqrt{g} x+i \sqrt{f}\right ) \log \left (\sqrt{g} x+i \sqrt{f}\right )\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (e \sqrt{f}+i d \sqrt{g}\right )}\right )+4 g \log \left (f+g x^2\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )-\frac{2 f g \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{f+g x^2}-\frac{2 f \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{x^2}-8 g \log (x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{4 f^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])/(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]))/x^2 - (2*f*g*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n
]))/(f + g*x^2) - 8*g*Log[x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n]) + 4*g*(a - b*n*Log[d + e*x] + b*Log
[c*(d + e*x)^n])*Log[f + g*x^2] + b*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*Sqr
t[f] + Sqrt[g]*x)*Log[I*Sqrt[f] + Sqrt[g]*x]))/((e*Sqrt[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*S
qrt[f] - I*d*Sqrt[g])]) + 4*g*(Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])] + PolyL
og[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])))/(4*f^3)

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Maple [C]  time = 0.431, size = 1165, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*e^2*n*ln(x)/d^2/f^2-1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/f^2*g/(g*x^2+f)+1/2*b*e*n/f^2*g^
2/(d^2*g+e^2*f)*d/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+2*b*n/f^3*g*ln(x)*ln((e*x+d)/d)-1/2*b*ln((e*x+d)^n)/f^2/
x^2-1/2*b*ln(c)/f^2/x^2-1/2*a/f^2/x^2-1/2*b*ln(c)/f^2*g/(g*x^2+f)+b*ln(c)/f^3*g*ln(g*x^2+f)-2*b*ln((e*x+d)^n)/
f^3*g*ln(x)-1/2*b*ln((e*x+d)^n)/f^2*g/(g*x^2+f)+b*ln((e*x+d)^n)/f^3*g*ln(g*x^2+f)+2*b*n/f^3*g*dilog((e*x+d)/d)
-2*b*ln(c)/f^3*g*ln(x)+b*n/f^3*g*ln(e*x+d)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))-b*n/f^3*g*l
n(e*x+d)*ln(g*x^2+f)+1/2*b*e^4*n/f/(d^2*g+e^2*f)/d^2*ln(e*x+d)-I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/f^3*g*ln
(x)-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/f^3*g*ln(x)-1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/f^2*
g/(g*x^2+f)+a/f^3*g*ln(g*x^2+f)-1/2*a/f^2*g/(g*x^2+f)+b*n/f^3*g*dilog((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)
^(1/2)+d*g))+b*n/f^3*g*dilog((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))-2*a/f^3*g*ln(x)+1/2*I*b*Pi*c
sgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/f^3*g*ln(g*x^2+f)+1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/f^2/x^2+b*n/f^3*g*ln
(e*x+d)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*
c*(e*x+d)^n)/f^2/x^2+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/f^3*g*ln(g*x^2+f)-1/4*I*b*Pi*csgn(I*(e*x+d)^n)
*csgn(I*c*(e*x+d)^n)^2/f^2/x^2+1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/f^2*g/(g*x^2+f)-1/4*I*b*Pi*csgn(I*c)*csgn(I*c*
(e*x+d)^n)^2/f^2/x^2-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/f^3*g*ln(g*x^2+f)+I*b*Pi*csgn(I*c*(e*x+d)^n)^3/f^3*g*ln(
x)+I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/f^3*g*ln(x)-1/2*b*e*n/d/f^2/x-1/2*I*b*Pi*csgn(I*c)*c
sgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/f^3*g*ln(g*x^2+f)+1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)
^n)/f^2*g/(g*x^2+f)+b*e^2*g*n*ln(e*x+d)/f^2/(d^2*g+e^2*f)-1/4*b*e^2*g*n*ln(g*x^2+f)/f^2/(d^2*g+e^2*f)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, a{\left (\frac{2 \, g x^{2} + f}{f^{2} g x^{4} + f^{3} x^{2}} - \frac{2 \, g \log \left (g x^{2} + f\right )}{f^{3}} + \frac{4 \, g \log \left (x\right )}{f^{3}}\right )} + b \int \frac{\log \left ({\left (e x + d\right )}^{n}\right ) + \log \left (c\right )}{g^{2} x^{7} + 2 \, f g x^{5} + f^{2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g^{2} x^{7} + 2 \, f g x^{5} + f^{2} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x^{2} + f\right )}^{2} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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