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

Optimal. Leaf size=560 \[ \frac{3 b \sqrt{g} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{4 (-f)^{5/2}}-\frac{3 b \sqrt{g} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{4 (-f)^{5/2}}+\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac{3 \sqrt{g} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 (-f)^{5/2}}+\frac{3 \sqrt{g} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 (-f)^{5/2}}-\frac{b e \sqrt{g} n \log (d+e x)}{4 f^2 \left (d \sqrt{g}+e \sqrt{-f}\right )}+\frac{b e \sqrt{g} n \log \left (\sqrt{-f}-\sqrt{g} x\right )}{4 f^2 \left (d \sqrt{g}+e \sqrt{-f}\right )}+\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2}-\frac{b e \sqrt{g} n \log (d+e x)}{4 f \left (d f \sqrt{g}+e (-f)^{3/2}\right )}+\frac{b e \sqrt{g} n \log \left (\sqrt{-f}+\sqrt{g} x\right )}{4 f \left (d f \sqrt{g}+e (-f)^{3/2}\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.819267, antiderivative size = 560, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 12, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {290, 325, 205, 2416, 2395, 36, 29, 31, 2409, 2394, 2393, 2391} \[ \frac{3 b \sqrt{g} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{4 (-f)^{5/2}}-\frac{3 b \sqrt{g} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{4 (-f)^{5/2}}+\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac{3 \sqrt{g} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 (-f)^{5/2}}+\frac{3 \sqrt{g} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 (-f)^{5/2}}-\frac{b e \sqrt{g} n \log (d+e x)}{4 f^2 \left (d \sqrt{g}+e \sqrt{-f}\right )}+\frac{b e \sqrt{g} n \log \left (\sqrt{-f}-\sqrt{g} x\right )}{4 f^2 \left (d \sqrt{g}+e \sqrt{-f}\right )}+\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2}-\frac{b e \sqrt{g} n \log (d+e x)}{4 f \left (d f \sqrt{g}+e (-f)^{3/2}\right )}+\frac{b e \sqrt{g} n \log \left (\sqrt{-f}+\sqrt{g} x\right )}{4 f \left (d f \sqrt{g}+e (-f)^{3/2}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 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 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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
 && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 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 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^2 \left (f+g x^2\right )^2} \, dx &=\int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f \left (f+g x^2\right )^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^2} \, dx}{f^2}-\frac{g \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x^2} \, dx}{f^2}-\frac{g \int \frac{a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx}{f}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac{g \int \left (\frac{\sqrt{-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\sqrt{-f} \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}-\frac{g \int \left (-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f \left (\sqrt{-f} \sqrt{g}-g x\right )^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f \left (\sqrt{-f} \sqrt{g}+g x\right )^2}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx}{f}+\frac{(b e n) \int \frac{1}{x (d+e x)} \, dx}{f^2}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac{g \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 (-f)^{5/2}}+\frac{g \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 (-f)^{5/2}}+\frac{g^2 \int \frac{a+b \log \left (c (d+e x)^n\right )}{\left (\sqrt{-f} \sqrt{g}-g x\right )^2} \, dx}{4 f^2}+\frac{g^2 \int \frac{a+b \log \left (c (d+e x)^n\right )}{\left (\sqrt{-f} \sqrt{g}+g x\right )^2} \, dx}{4 f^2}+\frac{g^2 \int \frac{a+b \log \left (c (d+e x)^n\right )}{-f g-g^2 x^2} \, dx}{2 f^2}+\frac{(b e n) \int \frac{1}{x} \, dx}{d f^2}-\frac{\left (b e^2 n\right ) \int \frac{1}{d+e x} \, dx}{d f^2}\\ &=\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{\sqrt{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 )}{2 (-f)^{5/2}}+\frac{\sqrt{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 )}{2 (-f)^{5/2}}+\frac{g^2 \int \left (-\frac{\sqrt{-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{2 f^2}+\frac{\left (b e \sqrt{g} n\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}}-\frac{\left (b e \sqrt{g} n\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}}-\frac{(b e g n) \int \frac{1}{(d+e x) \left (\sqrt{-f} \sqrt{g}-g x\right )} \, dx}{4 f^2}+\frac{(b e g n) \int \frac{1}{(d+e x) \left (\sqrt{-f} \sqrt{g}+g x\right )} \, dx}{4 f^2}\\ &=\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{\sqrt{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 )}{2 (-f)^{5/2}}+\frac{\sqrt{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 )}{2 (-f)^{5/2}}+\frac{g \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{4 (-f)^{5/2}}+\frac{g \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{4 (-f)^{5/2}}-\frac{\left (b \sqrt{g} n\right ) \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 )}{2 (-f)^{5/2}}+\frac{\left (b \sqrt{g} n\right ) \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 )}{2 (-f)^{5/2}}+\frac{\left (b e^2 \sqrt{g} n\right ) \int \frac{1}{d+e x} \, dx}{4 f^2 \left (e \sqrt{-f}-d \sqrt{g}\right )}-\frac{\left (b e^2 \sqrt{g} n\right ) \int \frac{1}{d+e x} \, dx}{4 f^2 \left (e \sqrt{-f}+d \sqrt{g}\right )}-\frac{\left (b e g^{3/2} n\right ) \int \frac{1}{\sqrt{-f} \sqrt{g}+g x} \, dx}{4 f^2 \left (e \sqrt{-f}-d \sqrt{g}\right )}-\frac{\left (b e g^{3/2} n\right ) \int \frac{1}{\sqrt{-f} \sqrt{g}-g x} \, dx}{4 f^2 \left (e \sqrt{-f}+d \sqrt{g}\right )}\\ &=\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2}+\frac{b e \sqrt{g} n \log (d+e x)}{4 f^2 \left (e \sqrt{-f}-d \sqrt{g}\right )}-\frac{b e \sqrt{g} n \log (d+e x)}{4 f^2 \left (e \sqrt{-f}+d \sqrt{g}\right )}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{b e \sqrt{g} n \log \left (\sqrt{-f}-\sqrt{g} x\right )}{4 f^2 \left (e \sqrt{-f}+d \sqrt{g}\right )}-\frac{3 \sqrt{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 )}{4 (-f)^{5/2}}-\frac{b e \sqrt{g} n \log \left (\sqrt{-f}+\sqrt{g} x\right )}{4 f^2 \left (e \sqrt{-f}-d \sqrt{g}\right )}+\frac{3 \sqrt{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 )}{4 (-f)^{5/2}}+\frac{b \sqrt{g} n \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{b \sqrt{g} n \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 (-f)^{5/2}}+\frac{\left (b e \sqrt{g} n\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}{4 (-f)^{5/2}}-\frac{\left (b e \sqrt{g} n\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}{4 (-f)^{5/2}}\\ &=\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2}+\frac{b e \sqrt{g} n \log (d+e x)}{4 f^2 \left (e \sqrt{-f}-d \sqrt{g}\right )}-\frac{b e \sqrt{g} n \log (d+e x)}{4 f^2 \left (e \sqrt{-f}+d \sqrt{g}\right )}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{b e \sqrt{g} n \log \left (\sqrt{-f}-\sqrt{g} x\right )}{4 f^2 \left (e \sqrt{-f}+d \sqrt{g}\right )}-\frac{3 \sqrt{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 )}{4 (-f)^{5/2}}-\frac{b e \sqrt{g} n \log \left (\sqrt{-f}+\sqrt{g} x\right )}{4 f^2 \left (e \sqrt{-f}-d \sqrt{g}\right )}+\frac{3 \sqrt{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 )}{4 (-f)^{5/2}}+\frac{b \sqrt{g} n \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{b \sqrt{g} n \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 (-f)^{5/2}}-\frac{\left (b \sqrt{g} n\right ) \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 )}{4 (-f)^{5/2}}+\frac{\left (b \sqrt{g} n\right ) \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 )}{4 (-f)^{5/2}}\\ &=\frac{b e n \log (x)}{d f^2}-\frac{b e n \log (d+e x)}{d f^2}+\frac{b e \sqrt{g} n \log (d+e x)}{4 f^2 \left (e \sqrt{-f}-d \sqrt{g}\right )}-\frac{b e \sqrt{g} n \log (d+e x)}{4 f^2 \left (e \sqrt{-f}+d \sqrt{g}\right )}-\frac{a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{b e \sqrt{g} n \log \left (\sqrt{-f}-\sqrt{g} x\right )}{4 f^2 \left (e \sqrt{-f}+d \sqrt{g}\right )}-\frac{3 \sqrt{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 )}{4 (-f)^{5/2}}-\frac{b e \sqrt{g} n \log \left (\sqrt{-f}+\sqrt{g} x\right )}{4 f^2 \left (e \sqrt{-f}-d \sqrt{g}\right )}+\frac{3 \sqrt{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 )}{4 (-f)^{5/2}}+\frac{3 b \sqrt{g} n \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{4 (-f)^{5/2}}-\frac{3 b \sqrt{g} n \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{4 (-f)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.954445, size = 476, normalized size = 0.85 \[ \frac{1}{4} \left (\frac{3 b \sqrt{g} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{(-f)^{5/2}}-\frac{3 b \sqrt{g} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{(-f)^{5/2}}+\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 \left (\sqrt{-f}-\sqrt{g} x\right )}-\frac{\sqrt{g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 \left (\sqrt{-f}+\sqrt{g} x\right )}-\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x}-\frac{3 \sqrt{g} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}+\frac{3 \sqrt{g} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(-f)^{5/2}}-\frac{b e \sqrt{g} n \left (\log (d+e x)-\log \left (\sqrt{-f}-\sqrt{g} x\right )\right )}{f^2 \left (d \sqrt{g}+e \sqrt{-f}\right )}+\frac{b e \sqrt{g} n \left (\log (d+e x)-\log \left (\sqrt{-f}+\sqrt{g} x\right )\right )}{f^2 \left (e \sqrt{-f}-d \sqrt{g}\right )}+\frac{4 b e n (\log (x)-\log (d+e x))}{d f^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.654, size = 2032, normalized size = 3.6 \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^2/(g*x^2+f)^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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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^{6} + 2 \, f g x^{4} + f^{2} x^{2}}, 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^2/(g*x^2+f)^2,x, algorithm="fricas")

[Out]

integral((b*log((e*x + d)^n*c) + a)/(g^2*x^6 + 2*f*g*x^4 + f^2*x^2), 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**2/(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^{2}}\,{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^2/(g*x^2+f)^2,x, algorithm="giac")

[Out]

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