3.253 \(\int \frac{\log (c (a+\frac{b}{x^2})^p)}{x^3 (d+e x)} \, dx\)

Optimal. Leaf size=414 \[ -\frac{e^2 p \text{PolyLog}\left (2,\frac{b}{a x^2}+1\right )}{2 d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{d^3}-\frac{2 e^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d^3}-\frac{e^2 \log \left (-\frac{b}{a x^2}\right ) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 d^3}-\frac{e^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d^3}+\frac{e \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d^2 x}-\frac{\left (a+\frac{b}{x^2}\right ) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 b d}+\frac{e^2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{d^3}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{-a} d-\sqrt{b} e}\right )}{d^3}-\frac{2 \sqrt{a} e p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{b} d^2}-\frac{2 e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}-\frac{2 e p}{d^2 x}+\frac{p}{2 d x^2} \]

[Out]

p/(2*d*x^2) - (2*e*p)/(d^2*x) - (2*Sqrt[a]*e*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[b]*d^2) - ((a + b/x^2)*Log[c
*(a + b/x^2)^p])/(2*b*d) + (e*Log[c*(a + b/x^2)^p])/(d^2*x) - (e^2*Log[c*(a + b/x^2)^p]*Log[-(b/(a*x^2))])/(2*
d^3) - (e^2*Log[c*(a + b/x^2)^p]*Log[d + e*x])/d^3 - (2*e^2*p*Log[-((e*x)/d)]*Log[d + e*x])/d^3 + (e^2*p*Log[(
e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/d^3 + (e^2*p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x)
)/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/d^3 - (e^2*p*PolyLog[2, 1 + b/(a*x^2)])/(2*d^3) + (e^2*p*PolyLog[2,
(Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/d^3 + (e^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sq
rt[b]*e)])/d^3 - (2*e^2*p*PolyLog[2, 1 + (e*x)/d])/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.546981, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 15, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.652, Rules used = {2466, 2454, 2389, 2295, 2455, 263, 325, 205, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ -\frac{e^2 p \text{PolyLog}\left (2,\frac{b}{a x^2}+1\right )}{2 d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{d^3}-\frac{2 e^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d^3}-\frac{e^2 \log \left (-\frac{b}{a x^2}\right ) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 d^3}-\frac{e^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d^3}+\frac{e \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d^2 x}-\frac{\left (a+\frac{b}{x^2}\right ) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 b d}+\frac{e^2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{d^3}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{-a} d-\sqrt{b} e}\right )}{d^3}-\frac{2 \sqrt{a} e p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{b} d^2}-\frac{2 e^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d^3}-\frac{2 e p}{d^2 x}+\frac{p}{2 d x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

p/(2*d*x^2) - (2*e*p)/(d^2*x) - (2*Sqrt[a]*e*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[b]*d^2) - ((a + b/x^2)*Log[c
*(a + b/x^2)^p])/(2*b*d) + (e*Log[c*(a + b/x^2)^p])/(d^2*x) - (e^2*Log[c*(a + b/x^2)^p]*Log[-(b/(a*x^2))])/(2*
d^3) - (e^2*Log[c*(a + b/x^2)^p]*Log[d + e*x])/d^3 - (2*e^2*p*Log[-((e*x)/d)]*Log[d + e*x])/d^3 + (e^2*p*Log[(
e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/d^3 + (e^2*p*Log[-((e*(Sqrt[b] + Sqrt[-a]*x)
)/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/d^3 - (e^2*p*PolyLog[2, 1 + b/(a*x^2)])/(2*d^3) + (e^2*p*PolyLog[2,
(Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/d^3 + (e^2*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sq
rt[b]*e)])/d^3 - (2*e^2*p*PolyLog[2, 1 + (e*x)/d])/d^3

Rule 2466

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

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2455

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

Rule 263

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

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 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 2462

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

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

Mathematica [A]  time = 0.289016, size = 364, normalized size = 0.88 \[ \frac{-e^2 \left (p \text{PolyLog}\left (2,\frac{b}{a x^2}+1\right )+\log \left (-\frac{b}{a x^2}\right ) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )\right )-2 e^2 p \left (-\text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )-\text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )+2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )-\log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )-\log (d+e x) \log \left (\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{b} e-\sqrt{-a} d}\right )+2 \log \left (-\frac{e x}{d}\right ) \log (d+e x)\right )+d^2 \left (\frac{p}{x^2}-\frac{\left (a+\frac{b}{x^2}\right ) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{b}\right )-2 e^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{2 d e \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{x}+4 d e p \left (\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} x}\right )}{\sqrt{b}}-\frac{1}{x}\right )}{2 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.769, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ( ex+d \right ) }\ln \left ( c \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(log(c*(a+b/x^2)^p)/x^3/(e*x+d),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{\log \left (c \left (\frac{a x^{2} + b}{x^{2}}\right )^{p}\right )}{e x^{4} + d x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log(c*((a*x^2 + b)/x^2)^p)/(e*x^4 + d*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(ln(c*(a+b/x**2)**p)/x**3/(e*x+d),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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