\(\int \frac {\log (c (a+\frac {b}{x^2})^p)}{x^3 (d+e x)} \, dx\) [253]

Optimal result

Integrand size = 23, antiderivative size = 414 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {p}{2 d x^2}-\frac {2 e p}{d^2 x}-\frac {2 \sqrt {a} e p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^2}\right )}{2 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}-\frac {2 e^2 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d^3} \]

[Out]

1/2*p/d/x^2-2*e*p/d^2/x-1/2*(a+b/x^2)*ln(c*(a+b/x^2)^p)/b/d+e*ln(c*(a+b/x^2)^p)/d^2/x-1/2*e^2*ln(c*(a+b/x^2)^p
)*ln(-b/a/x^2)/d^3-e^2*ln(c*(a+b/x^2)^p)*ln(e*x+d)/d^3-2*e^2*p*ln(-e*x/d)*ln(e*x+d)/d^3+e^2*p*ln(e*x+d)*ln(-e*
(x*(-a)^(1/2)+b^(1/2))/(d*(-a)^(1/2)-e*b^(1/2)))/d^3+e^2*p*ln(e*x+d)*ln(e*(-x*(-a)^(1/2)+b^(1/2))/(d*(-a)^(1/2
)+e*b^(1/2)))/d^3-1/2*e^2*p*polylog(2,1+b/a/x^2)/d^3-2*e^2*p*polylog(2,1+e*x/d)/d^3+e^2*p*polylog(2,(e*x+d)*(-
a)^(1/2)/(d*(-a)^(1/2)-e*b^(1/2)))/d^3+e^2*p*polylog(2,(e*x+d)*(-a)^(1/2)/(d*(-a)^(1/2)+e*b^(1/2)))/d^3-2*e*p*
arctan(x*a^(1/2)/b^(1/2))*a^(1/2)/d^2/b^(1/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {2516, 2504, 2436, 2332, 2505, 269, 331, 211, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3 (d+e x)} \, dx=-\frac {2 \sqrt {a} e p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {e^2 \log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}-\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {b}{a x^2}+1\right )}{2 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}-\frac {2 e^2 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}-\frac {2 e p}{d^2 x}+\frac {p}{2 d x^2} \]

[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 211

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

Rule 266

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 269

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 331

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 2332

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2436

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 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 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 2504

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 2505

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 2512

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 2516

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]

Rubi steps \begin{align*} \text {integral}& = \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 {\text {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,\frac {1}{x^2}\right )}{2 d}-\frac {e^2 \text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.17 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.93 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {-\frac {4 d e p}{x}+\frac {4 \sqrt {a} d e p \arctan \left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {b}}+\frac {2 d e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+d^2 \left (\frac {p}{x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{b}\right )-e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )-2 e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)-4 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)+2 e^2 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)+2 e^2 p \log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)-e^2 p \operatorname {PolyLog}\left (2,1+\frac {b}{a x^2}\right )+2 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )+2 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )-4 e^2 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 d^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.71 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.18

method	result	size
parts	\(-\frac {e^{2} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \ln \left (e x +d \right )}{d^{3}}-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {e \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{d^{2} x}+p b \left (\frac {1}{2 d b \,x^{2}}-\frac {2 e}{d^{2} b x}+\frac {a \ln \left (x \right )}{d \,b^{2}}-\frac {a \ln \left (x^{2} a +b \right )}{2 d \,b^{2}}-\frac {2 a e \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{d^{2} b \sqrt {a b}}+\frac {2 e^{2} \left (\frac {\ln \left (x \right )^{2}}{2 b}-\frac {a \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {-a x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {a x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 a}\right )}{b}\right )}{d^{3}}-\frac {2 e^{2} \left (-\frac {a \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\ln \left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )\right )}{2 a}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )}{2 a}\right )}{b}+\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b}\right )}{d^{3}}\right )\)	\(488\)

[In]

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

[Out]

-e^2*ln(c*(a+b/x^2)^p)*ln(e*x+d)/d^3-1/2*ln(c*(a+b/x^2)^p)/d/x^2+ln(c*(a+b/x^2)^p)*e^2/d^3*ln(x)+e*ln(c*(a+b/x
^2)^p)/d^2/x+p*b*(1/2/d/b/x^2-2/d^2/b*e/x+1/d*a/b^2*ln(x)-1/2/d/b^2*a*ln(a*x^2+b)-2/d^2/b*a*e/(a*b)^(1/2)*arct
an(a*x/(a*b)^(1/2))+2*e^2/d^3*(1/2/b*ln(x)^2-a/b*(1/2*ln(x)*(ln((-a*x+(-a*b)^(1/2))/(-a*b)^(1/2))+ln((a*x+(-a*
b)^(1/2))/(-a*b)^(1/2)))/a+1/2*(dilog((-a*x+(-a*b)^(1/2))/(-a*b)^(1/2))+dilog((a*x+(-a*b)^(1/2))/(-a*b)^(1/2))
)/a))-2*e^2/d^3*(-a/b*(1/2*ln(e*x+d)*(ln((e*(-a*b)^(1/2)+a*d-a*(e*x+d))/(e*(-a*b)^(1/2)+a*d))+ln((e*(-a*b)^(1/
2)-a*d+a*(e*x+d))/(e*(-a*b)^(1/2)-a*d)))/a+1/2*(dilog((e*(-a*b)^(1/2)+a*d-a*(e*x+d))/(e*(-a*b)^(1/2)+a*d))+dil
og((e*(-a*b)^(1/2)-a*d+a*(e*x+d))/(e*(-a*b)^(1/2)-a*d)))/a)+1/b*(dilog(-e*x/d)+ln(e*x+d)*ln(-e*x/d))))

Fricas [F]

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

[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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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