3.267 \(\int \frac{\log (c (d+\frac{e}{\sqrt{x}})^p)}{f+g x^2} \, dx\)

Optimal. Leaf size=561 \[ -\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-\sqrt{-f}} \left (d+\frac{e}{\sqrt{x}}\right )}{d \sqrt{-\sqrt{-f}}-e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt [4]{-f} \left (d+\frac{e}{\sqrt{x}}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-\sqrt{-f}} \left (d+\frac{e}{\sqrt{x}}\right )}{d \sqrt{-\sqrt{-f}}+e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt [4]{-f} \left (d+\frac{e}{\sqrt{x}}\right )}{d \sqrt [4]{-f}+e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{g}-\frac{\sqrt{-\sqrt{-f}}}{\sqrt{x}}\right )}{d \sqrt{-\sqrt{-f}}+e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (-\frac{e \left (\frac{\sqrt{-\sqrt{-f}}}{\sqrt{x}}+\sqrt [4]{g}\right )}{d \sqrt{-\sqrt{-f}}-e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{g}-\frac{\sqrt [4]{-f}}{\sqrt{x}}\right )}{d \sqrt [4]{-f}+e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (-\frac{e \left (\frac{\sqrt [4]{-f}}{\sqrt{x}}+\sqrt [4]{g}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}} \]

[Out]

-(Log[c*(d + e/Sqrt[x])^p]*Log[(e*(g^(1/4) - Sqrt[-Sqrt[-f]]/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] + e*g^(1/4))])/(2*Sq
rt[-f]*Sqrt[g]) - (Log[c*(d + e/Sqrt[x])^p]*Log[-((e*(g^(1/4) + Sqrt[-Sqrt[-f]]/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] -
e*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e/Sqrt[x])^p]*Log[(e*(g^(1/4) - (-f)^(1/4)/Sqrt[x]))/(d*(-f)
^(1/4) + e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e/Sqrt[x])^p]*Log[-((e*(g^(1/4) + (-f)^(1/4)/Sqrt[x])
)/(d*(-f)^(1/4) - e*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (Sqrt[-Sqrt[-f]]*(d + e/Sqrt[x]))/(d*Sqrt
[-Sqrt[-f]] - e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, ((-f)^(1/4)*(d + e/Sqrt[x]))/(d*(-f)^(1/4) - e
*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (Sqrt[-Sqrt[-f]]*(d + e/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] + e*g^(1
/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, ((-f)^(1/4)*(d + e/Sqrt[x]))/(d*(-f)^(1/4) + e*g^(1/4))])/(2*Sqrt[
-f]*Sqrt[g])

________________________________________________________________________________________

Rubi [A]  time = 1.10713, antiderivative size = 561, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2472, 2475, 263, 275, 205, 2416, 260, 2394, 2393, 2391} \[ -\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-\sqrt{-f}} \left (d+\frac{e}{\sqrt{x}}\right )}{d \sqrt{-\sqrt{-f}}-e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt [4]{-f} \left (d+\frac{e}{\sqrt{x}}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-\sqrt{-f}} \left (d+\frac{e}{\sqrt{x}}\right )}{d \sqrt{-\sqrt{-f}}+e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt [4]{-f} \left (d+\frac{e}{\sqrt{x}}\right )}{d \sqrt [4]{-f}+e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{g}-\frac{\sqrt{-\sqrt{-f}}}{\sqrt{x}}\right )}{d \sqrt{-\sqrt{-f}}+e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}-\frac{\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (-\frac{e \left (\frac{\sqrt{-\sqrt{-f}}}{\sqrt{x}}+\sqrt [4]{g}\right )}{d \sqrt{-\sqrt{-f}}-e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (\frac{e \left (\sqrt [4]{g}-\frac{\sqrt [4]{-f}}{\sqrt{x}}\right )}{d \sqrt [4]{-f}+e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}}+\frac{\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (-\frac{e \left (\frac{\sqrt [4]{-f}}{\sqrt{x}}+\sqrt [4]{g}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{2 \sqrt{-f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e/Sqrt[x])^p]/(f + g*x^2),x]

[Out]

-(Log[c*(d + e/Sqrt[x])^p]*Log[(e*(g^(1/4) - Sqrt[-Sqrt[-f]]/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] + e*g^(1/4))])/(2*Sq
rt[-f]*Sqrt[g]) - (Log[c*(d + e/Sqrt[x])^p]*Log[-((e*(g^(1/4) + Sqrt[-Sqrt[-f]]/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] -
e*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e/Sqrt[x])^p]*Log[(e*(g^(1/4) - (-f)^(1/4)/Sqrt[x]))/(d*(-f)
^(1/4) + e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e/Sqrt[x])^p]*Log[-((e*(g^(1/4) + (-f)^(1/4)/Sqrt[x])
)/(d*(-f)^(1/4) - e*g^(1/4)))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (Sqrt[-Sqrt[-f]]*(d + e/Sqrt[x]))/(d*Sqrt
[-Sqrt[-f]] - e*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, ((-f)^(1/4)*(d + e/Sqrt[x]))/(d*(-f)^(1/4) - e
*g^(1/4))])/(2*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (Sqrt[-Sqrt[-f]]*(d + e/Sqrt[x]))/(d*Sqrt[-Sqrt[-f]] + e*g^(1
/4))])/(2*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, ((-f)^(1/4)*(d + e/Sqrt[x]))/(d*(-f)^(1/4) + e*g^(1/4))])/(2*Sqrt[
-f]*Sqrt[g])

Rule 2472

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k - 1)*(f + g*x^(k*s))^r*(a + b*Log[c*(d + e*x^(k*n))^p])^q
, x], x, x^(1/k)], x] /; IntegerQ[k*s]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && FractionQ[n]

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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

Mathematica [C]  time = 0.530052, size = 912, normalized size = 1.63 \[ \frac{\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (-\sqrt [4]{g} \sqrt{x}-\sqrt [4]{-f}\right )-p \log \left (-\frac{\sqrt [4]{g} \left (\sqrt{x} d+e\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right ) \log \left (-\sqrt [4]{g} \sqrt{x}-\sqrt [4]{-f}\right )+p \log \left (\frac{f \sqrt [4]{g} \sqrt{x}}{(-f)^{5/4}}\right ) \log \left (-\sqrt [4]{g} \sqrt{x}-\sqrt [4]{-f}\right )-\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (-\sqrt [4]{g} \sqrt{x}-i \sqrt [4]{-f}\right )+p \log \left (\frac{i \sqrt [4]{g} \left (\sqrt{x} d+e\right )}{\sqrt [4]{-f} d+i e \sqrt [4]{g}}\right ) \log \left (-\sqrt [4]{g} \sqrt{x}-i \sqrt [4]{-f}\right )-\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (i \sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )+p \log \left (\frac{\sqrt [4]{g} \left (\sqrt{x} d+e\right )}{i \sqrt [4]{-f} d+e \sqrt [4]{g}}\right ) \log \left (i \sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )+\log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^p\right ) \log \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )-p \log \left (\frac{\sqrt [4]{g} \left (\sqrt{x} d+e\right )}{\sqrt [4]{-f} d+e \sqrt [4]{g}}\right ) \log \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )-p \log \left (i \sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right ) \log \left (-\frac{i \sqrt [4]{g} \sqrt{x}}{\sqrt [4]{-f}}\right )-p \log \left (-\sqrt [4]{g} \sqrt{x}-i \sqrt [4]{-f}\right ) \log \left (\frac{i \sqrt [4]{g} \sqrt{x}}{\sqrt [4]{-f}}\right )+p \log \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right ) \log \left (\frac{\sqrt [4]{g} \sqrt{x}}{\sqrt [4]{-f}}\right )-p \text{PolyLog}\left (2,\frac{d \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt{x}\right )}{\sqrt [4]{-f} d+e \sqrt [4]{g}}\right )+p \text{PolyLog}\left (2,\frac{d \left (\sqrt [4]{-f}-i \sqrt [4]{g} \sqrt{x}\right )}{\sqrt [4]{-f} d+i e \sqrt [4]{g}}\right )+p \text{PolyLog}\left (2,\frac{d \left (i \sqrt [4]{g} \sqrt{x}+\sqrt [4]{-f}\right )}{d \sqrt [4]{-f}-i e \sqrt [4]{g}}\right )-p \text{PolyLog}\left (2,\frac{d \left (\sqrt [4]{g} \sqrt{x}+\sqrt [4]{-f}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )-p \text{PolyLog}\left (2,1-\frac{i \sqrt [4]{g} \sqrt{x}}{\sqrt [4]{-f}}\right )-p \text{PolyLog}\left (2,\frac{i \sqrt [4]{g} \sqrt{x}}{\sqrt [4]{-f}}+1\right )+p \text{PolyLog}\left (2,\frac{\sqrt [4]{g} \sqrt{x}}{\sqrt [4]{-f}}+1\right )+p \text{PolyLog}\left (2,\frac{\sqrt [4]{g} \sqrt{x} f}{(-f)^{5/4}}+1\right )}{2 \sqrt{-f} \sqrt{g}} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e/Sqrt[x])^p]/(f + g*x^2),x]

[Out]

(Log[c*(d + e/Sqrt[x])^p]*Log[-(-f)^(1/4) - g^(1/4)*Sqrt[x]] - p*Log[-((g^(1/4)*(e + d*Sqrt[x]))/(d*(-f)^(1/4)
- e*g^(1/4)))]*Log[-(-f)^(1/4) - g^(1/4)*Sqrt[x]] - Log[c*(d + e/Sqrt[x])^p]*Log[(-I)*(-f)^(1/4) - g^(1/4)*Sq
rt[x]] + p*Log[(I*g^(1/4)*(e + d*Sqrt[x]))/(d*(-f)^(1/4) + I*e*g^(1/4))]*Log[(-I)*(-f)^(1/4) - g^(1/4)*Sqrt[x]
] - Log[c*(d + e/Sqrt[x])^p]*Log[I*(-f)^(1/4) - g^(1/4)*Sqrt[x]] + p*Log[(g^(1/4)*(e + d*Sqrt[x]))/(I*d*(-f)^(
1/4) + e*g^(1/4))]*Log[I*(-f)^(1/4) - g^(1/4)*Sqrt[x]] + Log[c*(d + e/Sqrt[x])^p]*Log[(-f)^(1/4) - g^(1/4)*Sqr
t[x]] - p*Log[(g^(1/4)*(e + d*Sqrt[x]))/(d*(-f)^(1/4) + e*g^(1/4))]*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]] - p*Log[
I*(-f)^(1/4) - g^(1/4)*Sqrt[x]]*Log[((-I)*g^(1/4)*Sqrt[x])/(-f)^(1/4)] - p*Log[(-I)*(-f)^(1/4) - g^(1/4)*Sqrt[
x]]*Log[(I*g^(1/4)*Sqrt[x])/(-f)^(1/4)] + p*Log[(-f)^(1/4) - g^(1/4)*Sqrt[x]]*Log[(g^(1/4)*Sqrt[x])/(-f)^(1/4)
] + p*Log[-(-f)^(1/4) - g^(1/4)*Sqrt[x]]*Log[(f*g^(1/4)*Sqrt[x])/(-f)^(5/4)] - p*PolyLog[2, (d*((-f)^(1/4) - g
^(1/4)*Sqrt[x]))/(d*(-f)^(1/4) + e*g^(1/4))] + p*PolyLog[2, (d*((-f)^(1/4) - I*g^(1/4)*Sqrt[x]))/(d*(-f)^(1/4)
+ I*e*g^(1/4))] + p*PolyLog[2, (d*((-f)^(1/4) + I*g^(1/4)*Sqrt[x]))/(d*(-f)^(1/4) - I*e*g^(1/4))] - p*PolyLog
[2, (d*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(d*(-f)^(1/4) - e*g^(1/4))] - p*PolyLog[2, 1 - (I*g^(1/4)*Sqrt[x])/(-f)
^(1/4)] - p*PolyLog[2, 1 + (I*g^(1/4)*Sqrt[x])/(-f)^(1/4)] + p*PolyLog[2, 1 + (g^(1/4)*Sqrt[x])/(-f)^(1/4)] +
p*PolyLog[2, 1 + (f*g^(1/4)*Sqrt[x])/(-f)^(5/4)])/(2*Sqrt[-f]*Sqrt[g])

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Maple [F]  time = 0.747, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{g{x}^{2}+f}\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) ^{p} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(d+e/x^(1/2))^p)/(g*x^2+f),x)

[Out]

int(ln(c*(d+e/x^(1/2))^p)/(g*x^2+f),x)

________________________________________________________________________________________

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*(d+e/x^(1/2))^p)/(g*x^2+f),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{d x + e \sqrt{x}}{x}\right )^{p}\right )}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e/x^(1/2))^p)/(g*x^2+f),x, algorithm="fricas")

[Out]

integral(log(c*((d*x + e*sqrt(x))/x)^p)/(g*x^2 + f), 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*(d+e/x**(1/2))**p)/(g*x**2+f),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(d+e/x^(1/2))^p)/(g*x^2+f),x, algorithm="giac")

[Out]

integrate(log(c*(d + e/sqrt(x))^p)/(g*x^2 + f), x)