\(\int \frac {\log (c (d+\frac {e}{\sqrt {x}})^p)}{f+g x^2} \, dx\) [267]

Optimal result

Integrand size = 24, antiderivative size = 561 \[ \int \frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f+g x^2} \, dx=-\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{g}-\frac {\sqrt {-\sqrt {-f}}}{\sqrt {x}}\right )}{d \sqrt {-\sqrt {-f}}+e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (-\frac {e \left (\sqrt [4]{g}+\frac {\sqrt {-\sqrt {-f}}}{\sqrt {x}}\right )}{d \sqrt {-\sqrt {-f}}-e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{g}-\frac {\sqrt [4]{-f}}{\sqrt {x}}\right )}{d \sqrt [4]{-f}+e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (-\frac {e \left (\sqrt [4]{g}+\frac {\sqrt [4]{-f}}{\sqrt {x}}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-f}} \left (d+\frac {e}{\sqrt {x}}\right )}{d \sqrt {-\sqrt {-f}}-e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-f} \left (d+\frac {e}{\sqrt {x}}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-f}} \left (d+\frac {e}{\sqrt {x}}\right )}{d \sqrt {-\sqrt {-f}}+e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-f} \left (d+\frac {e}{\sqrt {x}}\right )}{d \sqrt [4]{-f}+e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]

[Out]

1/2*ln(c*(d+e/x^(1/2))^p)*ln(e*(g^(1/4)-(-f)^(1/4)/x^(1/2))/(d*(-f)^(1/4)+e*g^(1/4)))/(-f)^(1/2)/g^(1/2)+1/2*l
n(c*(d+e/x^(1/2))^p)*ln(-e*(g^(1/4)+(-f)^(1/4)/x^(1/2))/(d*(-f)^(1/4)-e*g^(1/4)))/(-f)^(1/2)/g^(1/2)-1/2*ln(c*
(d+e/x^(1/2))^p)*ln(e*(g^(1/4)-(-(-f)^(1/2))^(1/2)/x^(1/2))/(e*g^(1/4)+d*(-(-f)^(1/2))^(1/2)))/(-f)^(1/2)/g^(1
/2)-1/2*ln(c*(d+e/x^(1/2))^p)*ln(-e*(g^(1/4)+(-(-f)^(1/2))^(1/2)/x^(1/2))/(-e*g^(1/4)+d*(-(-f)^(1/2))^(1/2)))/
(-f)^(1/2)/g^(1/2)+1/2*p*polylog(2,(-f)^(1/4)*(d+e/x^(1/2))/(d*(-f)^(1/4)-e*g^(1/4)))/(-f)^(1/2)/g^(1/2)+1/2*p
*polylog(2,(-f)^(1/4)*(d+e/x^(1/2))/(d*(-f)^(1/4)+e*g^(1/4)))/(-f)^(1/2)/g^(1/2)-1/2*p*polylog(2,(d+e/x^(1/2))
*(-(-f)^(1/2))^(1/2)/(-e*g^(1/4)+d*(-(-f)^(1/2))^(1/2)))/(-f)^(1/2)/g^(1/2)-1/2*p*polylog(2,(d+e/x^(1/2))*(-(-
f)^(1/2))^(1/2)/(e*g^(1/4)+d*(-(-f)^(1/2))^(1/2)))/(-f)^(1/2)/g^(1/2)

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2522, 2525, 269, 281, 211, 2463, 266, 2441, 2440, 2438} \[ \int \frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f+g x^2} \, dx=-\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{g}-\frac {\sqrt {-\sqrt {-f}}}{\sqrt {x}}\right )}{d \sqrt {-\sqrt {-f}}+e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (-\frac {e \left (\frac {\sqrt {-\sqrt {-f}}}{\sqrt {x}}+\sqrt [4]{g}\right )}{d \sqrt {-\sqrt {-f}}-e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{g}-\frac {\sqrt [4]{-f}}{\sqrt {x}}\right )}{d \sqrt [4]{-f}+e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (-\frac {e \left (\frac {\sqrt [4]{-f}}{\sqrt {x}}+\sqrt [4]{g}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-f}} \left (d+\frac {e}{\sqrt {x}}\right )}{d \sqrt {-\sqrt {-f}}-e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-f} \left (d+\frac {e}{\sqrt {x}}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-f}} \left (d+\frac {e}{\sqrt {x}}\right )}{\sqrt {-\sqrt {-f}} d+e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-f} \left (d+\frac {e}{\sqrt {x}}\right )}{\sqrt [4]{-f} d+e \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]

[In]

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

[Out]

-1/2*(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))])/(
Sqrt[-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*Sq
rt[-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*Sqr
t[-f]*Sqrt[g])

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 281

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 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 2522

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 2525

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])

Rubi steps \begin{align*} \text {integral}& = 2 \text {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 \text {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 \text {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} \text {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} \text {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} \text {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} \text {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 {\text {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 {\text {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 {\text {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 {\text {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) \text {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) \text {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) \text {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) \text {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 \text {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 \text {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 \text {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 \text {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] (verified)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 912, normalized size of antiderivative = 1.63 \[ \int \frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f+g x^2} \, dx=\frac {\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 (e+d \sqrt {x}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right ) \log \left (-\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\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 {i \sqrt [4]{g} \left (e+d \sqrt {x}\right )}{d \sqrt [4]{-f}+i 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 (i \sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )+p \log \left (\frac {\sqrt [4]{g} \left (e+d \sqrt {x}\right )}{i d \sqrt [4]{-f}+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 (e+d \sqrt {x}\right )}{d \sqrt [4]{-f}+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 (-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]{-f}-\sqrt [4]{g} \sqrt {x}\right ) \log \left (\frac {\sqrt [4]{g} \sqrt {x}}{\sqrt [4]{-f}}\right )+p \log \left (-\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right ) \log \left (\frac {f \sqrt [4]{g} \sqrt {x}}{(-f)^{5/4}}\right )-p \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{-f}+e \sqrt [4]{g}}\right )+p \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt [4]{-f}-i \sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{-f}+i e \sqrt [4]{g}}\right )+p \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt [4]{-f}+i \sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{-f}-i e \sqrt [4]{g}}\right )-p \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )-p \operatorname {PolyLog}\left (2,1-\frac {i \sqrt [4]{g} \sqrt {x}}{\sqrt [4]{-f}}\right )-p \operatorname {PolyLog}\left (2,1+\frac {i \sqrt [4]{g} \sqrt {x}}{\sqrt [4]{-f}}\right )+p \operatorname {PolyLog}\left (2,1+\frac {\sqrt [4]{g} \sqrt {x}}{\sqrt [4]{-f}}\right )+p \operatorname {PolyLog}\left (2,1+\frac {f \sqrt [4]{g} \sqrt {x}}{(-f)^{5/4}}\right )}{2 \sqrt {-f} \sqrt {g}} \]

[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])

Maple [F]

\[\int \frac {\ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{p}\right )}{g \,x^{2}+f}d x\]

[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)

Fricas [F]

\[ \int \frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{p}\right )}{g x^{2} + f} \,d x } \]

[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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{p}\right )}{g x^{2} + f} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{p}\right )}{g x^{2} + f} \,d x } \]

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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