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

3.3.67.1 Optimal result

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

output

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*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*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)
 

3.3.67.2 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}} \]

input

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

output

(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)*Sqrt 
[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)*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]] - 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*Lo 
g[-(-f)^(1/4) - g^(1/4)*Sqrt[x]]*Log[(f*g^(1/4)*Sqrt[x])/(-f)^(5/4)] - p*P 
olyLog[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)...
 

3.3.67.3 Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2922, 2925, 2005, 2863, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2922

\(\displaystyle 2 \int \frac {\sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{g x^2+f}d\sqrt {x}\)

\(\Big \downarrow \) 2925

\(\displaystyle -2 \int \frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{\left (f+\frac {g}{x^2}\right ) x^{3/2}}d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2005

\(\displaystyle -2 \int \frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{\sqrt {x} \left (f x^2+g\right )}d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2863

\(\displaystyle -2 \int \left (-\frac {f \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{2 \sqrt {-f} \sqrt {g} \sqrt {x} \left (\sqrt {-f} \sqrt {g}-f x\right )}-\frac {f \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right )}{2 \sqrt {-f} \sqrt {g} \sqrt {x} \left (f x+\sqrt {-f} \sqrt {g}\right )}\right )d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{g}-\frac {\sqrt {-\sqrt {-f}}}{\sqrt {x}}\right )}{d \sqrt {-\sqrt {-f}}+e \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (-\frac {e \left (\frac {\sqrt {-\sqrt {-f}}}{\sqrt {x}}+\sqrt [4]{g}\right )}{d \sqrt {-\sqrt {-f}}-e \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{g}-\frac {\sqrt [4]{-f}}{\sqrt {x}}\right )}{d \sqrt [4]{-f}+e \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^p\right ) \log \left (-\frac {e \left (\frac {\sqrt [4]{-f}}{\sqrt {x}}+\sqrt [4]{g}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-f}} \left (d+\frac {e}{\sqrt {x}}\right )}{d \sqrt {-\sqrt {-f}}-e \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-f} \left (d+\frac {e}{\sqrt {x}}\right )}{d \sqrt [4]{-f}-e \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-f}} \left (d+\frac {e}{\sqrt {x}}\right )}{\sqrt {-\sqrt {-f}} d+e \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-f} \left (d+\frac {e}{\sqrt {x}}\right )}{\sqrt [4]{-f} d+e \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}\right )\)

input

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

output

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

3.3.67.3.1 Defintions of rubi rules used

rule 2005

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2863

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 2922

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + 
 (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{k = Denominator[n]}, Simp[k   S 
ubst[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 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(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] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

3.3.67.4 Maple [F]

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

input

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

output

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

3.3.67.5 Fricas [F]

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

input

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

output

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

3.3.67.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.3.67.7 Maxima [F]

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

input

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

output

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

3.3.67.8 Giac [F]

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

input

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

output

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

3.3.67.9 Mupad [F(-1)]

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

input

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

output

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