\(\int \frac {a+b \log (c (d f+d g x^2)^p)}{x \sqrt {f+g x^2}} \, dx\) [693]

Optimal result

Integrand size = 34, antiderivative size = 101 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x \sqrt {f+g x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right ) \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{\sqrt {f}}-\frac {b p \operatorname {PolyLog}\left (2,-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {b p \operatorname {PolyLog}\left (2,\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{\sqrt {f}} \] Output:

-arctanh((g*x^2+f)^(1/2)/f^(1/2))*(a+b*ln(c*(d*g*x^2+d*f)^p))/f^(1/2)-b*p* 
polylog(2,-(g*x^2+f)^(1/2)/f^(1/2))/f^(1/2)+b*p*polylog(2,(g*x^2+f)^(1/2)/ 
f^(1/2))/f^(1/2)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x \sqrt {f+g x^2}} \, dx=\frac {\left (a+b \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )\right ) \left (\log \left (1-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )-\log \left (1+\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )\right )-2 b p \operatorname {PolyLog}\left (2,-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )+2 b p \operatorname {PolyLog}\left (2,\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{2 \sqrt {f}} \] Input:

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

Output:

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

Rubi [F]

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.

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

\(\Big \downarrow \) 2929

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2929

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x/(g*x^2+f)^(1/2),x, algorithm="fri 
cas")
 

Output:

integral((sqrt(g*x^2 + f)*b*log((d*g*x^2 + d*f)^p*c) + sqrt(g*x^2 + f)*a)/ 
(g*x^3 + f*x), x)
 

Sympy [F]

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

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

Output:

Integral((a + b*log(c*(d*f + d*g*x**2)**p))/(x*sqrt(f + g*x**2)), x)
 

Maxima [F]

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

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x/(g*x^2+f)^(1/2),x, algorithm="max 
ima")
 

Output:

b*integrate((p*log(g*x^2 + f) + p*log(d) + log(c))/(sqrt(g*x^2 + f)*x), x) 
 - a*arcsinh(f/(sqrt(f*g)*abs(x)))/sqrt(f)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x \sqrt {f+g x^2}} \, dx=\frac {\sqrt {f}\, \mathrm {log}\left (\frac {\sqrt {g \,x^{2}+f}-\sqrt {f}+\sqrt {g}\, x}{\sqrt {f}}\right ) a -\sqrt {f}\, \mathrm {log}\left (\frac {\sqrt {g \,x^{2}+f}+\sqrt {f}+\sqrt {g}\, x}{\sqrt {f}}\right ) a +\left (\int \frac {\mathrm {log}\left (\left (d g \,x^{2}+d f \right )^{p} c \right )}{\sqrt {g \,x^{2}+f}\, x}d x \right ) b f}{f} \] Input:

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

Output:

(sqrt(f)*log((sqrt(f + g*x**2) - sqrt(f) + sqrt(g)*x)/sqrt(f))*a - sqrt(f) 
*log((sqrt(f + g*x**2) + sqrt(f) + sqrt(g)*x)/sqrt(f))*a + int(log((d*f + 
d*g*x**2)**p*c)/(sqrt(f + g*x**2)*x),x)*b*f)/f