\(\int \frac {a+b \log (c (4 d+d g x^2)^p)}{x \sqrt {4+g x^2}} \, dx\) [667]

Optimal result

Integrand size = 34, antiderivative size = 86 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x \sqrt {4+g x^2}} \, dx=-\frac {1}{2} \text {arctanh}\left (\frac {1}{2} \sqrt {4+g x^2}\right ) \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )-\frac {1}{2} b p \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt {4+g x^2}\right )+\frac {1}{2} b p \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt {4+g x^2}\right ) \] Output:

-1/2*arctanh(1/2*(g*x^2+4)^(1/2))*(a+b*ln(c*(d*g*x^2+4*d)^p))-1/2*b*p*poly 
log(2,-1/2*(g*x^2+4)^(1/2))+1/2*b*p*polylog(2,1/2*(g*x^2+4)^(1/2))
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.15 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x \sqrt {4+g x^2}} \, dx=\frac {1}{4} \left (\left (a+b \log \left (c \left (d \left (4+g x^2\right )\right )^p\right )\right ) \left (\log \left (2-\sqrt {4+g x^2}\right )-\log \left (2+\sqrt {4+g x^2}\right )\right )-2 b p \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt {4+g x^2}\right )+2 b p \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt {4+g x^2}\right )\right ) \] Input:

Integrate[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x*Sqrt[4 + g*x^2]),x]
 

Output:

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

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x \sqrt {g x^2+4}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x \sqrt {g x^2+4}}dx\)

Input:

Int[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x*Sqrt[4 + 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]

\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+4 d \right )^{p}\right )}{x \sqrt {g \,x^{2}+4}}d x\]

Input:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x/(g*x^2+4)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x/(g*x^2+4)^(1/2),x)
 

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x/(g*x^2+4)^(1/2),x, algorithm="fri 
cas")
 

Output:

integral((sqrt(g*x^2 + 4)*b*log((d*g*x^2 + 4*d)^p*c) + sqrt(g*x^2 + 4)*a)/ 
(g*x^3 + 4*x), x)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x \sqrt {4+g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d g x^{2} + 4 d\right )^{p} \right )}}{x \sqrt {g x^{2} + 4}}\, dx \] Input:

integrate((a+b*ln(c*(d*g*x**2+4*d)**p))/x/(g*x**2+4)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*g*x**2 + 4*d)**p))/(x*sqrt(g*x**2 + 4)), x)
 

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x/(g*x^2+4)^(1/2),x, algorithm="max 
ima")
 

Output:

-1/2*a*arcsinh(2/(sqrt(g)*abs(x))) + b*integrate((p*log(g*x^2 + 4) + p*log 
(d) + log(c))/(sqrt(g*x^2 + 4)*x), x)
 

Giac [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x/(g*x^2+4)^(1/2),x, algorithm="gia 
c")
 

Output:

integrate((b*log((d*g*x^2 + 4*d)^p*c) + a)/(sqrt(g*x^2 + 4)*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x \sqrt {4+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,g\,x^2+4\,d\right )}^p\right )}{x\,\sqrt {g\,x^2+4}} \,d x \] Input:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x*(g*x^2 + 4)^(1/2)),x)
 

Output:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x*(g*x^2 + 4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x \sqrt {4+g x^2}} \, dx=\left (\int \frac {\mathrm {log}\left (\left (d g \,x^{2}+4 d \right )^{p} c \right )}{\sqrt {g \,x^{2}+4}\, x}d x \right ) b +\frac {\mathrm {log}\left (\frac {\sqrt {g \,x^{2}+4}}{2}+\frac {\sqrt {g}\, x}{2}-1\right ) a}{2}-\frac {\mathrm {log}\left (\frac {\sqrt {g \,x^{2}+4}}{2}+\frac {\sqrt {g}\, x}{2}+1\right ) a}{2} \] Input:

int((a+b*log(c*(d*g*x^2+4*d)^p))/x/(g*x^2+4)^(1/2),x)
 

Output:

(2*int(log((d*g*x**2 + 4*d)**p*c)/(sqrt(g*x**2 + 4)*x),x)*b + log((sqrt(g* 
x**2 + 4) + sqrt(g)*x - 2)/2)*a - log((sqrt(g*x**2 + 4) + sqrt(g)*x + 2)/2 
)*a)/2