\(\int \frac {a+b \log (c (4 d+d g x^2)^p)}{x^3 \sqrt {4+g x^2}} \, dx\) [668]

Optimal result

Integrand size = 34, antiderivative size = 149 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^3 \sqrt {4+g x^2}} \, dx=-\frac {1}{8} b g p \text {arctanh}\left (\frac {1}{2} \sqrt {4+g x^2}\right )-\frac {\sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{8 x^2}+\frac {1}{16} g \text {arctanh}\left (\frac {1}{2} \sqrt {4+g x^2}\right ) \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )+\frac {1}{16} b g p \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt {4+g x^2}\right )-\frac {1}{16} b g p \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt {4+g x^2}\right ) \] Output:

-1/8*b*g*p*arctanh(1/2*(g*x^2+4)^(1/2))-1/8*(g*x^2+4)^(1/2)*(a+b*ln(c*(d*g 
*x^2+4*d)^p))/x^2+1/16*g*arctanh(1/2*(g*x^2+4)^(1/2))*(a+b*ln(c*(d*g*x^2+4 
*d)^p))+1/16*b*g*p*polylog(2,-1/2*(g*x^2+4)^(1/2))-1/16*b*g*p*polylog(2,1/ 
2*(g*x^2+4)^(1/2))
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.74 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^3 \sqrt {4+g x^2}} \, dx=-\frac {4 a \sqrt {4+g x^2}+4 b \sqrt {4+g x^2} \log \left (c \left (d \left (4+g x^2\right )\right )^p\right )+a g x^2 \log \left (2-\sqrt {4+g x^2}\right )-2 b g p x^2 \log \left (2-\sqrt {4+g x^2}\right )+b g x^2 \log \left (c \left (d \left (4+g x^2\right )\right )^p\right ) \log \left (2-\sqrt {4+g x^2}\right )-a g x^2 \log \left (2+\sqrt {4+g x^2}\right )+2 b g p x^2 \log \left (2+\sqrt {4+g x^2}\right )-b g x^2 \log \left (c \left (d \left (4+g x^2\right )\right )^p\right ) \log \left (2+\sqrt {4+g x^2}\right )-2 b g p x^2 \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt {4+g x^2}\right )+2 b g p x^2 \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt {4+g x^2}\right )}{32 x^2} \] Input:

Integrate[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x^3*Sqrt[4 + g*x^2]),x]
 

Output:

-1/32*(4*a*Sqrt[4 + g*x^2] + 4*b*Sqrt[4 + g*x^2]*Log[c*(d*(4 + g*x^2))^p] 
+ a*g*x^2*Log[2 - Sqrt[4 + g*x^2]] - 2*b*g*p*x^2*Log[2 - Sqrt[4 + g*x^2]] 
+ b*g*x^2*Log[c*(d*(4 + g*x^2))^p]*Log[2 - Sqrt[4 + g*x^2]] - a*g*x^2*Log[ 
2 + Sqrt[4 + g*x^2]] + 2*b*g*p*x^2*Log[2 + Sqrt[4 + g*x^2]] - b*g*x^2*Log[ 
c*(d*(4 + g*x^2))^p]*Log[2 + Sqrt[4 + g*x^2]] - 2*b*g*p*x^2*PolyLog[2, -1/ 
2*Sqrt[4 + g*x^2]] + 2*b*g*p*x^2*PolyLog[2, Sqrt[4 + g*x^2]/2])/x^2
 

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x^3 \sqrt {g x^2+4}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x^3 \sqrt {g x^2+4}}dx\)

Input:

Int[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x^3*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]

\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+4 d \right )^{p}\right )}{x^{3} \sqrt {g \,x^{2}+4}}d x\]

Input:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x^3/(g*x^2+4)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x^3/(g*x^2+4)^(1/2),x)
 

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^3 \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^3/(g*x^2+4)^(1/2),x, algorithm="f 
ricas")
 

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^5 + 4*x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^3 \sqrt {4+g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d g x^{2} + 4 d\right )^{p} \right )}}{x^{3} \sqrt {g x^{2} + 4}}\, dx \] Input:

integrate((a+b*ln(c*(d*g*x**2+4*d)**p))/x**3/(g*x**2+4)**(1/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^3 \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^3/(g*x^2+4)^(1/2),x, algorithm="m 
axima")
 

Output:

1/16*(g*arcsinh(2/(sqrt(g)*abs(x))) - 2*sqrt(g*x^2 + 4)/x^2)*a + b*integra 
te((p*log(g*x^2 + 4) + p*log(d) + log(c))/(sqrt(g*x^2 + 4)*x^3), x)
 

Giac [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^3 \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^3/(g*x^2+4)^(1/2),x, algorithm="g 
iac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^3 \sqrt {4+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,g\,x^2+4\,d\right )}^p\right )}{x^3\,\sqrt {g\,x^2+4}} \,d x \] Input:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x^3*(g*x^2 + 4)^(1/2)),x)
 

Output:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x^3*(g*x^2 + 4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^3 \sqrt {4+g x^2}} \, dx=\frac {-2 \sqrt {g \,x^{2}+4}\, a +16 \left (\int \frac {\mathrm {log}\left (\left (d g \,x^{2}+4 d \right )^{p} c \right )}{\sqrt {g \,x^{2}+4}\, x^{3}}d x \right ) b \,x^{2}-\mathrm {log}\left (\frac {\sqrt {g \,x^{2}+4}}{2}+\frac {\sqrt {g}\, x}{2}-1\right ) a g \,x^{2}+\mathrm {log}\left (\frac {\sqrt {g \,x^{2}+4}}{2}+\frac {\sqrt {g}\, x}{2}+1\right ) a g \,x^{2}}{16 x^{2}} \] Input:

int((a+b*log(c*(d*g*x^2+4*d)^p))/x^3/(g*x^2+4)^(1/2),x)
 

Output:

( - 2*sqrt(g*x**2 + 4)*a + 16*int(log((d*g*x**2 + 4*d)**p*c)/(sqrt(g*x**2 
+ 4)*x**3),x)*b*x**2 - log((sqrt(g*x**2 + 4) + sqrt(g)*x - 2)/2)*a*g*x**2 
+ log((sqrt(g*x**2 + 4) + sqrt(g)*x + 2)/2)*a*g*x**2)/(16*x**2)