\(\int \frac {a+b \log (c (4 d+d g x^2)^p)}{x^5 \sqrt {4+g x^2}} \, dx\) [669]

Optimal result

Integrand size = 34, antiderivative size = 216 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^5 \sqrt {4+g x^2}} \, dx=-\frac {b g p \sqrt {4+g x^2}}{64 x^2}+\frac {1}{32} b g^2 p \text {arctanh}\left (\frac {1}{2} \sqrt {4+g x^2}\right )-\frac {\sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{16 x^4}+\frac {3 g \sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{128 x^2}-\frac {3}{256} g^2 \text {arctanh}\left (\frac {1}{2} \sqrt {4+g x^2}\right ) \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )-\frac {3}{256} b g^2 p \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt {4+g x^2}\right )+\frac {3}{256} b g^2 p \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt {4+g x^2}\right ) \] Output:

-1/64*b*g*p*(g*x^2+4)^(1/2)/x^2+1/32*b*g^2*p*arctanh(1/2*(g*x^2+4)^(1/2))- 
1/16*(g*x^2+4)^(1/2)*(a+b*ln(c*(d*g*x^2+4*d)^p))/x^4+3/128*g*(g*x^2+4)^(1/ 
2)*(a+b*ln(c*(d*g*x^2+4*d)^p))/x^2-3/256*g^2*arctanh(1/2*(g*x^2+4)^(1/2))* 
(a+b*ln(c*(d*g*x^2+4*d)^p))-3/256*b*g^2*p*polylog(2,-1/2*(g*x^2+4)^(1/2))+ 
3/256*b*g^2*p*polylog(2,1/2*(g*x^2+4)^(1/2))
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.61 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^5 \sqrt {4+g x^2}} \, dx=-\frac {32 a \sqrt {4+g x^2}-12 a g x^2 \sqrt {4+g x^2}+8 b g p x^2 \sqrt {4+g x^2}+32 b \sqrt {4+g x^2} \log \left (c \left (d \left (4+g x^2\right )\right )^p\right )-12 b g x^2 \sqrt {4+g x^2} \log \left (c \left (d \left (4+g x^2\right )\right )^p\right )-3 a g^2 x^4 \log \left (2-\sqrt {4+g x^2}\right )+8 b g^2 p x^4 \log \left (2-\sqrt {4+g x^2}\right )-3 b g^2 x^4 \log \left (c \left (d \left (4+g x^2\right )\right )^p\right ) \log \left (2-\sqrt {4+g x^2}\right )+3 a g^2 x^4 \log \left (2+\sqrt {4+g x^2}\right )-8 b g^2 p x^4 \log \left (2+\sqrt {4+g x^2}\right )+3 b g^2 x^4 \log \left (c \left (d \left (4+g x^2\right )\right )^p\right ) \log \left (2+\sqrt {4+g x^2}\right )+6 b g^2 p x^4 \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt {4+g x^2}\right )-6 b g^2 p x^4 \operatorname {PolyLog}\left (2,\frac {1}{2} \sqrt {4+g x^2}\right )}{512 x^4} \] Input:

Integrate[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x^5*Sqrt[4 + g*x^2]),x]
 

Output:

-1/512*(32*a*Sqrt[4 + g*x^2] - 12*a*g*x^2*Sqrt[4 + g*x^2] + 8*b*g*p*x^2*Sq 
rt[4 + g*x^2] + 32*b*Sqrt[4 + g*x^2]*Log[c*(d*(4 + g*x^2))^p] - 12*b*g*x^2 
*Sqrt[4 + g*x^2]*Log[c*(d*(4 + g*x^2))^p] - 3*a*g^2*x^4*Log[2 - Sqrt[4 + g 
*x^2]] + 8*b*g^2*p*x^4*Log[2 - Sqrt[4 + g*x^2]] - 3*b*g^2*x^4*Log[c*(d*(4 
+ g*x^2))^p]*Log[2 - Sqrt[4 + g*x^2]] + 3*a*g^2*x^4*Log[2 + Sqrt[4 + g*x^2 
]] - 8*b*g^2*p*x^4*Log[2 + Sqrt[4 + g*x^2]] + 3*b*g^2*x^4*Log[c*(d*(4 + g* 
x^2))^p]*Log[2 + Sqrt[4 + g*x^2]] + 6*b*g^2*p*x^4*PolyLog[2, -1/2*Sqrt[4 + 
 g*x^2]] - 6*b*g^2*p*x^4*PolyLog[2, Sqrt[4 + g*x^2]/2])/x^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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x^5 \sqrt {g x^2+4}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x^5 \sqrt {g x^2+4}}dx\)

Input:

Int[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x^5*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]

\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+4 d \right )^{p}\right )}{x^{5} \sqrt {g \,x^{2}+4}}d x\]

Input:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x^5/(g*x^2+4)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x^5/(g*x^2+4)^(1/2),x)
 

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^5 \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x^{5}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^5/(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^7 + 4*x^5), x)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^5 \sqrt {4+g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d g x^{2} + 4 d\right )^{p} \right )}}{x^{5} \sqrt {g x^{2} + 4}}\, dx \] Input:

integrate((a+b*ln(c*(d*g*x**2+4*d)**p))/x**5/(g*x**2+4)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*g*x**2 + 4*d)**p))/(x**5*sqrt(g*x**2 + 4)), x)
 

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^5 \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x^{5}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^5/(g*x^2+4)^(1/2),x, algorithm="m 
axima")
 

Output:

-1/256*(3*g^2*arcsinh(2/(sqrt(g)*abs(x))) - 6*sqrt(g*x^2 + 4)*g/x^2 + 16*s 
qrt(g*x^2 + 4)/x^4)*a + b*integrate((p*log(g*x^2 + 4) + p*log(d) + log(c)) 
/(sqrt(g*x^2 + 4)*x^5), x)
 

Giac [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^5 \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x^{5}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^5/(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^5), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^5 \sqrt {4+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,g\,x^2+4\,d\right )}^p\right )}{x^5\,\sqrt {g\,x^2+4}} \,d x \] Input:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x^5*(g*x^2 + 4)^(1/2)),x)
 

Output:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x^5*(g*x^2 + 4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^5 \sqrt {4+g x^2}} \, dx=\frac {6 \sqrt {g \,x^{2}+4}\, a g \,x^{2}-16 \sqrt {g \,x^{2}+4}\, a +256 \left (\int \frac {\mathrm {log}\left (\left (d g \,x^{2}+4 d \right )^{p} c \right )}{\sqrt {g \,x^{2}+4}\, x^{5}}d x \right ) b \,x^{4}+3 \,\mathrm {log}\left (\frac {\sqrt {g \,x^{2}+4}}{2}+\frac {\sqrt {g}\, x}{2}-1\right ) a \,g^{2} x^{4}-3 \,\mathrm {log}\left (\frac {\sqrt {g \,x^{2}+4}}{2}+\frac {\sqrt {g}\, x}{2}+1\right ) a \,g^{2} x^{4}}{256 x^{4}} \] Input:

int((a+b*log(c*(d*g*x^2+4*d)^p))/x^5/(g*x^2+4)^(1/2),x)
 

Output:

(6*sqrt(g*x**2 + 4)*a*g*x**2 - 16*sqrt(g*x**2 + 4)*a + 256*int(log((d*g*x* 
*2 + 4*d)**p*c)/(sqrt(g*x**2 + 4)*x**5),x)*b*x**4 + 3*log((sqrt(g*x**2 + 4 
) + sqrt(g)*x - 2)/2)*a*g**2*x**4 - 3*log((sqrt(g*x**2 + 4) + sqrt(g)*x + 
2)/2)*a*g**2*x**4)/(256*x**4)