\(\int \frac {a+b \log (c (4 d+d g x^2)^p)}{x^6 \sqrt {4+g x^2}} \, dx\) [674]

Optimal result

Integrand size = 34, antiderivative size = 182 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^6 \sqrt {4+g x^2}} \, dx=-\frac {b g p \sqrt {4+g x^2}}{120 x^3}+\frac {b g^2 p \sqrt {4+g x^2}}{80 x}+\frac {1}{60} b g^{5/2} p \text {arcsinh}\left (\frac {\sqrt {g} x}{2}\right )-\frac {\sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{20 x^5}+\frac {g \sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{60 x^3}-\frac {g^2 \sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{120 x} \] Output:

-1/120*b*g*p*(g*x^2+4)^(1/2)/x^3+1/80*b*g^2*p*(g*x^2+4)^(1/2)/x+1/60*b*g^( 
5/2)*p*arcsinh(1/2*g^(1/2)*x)-1/20*(g*x^2+4)^(1/2)*(a+b*ln(c*(d*g*x^2+4*d) 
^p))/x^5+1/60*g*(g*x^2+4)^(1/2)*(a+b*ln(c*(d*g*x^2+4*d)^p))/x^3-1/120*g^2* 
(g*x^2+4)^(1/2)*(a+b*ln(c*(d*g*x^2+4*d)^p))/x
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.59 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^6 \sqrt {4+g x^2}} \, dx=\frac {1}{240} \left (4 b g^{5/2} p \text {arcsinh}\left (\frac {\sqrt {g} x}{2}\right )-\frac {\sqrt {4+g x^2} \left (b g p x^2 \left (2-3 g x^2\right )+2 a \left (6-2 g x^2+g^2 x^4\right )+2 b \left (6-2 g x^2+g^2 x^4\right ) \log \left (c \left (d \left (4+g x^2\right )\right )^p\right )\right )}{x^5}\right ) \] Input:

Integrate[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x^6*Sqrt[4 + g*x^2]),x]
 

Output:

(4*b*g^(5/2)*p*ArcSinh[(Sqrt[g]*x)/2] - (Sqrt[4 + g*x^2]*(b*g*p*x^2*(2 - 3 
*g*x^2) + 2*a*(6 - 2*g*x^2 + g^2*x^4) + 2*b*(6 - 2*g*x^2 + g^2*x^4)*Log[c* 
(d*(4 + g*x^2))^p]))/x^5)/240
 

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x^6 \sqrt {g x^2+4}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x^6 \sqrt {g x^2+4}}dx\)

Input:

Int[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x^6*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]

\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+4 d \right )^{p}\right )}{x^{6} \sqrt {g \,x^{2}+4}}d x\]

Input:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x^6/(g*x^2+4)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x^6/(g*x^2+4)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.63 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^6 \sqrt {4+g x^2}} \, dx=\left [\frac {4 \, b g^{\frac {5}{2}} p x^{5} \log \left (-\sqrt {g} x - \sqrt {g x^{2} + 4}\right ) + {\left ({\left (3 \, b g^{2} p - 2 \, a g^{2}\right )} x^{4} - 2 \, {\left (b g p - 2 \, a g\right )} x^{2} - 2 \, {\left (b g^{2} p x^{4} - 2 \, b g p x^{2} + 6 \, b p\right )} \log \left (d g x^{2} + 4 \, d\right ) - 2 \, {\left (b g^{2} x^{4} - 2 \, b g x^{2} + 6 \, b\right )} \log \left (c\right ) - 12 \, a\right )} \sqrt {g x^{2} + 4}}{240 \, x^{5}}, -\frac {8 \, b \sqrt {-g} g^{2} p x^{5} \arctan \left (\frac {\sqrt {g x^{2} + 4} \sqrt {-g} - 2 \, \sqrt {-g}}{g x}\right ) - {\left ({\left (3 \, b g^{2} p - 2 \, a g^{2}\right )} x^{4} - 2 \, {\left (b g p - 2 \, a g\right )} x^{2} - 2 \, {\left (b g^{2} p x^{4} - 2 \, b g p x^{2} + 6 \, b p\right )} \log \left (d g x^{2} + 4 \, d\right ) - 2 \, {\left (b g^{2} x^{4} - 2 \, b g x^{2} + 6 \, b\right )} \log \left (c\right ) - 12 \, a\right )} \sqrt {g x^{2} + 4}}{240 \, x^{5}}\right ] \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^6/(g*x^2+4)^(1/2),x, algorithm="f 
ricas")
 

Output:

[1/240*(4*b*g^(5/2)*p*x^5*log(-sqrt(g)*x - sqrt(g*x^2 + 4)) + ((3*b*g^2*p 
- 2*a*g^2)*x^4 - 2*(b*g*p - 2*a*g)*x^2 - 2*(b*g^2*p*x^4 - 2*b*g*p*x^2 + 6* 
b*p)*log(d*g*x^2 + 4*d) - 2*(b*g^2*x^4 - 2*b*g*x^2 + 6*b)*log(c) - 12*a)*s 
qrt(g*x^2 + 4))/x^5, -1/240*(8*b*sqrt(-g)*g^2*p*x^5*arctan((sqrt(g*x^2 + 4 
)*sqrt(-g) - 2*sqrt(-g))/(g*x)) - ((3*b*g^2*p - 2*a*g^2)*x^4 - 2*(b*g*p - 
2*a*g)*x^2 - 2*(b*g^2*p*x^4 - 2*b*g*p*x^2 + 6*b*p)*log(d*g*x^2 + 4*d) - 2* 
(b*g^2*x^4 - 2*b*g*x^2 + 6*b)*log(c) - 12*a)*sqrt(g*x^2 + 4))/x^5]
 

Sympy [F]

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

integrate((a+b*ln(c*(d*g*x**2+4*d)**p))/x**6/(g*x**2+4)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*g*x**2 + 4*d)**p))/(x**6*sqrt(g*x**2 + 4)), x)
 

Maxima [F]

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

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^6/(g*x^2+4)^(1/2),x, algorithm="m 
axima")
 

Output:

-1/120*(sqrt(g*x^2 + 4)*g^2/x - 2*sqrt(g*x^2 + 4)*g/x^3 + 6*sqrt(g*x^2 + 4 
)/x^5)*a + 1/240*(240*g^4*p*integrate(1/60*x^2/(g*x^2 + 4)^(3/2), x) - ((2 
*g^3*p*log(d) - 7*g^3*p + 2*g^3*log(c))*x^6 + 2*(2*g^2*p*log(d) - 5*g^2*p 
+ 2*g^2*log(c))*x^4 - 4*(g*p*log(d) - 2*g*p + g*log(c))*x^2 + 2*(g^3*p*x^6 
 + 2*g^2*p*x^4 - 2*g*p*x^2 + 24*p)*log(g*x^2 + 4) + 48*p*log(d) + 48*log(c 
))/(sqrt(g*x^2 + 4)*x^5))*b
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^6 \sqrt {4+g x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^6/(g*x^2+4)^(1/2),x, algorithm="g 
iac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^6 \sqrt {4+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,g\,x^2+4\,d\right )}^p\right )}{x^6\,\sqrt {g\,x^2+4}} \,d x \] Input:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x^6*(g*x^2 + 4)^(1/2)),x)
 

Output:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x^6*(g*x^2 + 4)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.34 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^6 \sqrt {4+g x^2}} \, dx=\frac {-10 \sqrt {g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,g^{2} x^{4}+20 \sqrt {g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b g \,x^{2}-60 \sqrt {g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b -10 \sqrt {g \,x^{2}+4}\, a \,g^{2} x^{4}+20 \sqrt {g \,x^{2}+4}\, a g \,x^{2}-60 \sqrt {g \,x^{2}+4}\, a +15 \sqrt {g \,x^{2}+4}\, b \,g^{2} p \,x^{4}-10 \sqrt {g \,x^{2}+4}\, b g p \,x^{2}+20 \sqrt {g}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x}{2}+\frac {g \,x^{2}}{2}+2\right ) b \,g^{2} p \,x^{5}-10 \sqrt {g}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,g^{2} x^{5}+10 \sqrt {g}\, a \,g^{2} x^{5}-11 \sqrt {g}\, b \,g^{2} p \,x^{5}}{1200 x^{5}} \] Input:

int((a+b*log(c*(d*g*x^2+4*d)^p))/x^6/(g*x^2+4)^(1/2),x)
 

Output:

( - 10*sqrt(g*x**2 + 4)*log((d**p*(sqrt(g)*sqrt(g*x**2 + 4)*x + g*x**2 + 4 
)**(2*p)*4**p*c)/((sqrt(g*x**2 + 4) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*g**2* 
x**4 + 20*sqrt(g*x**2 + 4)*log((d**p*(sqrt(g)*sqrt(g*x**2 + 4)*x + g*x**2 
+ 4)**(2*p)*4**p*c)/((sqrt(g*x**2 + 4) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*g* 
x**2 - 60*sqrt(g*x**2 + 4)*log((d**p*(sqrt(g)*sqrt(g*x**2 + 4)*x + g*x**2 
+ 4)**(2*p)*4**p*c)/((sqrt(g*x**2 + 4) + sqrt(g)*x)**(2*p)*2**(2*p)))*b - 
10*sqrt(g*x**2 + 4)*a*g**2*x**4 + 20*sqrt(g*x**2 + 4)*a*g*x**2 - 60*sqrt(g 
*x**2 + 4)*a + 15*sqrt(g*x**2 + 4)*b*g**2*p*x**4 - 10*sqrt(g*x**2 + 4)*b*g 
*p*x**2 + 20*sqrt(g)*log((sqrt(g)*sqrt(g*x**2 + 4)*x + g*x**2 + 4)/2)*b*g* 
*2*p*x**5 - 10*sqrt(g)*log((d**p*(sqrt(g)*sqrt(g*x**2 + 4)*x + g*x**2 + 4) 
**(2*p)*4**p*c)/((sqrt(g*x**2 + 4) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*g**2*x 
**5 + 10*sqrt(g)*a*g**2*x**5 - 11*sqrt(g)*b*g**2*p*x**5)/(1200*x**5)