\(\int \frac {a+b \log (c (4 d-d g x^2)^p)}{x^6 \sqrt {4-g x^2}} \, dx\) [687]

Optimal result

Integrand size = 36, antiderivative size = 190 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^6 \sqrt {4-g x^2}} \, dx=\frac {b g p \sqrt {4-g x^2}}{120 x^3}+\frac {b g^2 p \sqrt {4-g x^2}}{80 x}-\frac {1}{60} b g^{5/2} p \arcsin \left (\frac {\sqrt {g} x}{2}\right )-\frac {\sqrt {4-g x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{20 x^5}-\frac {g \sqrt {4-g x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{60 x^3}-\frac {g^2 \sqrt {4-g x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{120 x} \] 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*arcsin(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 = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.57 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^6 \sqrt {4-g x^2}} \, dx=\frac {1}{240} \left (-4 b g^{5/2} p \arcsin \left (\frac {\sqrt {g} x}{2}\right )+\frac {\sqrt {4-g x^2} \left (b g p x^2 \left (2+3 g x^2\right )-2 a \left (6+2 g x^2+g^2 x^4\right )-2 b \left (6+2 g x^2+g^2 x^4\right ) \log \left (c \left (d \left (4-g x^2\right )\right )^p\right )\right )}{x^5}\right ) \] 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*ArcSin[(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.

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

\(\Big \downarrow \) 2929

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

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 = 0.11 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.54 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^6 \sqrt {4-g x^2}} \, dx=\left [\frac {4 \, b \sqrt {-g} g^{2} p x^{5} \log \left (-\sqrt {-g} x - \sqrt {-g x^{2} + 4}\right ) + {\left ({\left (3 \, b g^{2} p - 2 \, a g^{2}\right )} x^{4} + 2 \, {\left (b g p - 2 \, a g\right )} x^{2} - 2 \, {\left (b g^{2} p x^{4} + 2 \, b g p x^{2} + 6 \, b p\right )} \log \left (-d g x^{2} + 4 \, d\right ) - 2 \, {\left (b g^{2} x^{4} + 2 \, b g x^{2} + 6 \, b\right )} \log \left (c\right ) - 12 \, a\right )} \sqrt {-g x^{2} + 4}}{240 \, x^{5}}, \frac {8 \, b g^{\frac {5}{2}} p x^{5} \arctan \left (\frac {\sqrt {-g x^{2} + 4} - 2}{\sqrt {g} x}\right ) + {\left ({\left (3 \, b g^{2} p - 2 \, a g^{2}\right )} x^{4} + 2 \, {\left (b g p - 2 \, a g\right )} x^{2} - 2 \, {\left (b g^{2} p x^{4} + 2 \, b g p x^{2} + 6 \, b p\right )} \log \left (-d g x^{2} + 4 \, d\right ) - 2 \, {\left (b g^{2} x^{4} + 2 \, b g x^{2} + 6 \, b\right )} \log \left (c\right ) - 12 \, a\right )} \sqrt {-g x^{2} + 4}}{240 \, x^{5}}\right ] \] Input:

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

Output:

[1/240*(4*b*sqrt(-g)*g^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)*sqrt(-g*x^2 + 4))/x^5, 1/240*(8*b*g^(5/2)*p*x^5*arctan((sqrt(-g*x^2 
 + 4) - 2)/(sqrt(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(-1)]

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

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

Output:

Timed out
 

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= 
"maxima")
 

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

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= 
"giac")
 

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.54 \[ \int \frac {a+b \log \left (c \left (4 d-d g x^2\right )^p\right )}{x^6 \sqrt {4-g x^2}} \, dx=\frac {-4 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right ) b \,g^{2} p \,x^{5}-2 \sqrt {-g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p} 4^{p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b \,g^{2} x^{4}-4 \sqrt {-g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p} 4^{p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b g \,x^{2}-12 \sqrt {-g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p} 4^{p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{2}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b -2 \sqrt {-g \,x^{2}+4}\, a \,g^{2} x^{4}-4 \sqrt {-g \,x^{2}+4}\, a g \,x^{2}-12 \sqrt {-g \,x^{2}+4}\, a +3 \sqrt {-g \,x^{2}+4}\, b \,g^{2} p \,x^{4}+2 \sqrt {-g \,x^{2}+4}\, b g p \,x^{2}}{240 x^{5}} \] Input:

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

Output:

( - 4*sqrt(g)*asin((sqrt(g)*x)/2)*b*g**2*p*x**5 - 2*sqrt( - g*x**2 + 4)*lo 
g((d**p*( - tan(asin((sqrt(g)*x)/2)/2)**2 + 1)**(2*p)*4**p*c)/(tan(asin((s 
qrt(g)*x)/2)/2)**2 + 1)**(2*p))*b*g**2*x**4 - 4*sqrt( - g*x**2 + 4)*log((d 
**p*( - tan(asin((sqrt(g)*x)/2)/2)**2 + 1)**(2*p)*4**p*c)/(tan(asin((sqrt( 
g)*x)/2)/2)**2 + 1)**(2*p))*b*g*x**2 - 12*sqrt( - g*x**2 + 4)*log((d**p*( 
- tan(asin((sqrt(g)*x)/2)/2)**2 + 1)**(2*p)*4**p*c)/(tan(asin((sqrt(g)*x)/ 
2)/2)**2 + 1)**(2*p))*b - 2*sqrt( - g*x**2 + 4)*a*g**2*x**4 - 4*sqrt( - g* 
x**2 + 4)*a*g*x**2 - 12*sqrt( - g*x**2 + 4)*a + 3*sqrt( - g*x**2 + 4)*b*g* 
*2*p*x**4 + 2*sqrt( - g*x**2 + 4)*b*g*p*x**2)/(240*x**5)