\(\int \frac {a+b \log (c (4 d+d g x^2)^p)}{x^4 \sqrt {4+g x^2}} \, dx\) [673]

Optimal result

Integrand size = 34, antiderivative size = 119 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^4 \sqrt {4+g x^2}} \, dx=-\frac {b g p \sqrt {4+g x^2}}{24 x}-\frac {1}{12} b g^{3/2} p \text {arcsinh}\left (\frac {\sqrt {g} x}{2}\right )-\frac {\sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{12 x^3}+\frac {g \sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{24 x} \] Output:

-1/24*b*g*p*(g*x^2+4)^(1/2)/x-1/12*b*g^(3/2)*p*arcsinh(1/2*g^(1/2)*x)-1/12 
*(g*x^2+4)^(1/2)*(a+b*ln(c*(d*g*x^2+4*d)^p))/x^3+1/24*g*(g*x^2+4)^(1/2)*(a 
+b*ln(c*(d*g*x^2+4*d)^p))/x
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^4 \sqrt {4+g x^2}} \, dx=\frac {1}{24} \left (-2 b g^{3/2} p \text {arcsinh}\left (\frac {\sqrt {g} x}{2}\right )+\frac {\sqrt {4+g x^2} \left (-2 a+a g x^2-b g p x^2+b \left (-2+g x^2\right ) \log \left (c \left (d \left (4+g x^2\right )\right )^p\right )\right )}{x^3}\right ) \] Input:

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

Output:

(-2*b*g^(3/2)*p*ArcSinh[(Sqrt[g]*x)/2] + (Sqrt[4 + g*x^2]*(-2*a + a*g*x^2 
- b*g*p*x^2 + b*(-2 + g*x^2)*Log[c*(d*(4 + g*x^2))^p]))/x^3)/24
 

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.

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

\(\Big \downarrow \) 2929

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

Input:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.84 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^4 \sqrt {4+g x^2}} \, dx=\left [\frac {2 \, b g^{\frac {3}{2}} p x^{3} \log \left (\sqrt {g} x - \sqrt {g x^{2} + 4}\right ) - {\left ({\left (b g p - a g\right )} x^{2} - {\left (b g p x^{2} - 2 \, b p\right )} \log \left (d g x^{2} + 4 \, d\right ) - {\left (b g x^{2} - 2 \, b\right )} \log \left (c\right ) + 2 \, a\right )} \sqrt {g x^{2} + 4}}{24 \, x^{3}}, \frac {4 \, b \sqrt {-g} g p x^{3} \arctan \left (\frac {\sqrt {g x^{2} + 4} \sqrt {-g} - 2 \, \sqrt {-g}}{g x}\right ) - {\left ({\left (b g p - a g\right )} x^{2} - {\left (b g p x^{2} - 2 \, b p\right )} \log \left (d g x^{2} + 4 \, d\right ) - {\left (b g x^{2} - 2 \, b\right )} \log \left (c\right ) + 2 \, a\right )} \sqrt {g x^{2} + 4}}{24 \, x^{3}}\right ] \] Input:

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

Output:

[1/24*(2*b*g^(3/2)*p*x^3*log(sqrt(g)*x - sqrt(g*x^2 + 4)) - ((b*g*p - a*g) 
*x^2 - (b*g*p*x^2 - 2*b*p)*log(d*g*x^2 + 4*d) - (b*g*x^2 - 2*b)*log(c) + 2 
*a)*sqrt(g*x^2 + 4))/x^3, 1/24*(4*b*sqrt(-g)*g*p*x^3*arctan((sqrt(g*x^2 + 
4)*sqrt(-g) - 2*sqrt(-g))/(g*x)) - ((b*g*p - a*g)*x^2 - (b*g*p*x^2 - 2*b*p 
)*log(d*g*x^2 + 4*d) - (b*g*x^2 - 2*b)*log(c) + 2*a)*sqrt(g*x^2 + 4))/x^3]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/24*(24*g^3*p*integrate(1/12*x^2/(g*x^2 + 4)^(3/2), x) - ((g^2*p*log(d) 
- 3*g^2*p + g^2*log(c))*x^4 + 2*(g*p*log(d) - 2*g*p + g*log(c))*x^2 + (g^2 
*p*x^4 + 2*g*p*x^2 - 8*p)*log(g*x^2 + 4) - 8*p*log(d) - 8*log(c))/(sqrt(g* 
x^2 + 4)*x^3))*b + 1/24*a*(sqrt(g*x^2 + 4)*g/x - 2*sqrt(g*x^2 + 4)/x^3)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.58 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^4 \sqrt {4+g x^2}} \, dx=\frac {3 \sqrt {g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b g \,x^{2}-6 \sqrt {g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b +3 \sqrt {g \,x^{2}+4}\, a g \,x^{2}-6 \sqrt {g \,x^{2}+4}\, a -3 \sqrt {g \,x^{2}+4}\, b g p \,x^{2}-6 \sqrt {g}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x}{2}+\frac {g \,x^{2}}{2}+2\right ) b g p \,x^{3}+3 \sqrt {g}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b g \,x^{3}-3 \sqrt {g}\, a g \,x^{3}+\sqrt {g}\, b g p \,x^{3}}{72 x^{3}} \] Input:

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

Output:

(3*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 - 
6*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 + 3*sqrt(g 
*x**2 + 4)*a*g*x**2 - 6*sqrt(g*x**2 + 4)*a - 3*sqrt(g*x**2 + 4)*b*g*p*x**2 
 - 6*sqrt(g)*log((sqrt(g)*sqrt(g*x**2 + 4)*x + g*x**2 + 4)/2)*b*g*p*x**3 + 
 3*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*x**3 - 3*sqrt(g) 
*a*g*x**3 + sqrt(g)*b*g*p*x**3)/(72*x**3)