\(\int \frac {a+b \log (c (4 d+d g x^2)^p)}{x^8 \sqrt {4+g x^2}} \, dx\) [675]

Optimal result

Integrand size = 34, antiderivative size = 245 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^8 \sqrt {4+g x^2}} \, dx=-\frac {b g p \sqrt {4+g x^2}}{280 x^5}+\frac {b g^2 p \sqrt {4+g x^2}}{336 x^3}-\frac {11 b g^3 p \sqrt {4+g x^2}}{3360 x}-\frac {1}{280} b g^{7/2} p \text {arcsinh}\left (\frac {\sqrt {g} x}{2}\right )-\frac {\sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{28 x^7}+\frac {3 g \sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{280 x^5}-\frac {g^2 \sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{280 x^3}+\frac {g^3 \sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{560 x} \] Output:

-1/280*b*g*p*(g*x^2+4)^(1/2)/x^5+1/336*b*g^2*p*(g*x^2+4)^(1/2)/x^3-11/3360 
*b*g^3*p*(g*x^2+4)^(1/2)/x-1/280*b*g^(7/2)*p*arcsinh(1/2*g^(1/2)*x)-1/28*( 
g*x^2+4)^(1/2)*(a+b*ln(c*(d*g*x^2+4*d)^p))/x^7+3/280*g*(g*x^2+4)^(1/2)*(a+ 
b*ln(c*(d*g*x^2+4*d)^p))/x^5-1/280*g^2*(g*x^2+4)^(1/2)*(a+b*ln(c*(d*g*x^2+ 
4*d)^p))/x^3+1/560*g^3*(g*x^2+4)^(1/2)*(a+b*ln(c*(d*g*x^2+4*d)^p))/x
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.53 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^8 \sqrt {4+g x^2}} \, dx=\frac {-12 b g^{7/2} p \text {arcsinh}\left (\frac {\sqrt {g} x}{2}\right )+\frac {\sqrt {4+g x^2} \left (b g p x^2 \left (-12+10 g x^2-11 g^2 x^4\right )+6 a \left (-20+6 g x^2-2 g^2 x^4+g^3 x^6\right )+6 b \left (-20+6 g x^2-2 g^2 x^4+g^3 x^6\right ) \log \left (c \left (d \left (4+g x^2\right )\right )^p\right )\right )}{x^7}}{3360} \] Input:

Integrate[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x^8*Sqrt[4 + g*x^2]),x]
 

Output:

(-12*b*g^(7/2)*p*ArcSinh[(Sqrt[g]*x)/2] + (Sqrt[4 + g*x^2]*(b*g*p*x^2*(-12 
 + 10*g*x^2 - 11*g^2*x^4) + 6*a*(-20 + 6*g*x^2 - 2*g^2*x^4 + g^3*x^6) + 6* 
b*(-20 + 6*g*x^2 - 2*g^2*x^4 + g^3*x^6)*Log[c*(d*(4 + g*x^2))^p]))/x^7)/33 
60
 

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x^8 \sqrt {g x^2+4}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (d g x^2+4 d\right )^p\right )}{x^8 \sqrt {g x^2+4}}dx\)

Input:

Int[(a + b*Log[c*(4*d + d*g*x^2)^p])/(x^8*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]

\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+4 d \right )^{p}\right )}{x^{8} \sqrt {g \,x^{2}+4}}d x\]

Input:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x^8/(g*x^2+4)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*g*x^2+4*d)^p))/x^8/(g*x^2+4)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^8 \sqrt {4+g x^2}} \, dx=\left [\frac {12 \, b g^{\frac {7}{2}} p x^{7} \log \left (\sqrt {g} x - \sqrt {g x^{2} + 4}\right ) - {\left ({\left (11 \, b g^{3} p - 6 \, a g^{3}\right )} x^{6} - 2 \, {\left (5 \, b g^{2} p - 6 \, a g^{2}\right )} x^{4} + 12 \, {\left (b g p - 3 \, a g\right )} x^{2} - 6 \, {\left (b g^{3} p x^{6} - 2 \, b g^{2} p x^{4} + 6 \, b g p x^{2} - 20 \, b p\right )} \log \left (d g x^{2} + 4 \, d\right ) - 6 \, {\left (b g^{3} x^{6} - 2 \, b g^{2} x^{4} + 6 \, b g x^{2} - 20 \, b\right )} \log \left (c\right ) + 120 \, a\right )} \sqrt {g x^{2} + 4}}{3360 \, x^{7}}, \frac {24 \, b \sqrt {-g} g^{3} p x^{7} \arctan \left (\frac {\sqrt {g x^{2} + 4} \sqrt {-g} - 2 \, \sqrt {-g}}{g x}\right ) - {\left ({\left (11 \, b g^{3} p - 6 \, a g^{3}\right )} x^{6} - 2 \, {\left (5 \, b g^{2} p - 6 \, a g^{2}\right )} x^{4} + 12 \, {\left (b g p - 3 \, a g\right )} x^{2} - 6 \, {\left (b g^{3} p x^{6} - 2 \, b g^{2} p x^{4} + 6 \, b g p x^{2} - 20 \, b p\right )} \log \left (d g x^{2} + 4 \, d\right ) - 6 \, {\left (b g^{3} x^{6} - 2 \, b g^{2} x^{4} + 6 \, b g x^{2} - 20 \, b\right )} \log \left (c\right ) + 120 \, a\right )} \sqrt {g x^{2} + 4}}{3360 \, x^{7}}\right ] \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^8/(g*x^2+4)^(1/2),x, algorithm="f 
ricas")
 

Output:

[1/3360*(12*b*g^(7/2)*p*x^7*log(sqrt(g)*x - sqrt(g*x^2 + 4)) - ((11*b*g^3* 
p - 6*a*g^3)*x^6 - 2*(5*b*g^2*p - 6*a*g^2)*x^4 + 12*(b*g*p - 3*a*g)*x^2 - 
6*(b*g^3*p*x^6 - 2*b*g^2*p*x^4 + 6*b*g*p*x^2 - 20*b*p)*log(d*g*x^2 + 4*d) 
- 6*(b*g^3*x^6 - 2*b*g^2*x^4 + 6*b*g*x^2 - 20*b)*log(c) + 120*a)*sqrt(g*x^ 
2 + 4))/x^7, 1/3360*(24*b*sqrt(-g)*g^3*p*x^7*arctan((sqrt(g*x^2 + 4)*sqrt( 
-g) - 2*sqrt(-g))/(g*x)) - ((11*b*g^3*p - 6*a*g^3)*x^6 - 2*(5*b*g^2*p - 6* 
a*g^2)*x^4 + 12*(b*g*p - 3*a*g)*x^2 - 6*(b*g^3*p*x^6 - 2*b*g^2*p*x^4 + 6*b 
*g*p*x^2 - 20*b*p)*log(d*g*x^2 + 4*d) - 6*(b*g^3*x^6 - 2*b*g^2*x^4 + 6*b*g 
*x^2 - 20*b)*log(c) + 120*a)*sqrt(g*x^2 + 4))/x^7]
 

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^8 \sqrt {4+g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d g x^{2} + 4 d\right )^{p} \right )}}{x^{8} \sqrt {g x^{2} + 4}}\, dx \] Input:

integrate((a+b*ln(c*(d*g*x**2+4*d)**p))/x**8/(g*x**2+4)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*g*x**2 + 4*d)**p))/(x**8*sqrt(g*x**2 + 4)), x)
 

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^8 \sqrt {4+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + 4} x^{8}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^8/(g*x^2+4)^(1/2),x, algorithm="m 
axima")
 

Output:

1/560*(sqrt(g*x^2 + 4)*g^3/x - 2*sqrt(g*x^2 + 4)*g^2/x^3 + 6*sqrt(g*x^2 + 
4)*g/x^5 - 20*sqrt(g*x^2 + 4)/x^7)*a - 1/3360*(3360*g^5*p*integrate(1/280* 
x^2/(g*x^2 + 4)^(3/2), x) - ((6*g^4*p*log(d) - 23*g^4*p + 6*g^4*log(c))*x^ 
8 + 2*(6*g^3*p*log(d) - 17*g^3*p + 6*g^3*log(c))*x^6 - 4*(3*g^2*p*log(d) - 
 7*g^2*p + 3*g^2*log(c))*x^4 + 24*(g*p*log(d) - 2*g*p + g*log(c))*x^2 + 6* 
(g^4*p*x^8 + 2*g^3*p*x^6 - 2*g^2*p*x^4 + 4*g*p*x^2 - 80*p)*log(g*x^2 + 4) 
- 480*p*log(d) - 480*log(c))/(sqrt(g*x^2 + 4)*x^7))*b
 

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^8 \sqrt {4+g x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*log(c*(d*g*x^2+4*d)^p))/x^8/(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. \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^8 \sqrt {4+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,g\,x^2+4\,d\right )}^p\right )}{x^8\,\sqrt {g\,x^2+4}} \,d x \] Input:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x^8*(g*x^2 + 4)^(1/2)),x)
 

Output:

int((a + b*log(c*(4*d + d*g*x^2)^p))/(x^8*(g*x^2 + 4)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.19 \[ \int \frac {a+b \log \left (c \left (4 d+d g x^2\right )^p\right )}{x^8 \sqrt {4+g x^2}} \, dx=\frac {42 \sqrt {g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,g^{3} x^{6}-84 \sqrt {g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,g^{2} x^{4}+252 \sqrt {g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b g \,x^{2}-840 \sqrt {g \,x^{2}+4}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b +42 \sqrt {g \,x^{2}+4}\, a \,g^{3} x^{6}-84 \sqrt {g \,x^{2}+4}\, a \,g^{2} x^{4}+252 \sqrt {g \,x^{2}+4}\, a g \,x^{2}-840 \sqrt {g \,x^{2}+4}\, a -77 \sqrt {g \,x^{2}+4}\, b \,g^{3} p \,x^{6}+70 \sqrt {g \,x^{2}+4}\, b \,g^{2} p \,x^{4}-84 \sqrt {g \,x^{2}+4}\, b g p \,x^{2}-84 \sqrt {g}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x}{2}+\frac {g \,x^{2}}{2}+2\right ) b \,g^{3} p \,x^{7}+42 \sqrt {g}\, \mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (\sqrt {g \,x^{2}+4}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,g^{3} x^{7}-42 \sqrt {g}\, a \,g^{3} x^{7}+65 \sqrt {g}\, b \,g^{3} p \,x^{7}}{23520 x^{7}} \] Input:

int((a+b*log(c*(d*g*x^2+4*d)^p))/x^8/(g*x^2+4)^(1/2),x)
 

Output:

(42*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**3*x** 
6 - 84*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 + 252*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 - 840*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 
+ 42*sqrt(g*x**2 + 4)*a*g**3*x**6 - 84*sqrt(g*x**2 + 4)*a*g**2*x**4 + 252* 
sqrt(g*x**2 + 4)*a*g*x**2 - 840*sqrt(g*x**2 + 4)*a - 77*sqrt(g*x**2 + 4)*b 
*g**3*p*x**6 + 70*sqrt(g*x**2 + 4)*b*g**2*p*x**4 - 84*sqrt(g*x**2 + 4)*b*g 
*p*x**2 - 84*sqrt(g)*log((sqrt(g)*sqrt(g*x**2 + 4)*x + g*x**2 + 4)/2)*b*g* 
*3*p*x**7 + 42*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**3*x 
**7 - 42*sqrt(g)*a*g**3*x**7 + 65*sqrt(g)*b*g**3*p*x**7)/(23520*x**7)