\(\int \frac {a+b \log (c (d f+d g x^2)^p)}{x^8 \sqrt {f+g x^2}} \, dx\) [701]

Optimal result

Integrand size = 34, antiderivative size = 277 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^8 \sqrt {f+g x^2}} \, dx=-\frac {2 b g p \sqrt {f+g x^2}}{35 f^2 x^5}+\frac {4 b g^2 p \sqrt {f+g x^2}}{21 f^3 x^3}-\frac {88 b g^3 p \sqrt {f+g x^2}}{105 f^4 x}-\frac {32 b g^{7/2} p \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f+g x^2}}\right )}{35 f^4}-\frac {\sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{7 f x^7}+\frac {6 g \sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{35 f^2 x^5}-\frac {8 g^2 \sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{35 f^3 x^3}+\frac {16 g^3 \sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{35 f^4 x} \] Output:

-2/35*b*g*p*(g*x^2+f)^(1/2)/f^2/x^5+4/21*b*g^2*p*(g*x^2+f)^(1/2)/f^3/x^3-8 
8/105*b*g^3*p*(g*x^2+f)^(1/2)/f^4/x-32/35*b*g^(7/2)*p*arctanh(g^(1/2)*x/(g 
*x^2+f)^(1/2))/f^4-1/7*(g*x^2+f)^(1/2)*(a+b*ln(c*(d*g*x^2+d*f)^p))/f/x^7+6 
/35*g*(g*x^2+f)^(1/2)*(a+b*ln(c*(d*g*x^2+d*f)^p))/f^2/x^5-8/35*g^2*(g*x^2+ 
f)^(1/2)*(a+b*ln(c*(d*g*x^2+d*f)^p))/f^3/x^3+16/35*g^3*(g*x^2+f)^(1/2)*(a+ 
b*ln(c*(d*g*x^2+d*f)^p))/f^4/x
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.66 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^8 \sqrt {f+g x^2}} \, dx=\frac {\sqrt {f+g x^2} \left (-2 b g p x^2 \left (3 f^2-10 f g x^2+44 g^2 x^4\right )-3 a \left (5 f^3-6 f^2 g x^2+8 f g^2 x^4-16 g^3 x^6\right )\right )-3 b \sqrt {f+g x^2} \left (5 f^3-6 f^2 g x^2+8 f g^2 x^4-16 g^3 x^6\right ) \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )-96 b g^{7/2} p x^7 \log \left (g x+\sqrt {g} \sqrt {f+g x^2}\right )}{105 f^4 x^7} \] Input:

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

Output:

(Sqrt[f + g*x^2]*(-2*b*g*p*x^2*(3*f^2 - 10*f*g*x^2 + 44*g^2*x^4) - 3*a*(5* 
f^3 - 6*f^2*g*x^2 + 8*f*g^2*x^4 - 16*g^3*x^6)) - 3*b*Sqrt[f + g*x^2]*(5*f^ 
3 - 6*f^2*g*x^2 + 8*f*g^2*x^4 - 16*g^3*x^6)*Log[c*(d*(f + g*x^2))^p] - 96* 
b*g^(7/2)*p*x^7*Log[g*x + Sqrt[g]*Sqrt[f + g*x^2]])/(105*f^4*x^7)
 

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.

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

\(\Big \downarrow \) 2929

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

Input:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.55 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^8 \sqrt {f+g x^2}} \, dx=\left [\frac {48 \, b g^{\frac {7}{2}} p x^{7} \log \left (-2 \, g x^{2} + 2 \, \sqrt {g x^{2} + f} \sqrt {g} x - f\right ) - {\left (8 \, {\left (11 \, b g^{3} p - 6 \, a g^{3}\right )} x^{6} - 4 \, {\left (5 \, b f g^{2} p - 6 \, a f g^{2}\right )} x^{4} + 15 \, a f^{3} + 6 \, {\left (b f^{2} g p - 3 \, a f^{2} g\right )} x^{2} - 3 \, {\left (16 \, b g^{3} p x^{6} - 8 \, b f g^{2} p x^{4} + 6 \, b f^{2} g p x^{2} - 5 \, b f^{3} p\right )} \log \left (d g x^{2} + d f\right ) - 3 \, {\left (16 \, b g^{3} x^{6} - 8 \, b f g^{2} x^{4} + 6 \, b f^{2} g x^{2} - 5 \, b f^{3}\right )} \log \left (c\right )\right )} \sqrt {g x^{2} + f}}{105 \, f^{4} x^{7}}, \frac {96 \, b \sqrt {-g} g^{3} p x^{7} \arctan \left (\frac {\sqrt {-g} x}{\sqrt {g x^{2} + f}}\right ) - {\left (8 \, {\left (11 \, b g^{3} p - 6 \, a g^{3}\right )} x^{6} - 4 \, {\left (5 \, b f g^{2} p - 6 \, a f g^{2}\right )} x^{4} + 15 \, a f^{3} + 6 \, {\left (b f^{2} g p - 3 \, a f^{2} g\right )} x^{2} - 3 \, {\left (16 \, b g^{3} p x^{6} - 8 \, b f g^{2} p x^{4} + 6 \, b f^{2} g p x^{2} - 5 \, b f^{3} p\right )} \log \left (d g x^{2} + d f\right ) - 3 \, {\left (16 \, b g^{3} x^{6} - 8 \, b f g^{2} x^{4} + 6 \, b f^{2} g x^{2} - 5 \, b f^{3}\right )} \log \left (c\right )\right )} \sqrt {g x^{2} + f}}{105 \, f^{4} x^{7}}\right ] \] Input:

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

Output:

[1/105*(48*b*g^(7/2)*p*x^7*log(-2*g*x^2 + 2*sqrt(g*x^2 + f)*sqrt(g)*x - f) 
 - (8*(11*b*g^3*p - 6*a*g^3)*x^6 - 4*(5*b*f*g^2*p - 6*a*f*g^2)*x^4 + 15*a* 
f^3 + 6*(b*f^2*g*p - 3*a*f^2*g)*x^2 - 3*(16*b*g^3*p*x^6 - 8*b*f*g^2*p*x^4 
+ 6*b*f^2*g*p*x^2 - 5*b*f^3*p)*log(d*g*x^2 + d*f) - 3*(16*b*g^3*x^6 - 8*b* 
f*g^2*x^4 + 6*b*f^2*g*x^2 - 5*b*f^3)*log(c))*sqrt(g*x^2 + f))/(f^4*x^7), 1 
/105*(96*b*sqrt(-g)*g^3*p*x^7*arctan(sqrt(-g)*x/sqrt(g*x^2 + f)) - (8*(11* 
b*g^3*p - 6*a*g^3)*x^6 - 4*(5*b*f*g^2*p - 6*a*f*g^2)*x^4 + 15*a*f^3 + 6*(b 
*f^2*g*p - 3*a*f^2*g)*x^2 - 3*(16*b*g^3*p*x^6 - 8*b*f*g^2*p*x^4 + 6*b*f^2* 
g*p*x^2 - 5*b*f^3*p)*log(d*g*x^2 + d*f) - 3*(16*b*g^3*x^6 - 8*b*f*g^2*x^4 
+ 6*b*f^2*g*x^2 - 5*b*f^3)*log(c))*sqrt(g*x^2 + f))/(f^4*x^7)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/105*(3360*g^5*p*integrate(1/35*x^2/((f^4*g*x^2 + f^5)*sqrt(g*x^2 + f)), 
 x) - (8*(6*g^4*p*log(d) - 23*g^4*p + 6*g^4*log(c))*x^8 + 4*(6*f*g^3*p*log 
(d) - 17*f*g^3*p + 6*f*g^3*log(c))*x^6 - 15*f^4*p*log(d) - 2*(3*f^2*g^2*p* 
log(d) - 7*f^2*g^2*p + 3*f^2*g^2*log(c))*x^4 - 15*f^4*log(c) + 3*(f^3*g*p* 
log(d) - 2*f^3*g*p + f^3*g*log(c))*x^2 + 3*(16*g^4*p*x^8 + 8*f*g^3*p*x^6 - 
 2*f^2*g^2*p*x^4 + f^3*g*p*x^2 - 5*f^4*p)*log(g*x^2 + f))/(sqrt(g*x^2 + f) 
*f^4*x^7))*b + 1/35*a*(16*sqrt(g*x^2 + f)*g^3/(f^4*x) - 8*sqrt(g*x^2 + f)* 
g^2/(f^3*x^3) + 6*sqrt(g*x^2 + f)*g/(f^2*x^5) - 5*sqrt(g*x^2 + f)/(f*x^7))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (241) = 482\).

Time = 1.24 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.86 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^8 \sqrt {f+g x^2}} \, dx=\frac {16}{105} \, b p {\left (\frac {3 \, g^{\frac {7}{2}} \log \left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2}\right )}{f^{4}} - \frac {3 \, g^{\frac {7}{2}} \log \left ({\left | g x^{2} + f \right |}\right )}{f^{4}} + \frac {6 \, {\left (35 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{6} - 21 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{4} f + 7 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} f^{2} - f^{3}\right )} g^{\frac {7}{2}} \log \left (d g x^{2} + d f\right )}{{\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{7}} + \frac {6 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{8} g^{\frac {7}{2}} - 33 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{6} f g^{\frac {7}{2}} + 77 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{4} f^{2} g^{\frac {7}{2}} - 49 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} f^{3} g^{\frac {7}{2}} + 11 \, f^{4} g^{\frac {7}{2}}}{{\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{5} f^{3}}\right )} + \frac {32 \, {\left (35 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{6} - 21 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{4} f + 7 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} f^{2} - f^{3}\right )} b g^{\frac {7}{2}} \log \left (c\right )}{35 \, {\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{7}} + \frac {32 \, {\left (35 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{6} - 21 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{4} f + 7 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} f^{2} - f^{3}\right )} a g^{\frac {7}{2}}}{35 \, {\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{7}} \] Input:

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

Output:

16/105*b*p*(3*g^(7/2)*log((sqrt(g)*x - sqrt(g*x^2 + f))^2)/f^4 - 3*g^(7/2) 
*log(abs(g*x^2 + f))/f^4 + 6*(35*(sqrt(g)*x - sqrt(g*x^2 + f))^6 - 21*(sqr 
t(g)*x - sqrt(g*x^2 + f))^4*f + 7*(sqrt(g)*x - sqrt(g*x^2 + f))^2*f^2 - f^ 
3)*g^(7/2)*log(d*g*x^2 + d*f)/((sqrt(g)*x - sqrt(g*x^2 + f))^2 - f)^7 + (6 
*(sqrt(g)*x - sqrt(g*x^2 + f))^8*g^(7/2) - 33*(sqrt(g)*x - sqrt(g*x^2 + f) 
)^6*f*g^(7/2) + 77*(sqrt(g)*x - sqrt(g*x^2 + f))^4*f^2*g^(7/2) - 49*(sqrt( 
g)*x - sqrt(g*x^2 + f))^2*f^3*g^(7/2) + 11*f^4*g^(7/2))/(((sqrt(g)*x - sqr 
t(g*x^2 + f))^2 - f)^5*f^3)) + 32/35*(35*(sqrt(g)*x - sqrt(g*x^2 + f))^6 - 
 21*(sqrt(g)*x - sqrt(g*x^2 + f))^4*f + 7*(sqrt(g)*x - sqrt(g*x^2 + f))^2* 
f^2 - f^3)*b*g^(7/2)*log(c)/((sqrt(g)*x - sqrt(g*x^2 + f))^2 - f)^7 + 32/3 
5*(35*(sqrt(g)*x - sqrt(g*x^2 + f))^6 - 21*(sqrt(g)*x - sqrt(g*x^2 + f))^4 
*f + 7*(sqrt(g)*x - sqrt(g*x^2 + f))^2*f^2 - f^3)*a*g^(7/2)/((sqrt(g)*x - 
sqrt(g*x^2 + f))^2 - f)^7
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.05 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^8 \sqrt {f+g x^2}} \, dx=\frac {-105 \sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,f^{3}+126 \sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,f^{2} g \,x^{2}-168 \sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b f \,g^{2} x^{4}+336 \sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,g^{3} x^{6}-105 \sqrt {g \,x^{2}+f}\, a \,f^{3}+126 \sqrt {g \,x^{2}+f}\, a \,f^{2} g \,x^{2}-168 \sqrt {g \,x^{2}+f}\, a f \,g^{2} x^{4}+336 \sqrt {g \,x^{2}+f}\, a \,g^{3} x^{6}-42 \sqrt {g \,x^{2}+f}\, b \,f^{2} g p \,x^{2}+140 \sqrt {g \,x^{2}+f}\, b f \,g^{2} p \,x^{4}-616 \sqrt {g \,x^{2}+f}\, b \,g^{3} p \,x^{6}-672 \sqrt {g}\, \mathrm {log}\left (\frac {2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}}{f}\right ) b \,g^{3} p \,x^{7}+336 \sqrt {g}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,g^{3} x^{7}-336 \sqrt {g}\, a \,g^{3} x^{7}+520 \sqrt {g}\, b \,g^{3} p \,x^{7}}{735 f^{4} x^{7}} \] Input:

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

Output:

( - 105*sqrt(f + g*x**2)*log((d**p*(2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 2 
*g*x**2)**(2*p)*c)/((sqrt(f + g*x**2) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*f** 
3 + 126*sqrt(f + g*x**2)*log((d**p*(2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 2 
*g*x**2)**(2*p)*c)/((sqrt(f + g*x**2) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*f** 
2*g*x**2 - 168*sqrt(f + g*x**2)*log((d**p*(2*sqrt(g)*sqrt(f + g*x**2)*x + 
2*f + 2*g*x**2)**(2*p)*c)/((sqrt(f + g*x**2) + sqrt(g)*x)**(2*p)*2**(2*p)) 
)*b*f*g**2*x**4 + 336*sqrt(f + g*x**2)*log((d**p*(2*sqrt(g)*sqrt(f + g*x** 
2)*x + 2*f + 2*g*x**2)**(2*p)*c)/((sqrt(f + g*x**2) + sqrt(g)*x)**(2*p)*2* 
*(2*p)))*b*g**3*x**6 - 105*sqrt(f + g*x**2)*a*f**3 + 126*sqrt(f + g*x**2)* 
a*f**2*g*x**2 - 168*sqrt(f + g*x**2)*a*f*g**2*x**4 + 336*sqrt(f + g*x**2)* 
a*g**3*x**6 - 42*sqrt(f + g*x**2)*b*f**2*g*p*x**2 + 140*sqrt(f + g*x**2)*b 
*f*g**2*p*x**4 - 616*sqrt(f + g*x**2)*b*g**3*p*x**6 - 672*sqrt(g)*log((2*s 
qrt(g)*sqrt(f + g*x**2)*x + 2*f + 2*g*x**2)/f)*b*g**3*p*x**7 + 336*sqrt(g) 
*log((d**p*(2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 2*g*x**2)**(2*p)*c)/((sqr 
t(f + g*x**2) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*g**3*x**7 - 336*sqrt(g)*a*g 
**3*x**7 + 520*sqrt(g)*b*g**3*p*x**7)/(735*f**4*x**7)