\(\int \frac {a+b \log (c (d f-d g x^2)^p)}{x^8 \sqrt {f-g x^2}} \, dx\) [714]

Optimal result

Integrand size = 36, antiderivative size = 289 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^8 \sqrt {f-g x^2}} \, dx=\frac {2 b g p \sqrt {f-g x^2}}{35 f^2 x^5}+\frac {4 b g^2 p \sqrt {f-g x^2}}{21 f^3 x^3}+\frac {88 b g^3 p \sqrt {f-g x^2}}{105 f^4 x}-\frac {32 b g^{7/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f-g x^2}}\right )}{35 f^4}-\frac {\sqrt {f-g x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{7 f x^7}-\frac {6 g \sqrt {f-g x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{35 f^2 x^5}-\frac {8 g^2 \sqrt {f-g x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{35 f^3 x^3}-\frac {16 g^3 \sqrt {f-g x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{35 f^4 x} \] 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+ 
88/105*b*g^3*p*(-g*x^2+f)^(1/2)/f^4/x-32/35*b*g^(7/2)*p*arctan(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 = 0.32 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.59 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^8 \sqrt {f-g x^2}} \, dx=\frac {-96 b g^{7/2} p x^7 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f-g x^2}}\right )+\sqrt {f-g x^2} \left (2 b g p x^2 \left (3 f^2+10 f g x^2+44 g^2 x^4\right )-3 a \left (5 f^3+6 f^2 g x^2+8 f g^2 x^4+16 g^3 x^6\right )-3 b \left (5 f^3+6 f^2 g x^2+8 f g^2 x^4+16 g^3 x^6\right ) \log \left (c \left (d \left (f-g x^2\right )\right )^p\right )\right )}{105 f^4 x^7} \] Input:

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

Output:

(-96*b*g^(7/2)*p*x^7*ArcTan[(Sqrt[g]*x)/Sqrt[f - g*x^2]] + 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*(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]))/(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 = 0.12 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.54 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^8 \sqrt {f-g x^2}} \, dx=\left [\frac {48 \, b \sqrt {-g} g^{3} p x^{7} \log \left (2 \, g x^{2} - 2 \, \sqrt {-g x^{2} + f} \sqrt {-g} x - f\right ) + {\left (8 \, {\left (11 \, b g^{3} p - 6 \, a g^{3}\right )} x^{6} + 4 \, {\left (5 \, b f g^{2} p - 6 \, a f g^{2}\right )} x^{4} - 15 \, a f^{3} + 6 \, {\left (b f^{2} g p - 3 \, a f^{2} g\right )} x^{2} - 3 \, {\left (16 \, b g^{3} p x^{6} + 8 \, b f g^{2} p x^{4} + 6 \, b f^{2} g p x^{2} + 5 \, b f^{3} p\right )} \log \left (-d g x^{2} + d f\right ) - 3 \, {\left (16 \, b g^{3} x^{6} + 8 \, b f g^{2} x^{4} + 6 \, b f^{2} g x^{2} + 5 \, b f^{3}\right )} \log \left (c\right )\right )} \sqrt {-g x^{2} + f}}{105 \, f^{4} x^{7}}, \frac {96 \, b g^{\frac {7}{2}} p x^{7} \arctan \left (\frac {\sqrt {-g x^{2} + f} \sqrt {g} x}{g x^{2} - f}\right ) + {\left (8 \, {\left (11 \, b g^{3} p - 6 \, a g^{3}\right )} x^{6} + 4 \, {\left (5 \, b f g^{2} p - 6 \, a f g^{2}\right )} x^{4} - 15 \, a f^{3} + 6 \, {\left (b f^{2} g p - 3 \, a f^{2} g\right )} x^{2} - 3 \, {\left (16 \, b g^{3} p x^{6} + 8 \, b f g^{2} p x^{4} + 6 \, b f^{2} g p x^{2} + 5 \, b f^{3} p\right )} \log \left (-d g x^{2} + d f\right ) - 3 \, {\left (16 \, b g^{3} x^{6} + 8 \, b f g^{2} x^{4} + 6 \, b f^{2} g x^{2} + 5 \, b f^{3}\right )} \log \left (c\right )\right )} \sqrt {-g x^{2} + f}}{105 \, f^{4} x^{7}}\right ] \] Input:

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

Output:

[1/105*(48*b*sqrt(-g)*g^3*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*g^(7/2)*p*x^7*arctan(sqrt(-g*x^2 + f)*sqrt(g)*x/(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(-1)]

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

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

Output:

Timed out
 

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= 
"maxima")
 

Output:

-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)) - 1/35*b* 
(integrate((32*g^4*p*x^8 + 16*f*g^3*p*x^6 + 12*f^2*g^2*p*x^4 + 10*f^3*g*p* 
x^2 - 35*f^4*p*log(d) - 35*f^4*log(c))/(sqrt(-g*x^2 + f)*x^8), x)/f^4 - (1 
6*g^4*x^8 - 8*f*g^3*x^6 - 2*f^2*g^2*x^4 - f^3*g*x^2 - 5*f^4)*log((-g*x^2 + 
 f)^p)/(sqrt(-g*x^2 + f)*f^4*x^7))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (253) = 506\).

Time = 1.19 (sec) , antiderivative size = 623, normalized size of antiderivative = 2.16 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^8 \sqrt {f-g x^2}} \, dx=-\frac {16}{105} \, b p {\left (\frac {3 \, \sqrt {-g} g^{3} \log \left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2}\right )}{f^{4}} - \frac {3 \, \sqrt {-g} g^{3} \log \left ({\left | g x^{2} - f \right |}\right )}{f^{4}} + \frac {6 \, {\left (35 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{6} - 21 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{4} f + 7 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} f^{2} - f^{3}\right )} \sqrt {-g} g^{3} \log \left (-d g x^{2} + d f\right )}{{\left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )}^{7}} + \frac {6 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{8} \sqrt {-g} g^{3} - 33 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{6} f \sqrt {-g} g^{3} + 77 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{4} f^{2} \sqrt {-g} g^{3} - 49 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} f^{3} \sqrt {-g} g^{3} + 11 \, f^{4} \sqrt {-g} g^{3}}{{\left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )}^{5} f^{3}}\right )} - \frac {32 \, {\left (35 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{6} - 21 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{4} f + 7 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} f^{2} - f^{3}\right )} b \sqrt {-g} g^{3} \log \left (c\right )}{35 \, {\left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )}^{7}} - \frac {32 \, {\left (35 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{6} - 21 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{4} f + 7 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} f^{2} - f^{3}\right )} a \sqrt {-g} g^{3}}{35 \, {\left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )}^{7}} \] Input:

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

Output:

-16/105*b*p*(3*sqrt(-g)*g^3*log((sqrt(-g)*x - sqrt(-g*x^2 + f))^2)/f^4 - 3 
*sqrt(-g)*g^3*log(abs(g*x^2 - f))/f^4 + 6*(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)*sqrt(-g)*g^3*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*sqrt(-g)*g^3 
 - 33*(sqrt(-g)*x - sqrt(-g*x^2 + f))^6*f*sqrt(-g)*g^3 + 77*(sqrt(-g)*x - 
sqrt(-g*x^2 + f))^4*f^2*sqrt(-g)*g^3 - 49*(sqrt(-g)*x - sqrt(-g*x^2 + f))^ 
2*f^3*sqrt(-g)*g^3 + 11*f^4*sqrt(-g)*g^3)/(((sqrt(-g)*x - sqrt(-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*sqrt(-g)*g^3*log(c)/((sqrt(-g)*x - sqrt(-g*x^2 + f))^2 - f)^7 - 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)*a*sqrt(-g)*g^3/(( 
sqrt(-g)*x - sqrt(-g*x^2 + f))^2 - f)^7
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^8 \sqrt {f-g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,f-d\,g\,x^2\right )}^p\right )}{x^8\,\sqrt {f-g\,x^2}} \,d x \] 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 = 0.17 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.53 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^8 \sqrt {f-g x^2}} \, dx=\frac {-96 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right ) b \,g^{3} p \,x^{7}-15 \sqrt {-g \,x^{2}+f}\, \mathrm {log}\left (\frac {f^{p} d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b \,f^{3}-18 \sqrt {-g \,x^{2}+f}\, \mathrm {log}\left (\frac {f^{p} d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b \,f^{2} g \,x^{2}-24 \sqrt {-g \,x^{2}+f}\, \mathrm {log}\left (\frac {f^{p} d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b f \,g^{2} x^{4}-48 \sqrt {-g \,x^{2}+f}\, \mathrm {log}\left (\frac {f^{p} d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b \,g^{3} x^{6}-15 \sqrt {-g \,x^{2}+f}\, a \,f^{3}-18 \sqrt {-g \,x^{2}+f}\, a \,f^{2} g \,x^{2}-24 \sqrt {-g \,x^{2}+f}\, a f \,g^{2} x^{4}-48 \sqrt {-g \,x^{2}+f}\, a \,g^{3} x^{6}+6 \sqrt {-g \,x^{2}+f}\, b \,f^{2} g p \,x^{2}+20 \sqrt {-g \,x^{2}+f}\, b f \,g^{2} p \,x^{4}+88 \sqrt {-g \,x^{2}+f}\, b \,g^{3} p \,x^{6}}{105 f^{4} x^{7}} \] Input:

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

Output:

( - 96*sqrt(g)*asin((sqrt(g)*x)/sqrt(f))*b*g**3*p*x**7 - 15*sqrt(f - g*x** 
2)*log((f**p*d**p*( - tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p)*c)/( 
tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p))*b*f**3 - 18*sqrt(f - g*x* 
*2)*log((f**p*d**p*( - tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p)*c)/ 
(tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p))*b*f**2*g*x**2 - 24*sqrt( 
f - g*x**2)*log((f**p*d**p*( - tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**( 
2*p)*c)/(tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p))*b*f*g**2*x**4 - 
48*sqrt(f - g*x**2)*log((f**p*d**p*( - tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 
 + 1)**(2*p)*c)/(tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p))*b*g**3*x 
**6 - 15*sqrt(f - g*x**2)*a*f**3 - 18*sqrt(f - g*x**2)*a*f**2*g*x**2 - 24* 
sqrt(f - g*x**2)*a*f*g**2*x**4 - 48*sqrt(f - g*x**2)*a*g**3*x**6 + 6*sqrt( 
f - g*x**2)*b*f**2*g*p*x**2 + 20*sqrt(f - g*x**2)*b*f*g**2*p*x**4 + 88*sqr 
t(f - g*x**2)*b*g**3*p*x**6)/(105*f**4*x**7)