\(\int \frac {a+b \log (c (d f+d g x^2)^p)}{x^6 \sqrt {f+g x^2}} \, dx\) [700]

Optimal result

Integrand size = 34, antiderivative size = 208 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^6 \sqrt {f+g x^2}} \, dx=-\frac {2 b g p \sqrt {f+g x^2}}{15 f^2 x^3}+\frac {4 b g^2 p \sqrt {f+g x^2}}{5 f^3 x}+\frac {16 b g^{5/2} p \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f+g x^2}}\right )}{15 f^3}-\frac {\sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{5 f x^5}+\frac {4 g \sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{15 f^2 x^3}-\frac {8 g^2 \sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{15 f^3 x} \] Output:

-2/15*b*g*p*(g*x^2+f)^(1/2)/f^2/x^3+4/5*b*g^2*p*(g*x^2+f)^(1/2)/f^3/x+16/1 
5*b*g^(5/2)*p*arctanh(g^(1/2)*x/(g*x^2+f)^(1/2))/f^3-1/5*(g*x^2+f)^(1/2)*( 
a+b*ln(c*(d*g*x^2+d*f)^p))/f/x^5+4/15*g*(g*x^2+f)^(1/2)*(a+b*ln(c*(d*g*x^2 
+d*f)^p))/f^2/x^3-8/15*g^2*(g*x^2+f)^(1/2)*(a+b*ln(c*(d*g*x^2+d*f)^p))/f^3 
/x
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.71 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^6 \sqrt {f+g x^2}} \, dx=\frac {\sqrt {f+g x^2} \left (2 b g p x^2 \left (-f+6 g x^2\right )+a \left (-3 f^2+4 f g x^2-8 g^2 x^4\right )\right )+b \sqrt {f+g x^2} \left (-3 f^2+4 f g x^2-8 g^2 x^4\right ) \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )+16 b g^{5/2} p x^5 \log \left (g x+\sqrt {g} \sqrt {f+g x^2}\right )}{15 f^3 x^5} \] Input:

Integrate[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^6*Sqrt[f + g*x^2]),x]
 

Output:

(Sqrt[f + g*x^2]*(2*b*g*p*x^2*(-f + 6*g*x^2) + a*(-3*f^2 + 4*f*g*x^2 - 8*g 
^2*x^4)) + b*Sqrt[f + g*x^2]*(-3*f^2 + 4*f*g*x^2 - 8*g^2*x^4)*Log[c*(d*(f 
+ g*x^2))^p] + 16*b*g^(5/2)*p*x^5*Log[g*x + Sqrt[g]*Sqrt[f + g*x^2]])/(15* 
f^3*x^5)
 

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.

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

\(\Big \downarrow \) 2929

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

Input:

Int[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^6*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]

\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+d f \right )^{p}\right )}{x^{6} \sqrt {g \,x^{2}+f}}d x\]

Input:

int((a+b*ln(c*(d*g*x^2+d*f)^p))/x^6/(g*x^2+f)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*g*x^2+d*f)^p))/x^6/(g*x^2+f)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.58 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^6 \sqrt {f+g x^2}} \, dx=\left [\frac {8 \, b g^{\frac {5}{2}} p x^{5} \log \left (-2 \, g x^{2} - 2 \, \sqrt {g x^{2} + f} \sqrt {g} x - f\right ) + {\left (4 \, {\left (3 \, b g^{2} p - 2 \, a g^{2}\right )} x^{4} - 3 \, a f^{2} - 2 \, {\left (b f g p - 2 \, a f g\right )} x^{2} - {\left (8 \, b g^{2} p x^{4} - 4 \, b f g p x^{2} + 3 \, b f^{2} p\right )} \log \left (d g x^{2} + d f\right ) - {\left (8 \, b g^{2} x^{4} - 4 \, b f g x^{2} + 3 \, b f^{2}\right )} \log \left (c\right )\right )} \sqrt {g x^{2} + f}}{15 \, f^{3} x^{5}}, -\frac {16 \, b \sqrt {-g} g^{2} p x^{5} \arctan \left (\frac {\sqrt {-g} x}{\sqrt {g x^{2} + f}}\right ) - {\left (4 \, {\left (3 \, b g^{2} p - 2 \, a g^{2}\right )} x^{4} - 3 \, a f^{2} - 2 \, {\left (b f g p - 2 \, a f g\right )} x^{2} - {\left (8 \, b g^{2} p x^{4} - 4 \, b f g p x^{2} + 3 \, b f^{2} p\right )} \log \left (d g x^{2} + d f\right ) - {\left (8 \, b g^{2} x^{4} - 4 \, b f g x^{2} + 3 \, b f^{2}\right )} \log \left (c\right )\right )} \sqrt {g x^{2} + f}}{15 \, f^{3} x^{5}}\right ] \] Input:

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

Output:

[1/15*(8*b*g^(5/2)*p*x^5*log(-2*g*x^2 - 2*sqrt(g*x^2 + f)*sqrt(g)*x - f) + 
 (4*(3*b*g^2*p - 2*a*g^2)*x^4 - 3*a*f^2 - 2*(b*f*g*p - 2*a*f*g)*x^2 - (8*b 
*g^2*p*x^4 - 4*b*f*g*p*x^2 + 3*b*f^2*p)*log(d*g*x^2 + d*f) - (8*b*g^2*x^4 
- 4*b*f*g*x^2 + 3*b*f^2)*log(c))*sqrt(g*x^2 + f))/(f^3*x^5), -1/15*(16*b*s 
qrt(-g)*g^2*p*x^5*arctan(sqrt(-g)*x/sqrt(g*x^2 + f)) - (4*(3*b*g^2*p - 2*a 
*g^2)*x^4 - 3*a*f^2 - 2*(b*f*g*p - 2*a*f*g)*x^2 - (8*b*g^2*p*x^4 - 4*b*f*g 
*p*x^2 + 3*b*f^2*p)*log(d*g*x^2 + d*f) - (8*b*g^2*x^4 - 4*b*f*g*x^2 + 3*b* 
f^2)*log(c))*sqrt(g*x^2 + f))/(f^3*x^5)]
 

Sympy [F]

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

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

Output:

Integral((a + b*log(c*(d*f + d*g*x**2)**p))/(x**6*sqrt(f + g*x**2)), x)
 

Maxima [F]

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

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^6/(g*x^2+f)^(1/2),x, algorithm="m 
axima")
 

Output:

1/15*(240*g^4*p*integrate(1/15*x^2/((f^3*g*x^2 + f^4)*sqrt(g*x^2 + f)), x) 
 - (4*(2*g^3*p*log(d) - 7*g^3*p + 2*g^3*log(c))*x^6 + 2*(2*f*g^2*p*log(d) 
- 5*f*g^2*p + 2*f*g^2*log(c))*x^4 + 3*f^3*p*log(d) + 3*f^3*log(c) - (f^2*g 
*p*log(d) - 2*f^2*g*p + f^2*g*log(c))*x^2 + (8*g^3*p*x^6 + 4*f*g^2*p*x^4 - 
 f^2*g*p*x^2 + 3*f^3*p)*log(g*x^2 + f))/(sqrt(g*x^2 + f)*f^3*x^5))*b - 1/1 
5*a*(8*sqrt(g*x^2 + f)*g^2/(f^3*x) - 4*sqrt(g*x^2 + f)*g/(f^2*x^3) + 3*sqr 
t(g*x^2 + f)/(f*x^5))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (180) = 360\).

Time = 0.98 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.84 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^6 \sqrt {f+g x^2}} \, dx=-\frac {8}{15} \, b p {\left (\frac {g^{\frac {5}{2}} \log \left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2}\right )}{f^{3}} - \frac {g^{\frac {5}{2}} \log \left ({\left | g x^{2} + f \right |}\right )}{f^{3}} - \frac {2 \, {\left (10 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{4} - 5 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} f + f^{2}\right )} g^{\frac {5}{2}} \log \left (d g x^{2} + d f\right )}{{\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{5}} + \frac {2 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{4} g^{\frac {5}{2}} - 7 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} f g^{\frac {5}{2}} + 3 \, f^{2} g^{\frac {5}{2}}}{{\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{3} f^{2}}\right )} + \frac {16 \, {\left (10 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{4} - 5 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} f + f^{2}\right )} b g^{\frac {5}{2}} \log \left (c\right )}{15 \, {\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{5}} + \frac {16 \, {\left (10 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{4} - 5 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} f + f^{2}\right )} a g^{\frac {5}{2}}}{15 \, {\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{5}} \] Input:

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

Output:

-8/15*b*p*(g^(5/2)*log((sqrt(g)*x - sqrt(g*x^2 + f))^2)/f^3 - g^(5/2)*log( 
abs(g*x^2 + f))/f^3 - 2*(10*(sqrt(g)*x - sqrt(g*x^2 + f))^4 - 5*(sqrt(g)*x 
 - sqrt(g*x^2 + f))^2*f + f^2)*g^(5/2)*log(d*g*x^2 + d*f)/((sqrt(g)*x - sq 
rt(g*x^2 + f))^2 - f)^5 + (2*(sqrt(g)*x - sqrt(g*x^2 + f))^4*g^(5/2) - 7*( 
sqrt(g)*x - sqrt(g*x^2 + f))^2*f*g^(5/2) + 3*f^2*g^(5/2))/(((sqrt(g)*x - s 
qrt(g*x^2 + f))^2 - f)^3*f^2)) + 16/15*(10*(sqrt(g)*x - sqrt(g*x^2 + f))^4 
 - 5*(sqrt(g)*x - sqrt(g*x^2 + f))^2*f + f^2)*b*g^(5/2)*log(c)/((sqrt(g)*x 
 - sqrt(g*x^2 + f))^2 - f)^5 + 16/15*(10*(sqrt(g)*x - sqrt(g*x^2 + f))^4 - 
 5*(sqrt(g)*x - sqrt(g*x^2 + f))^2*f + f^2)*a*g^(5/2)/((sqrt(g)*x - sqrt(g 
*x^2 + f))^2 - f)^5
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.15 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^6 \sqrt {f+g x^2}} \, dx=\frac {-15 \sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,f^{2}+20 \sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b f g \,x^{2}-40 \sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,g^{2} x^{4}-15 \sqrt {g \,x^{2}+f}\, a \,f^{2}+20 \sqrt {g \,x^{2}+f}\, a f g \,x^{2}-40 \sqrt {g \,x^{2}+f}\, a \,g^{2} x^{4}-10 \sqrt {g \,x^{2}+f}\, b f g p \,x^{2}+60 \sqrt {g \,x^{2}+f}\, b \,g^{2} p \,x^{4}+80 \sqrt {g}\, \mathrm {log}\left (\frac {2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}}{f}\right ) b \,g^{2} p \,x^{5}-40 \sqrt {g}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b \,g^{2} x^{5}+40 \sqrt {g}\, a \,g^{2} x^{5}-44 \sqrt {g}\, b \,g^{2} p \,x^{5}}{75 f^{3} x^{5}} \] Input:

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

Output:

( - 15*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 
 + 20*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*x 
**2 - 40*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* 
*2*x**4 - 15*sqrt(f + g*x**2)*a*f**2 + 20*sqrt(f + g*x**2)*a*f*g*x**2 - 40 
*sqrt(f + g*x**2)*a*g**2*x**4 - 10*sqrt(f + g*x**2)*b*f*g*p*x**2 + 60*sqrt 
(f + g*x**2)*b*g**2*p*x**4 + 80*sqrt(g)*log((2*sqrt(g)*sqrt(f + g*x**2)*x 
+ 2*f + 2*g*x**2)/f)*b*g**2*p*x**5 - 40*sqrt(g)*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**2*x**5 + 40*sqrt(g)*a*g**2*x**5 - 44*sqrt(g)*b*g**2 
*p*x**5)/(75*f**3*x**5)