\(\int \frac {a+b \log (c (d f-d g x^2)^p)}{x^4 \sqrt {f-g x^2}} \, dx\) [712]

Optimal result

Integrand size = 36, antiderivative size = 145 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^4 \sqrt {f-g x^2}} \, dx=\frac {2 b g p \sqrt {f-g x^2}}{3 f^2 x}-\frac {4 b g^{3/2} p \arctan \left (\frac {\sqrt {g} x}{\sqrt {f-g x^2}}\right )}{3 f^2}-\frac {\sqrt {f-g x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{3 f x^3}-\frac {2 g \sqrt {f-g x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{3 f^2 x} \] Output:

2/3*b*g*p*(-g*x^2+f)^(1/2)/f^2/x-4/3*b*g^(3/2)*p*arctan(g^(1/2)*x/(-g*x^2+ 
f)^(1/2))/f^2-1/3*(-g*x^2+f)^(1/2)*(a+b*ln(c*(-d*g*x^2+d*f)^p))/f/x^3-2/3* 
g*(-g*x^2+f)^(1/2)*(a+b*ln(c*(-d*g*x^2+d*f)^p))/f^2/x
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.69 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^4 \sqrt {f-g x^2}} \, dx=-\frac {4 b g^{3/2} p x^3 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f-g x^2}}\right )+\sqrt {f-g x^2} \left (a f+2 a g x^2-2 b g p x^2+b \left (f+2 g x^2\right ) \log \left (c \left (d \left (f-g x^2\right )\right )^p\right )\right )}{3 f^2 x^3} \] Input:

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

Output:

-1/3*(4*b*g^(3/2)*p*x^3*ArcTan[(Sqrt[g]*x)/Sqrt[f - g*x^2]] + Sqrt[f - g*x 
^2]*(a*f + 2*a*g*x^2 - 2*b*g*p*x^2 + b*(f + 2*g*x^2)*Log[c*(d*(f - g*x^2)) 
^p]))/(f^2*x^3)
 

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.

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

\(\Big \downarrow \) 2929

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

Input:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.67 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^4 \sqrt {f-g x^2}} \, dx=\left [\frac {2 \, b \sqrt {-g} g p x^{3} \log \left (2 \, g x^{2} - 2 \, \sqrt {-g x^{2} + f} \sqrt {-g} x - f\right ) + {\left (2 \, {\left (b g p - a g\right )} x^{2} - a f - {\left (2 \, b g p x^{2} + b f p\right )} \log \left (-d g x^{2} + d f\right ) - {\left (2 \, b g x^{2} + b f\right )} \log \left (c\right )\right )} \sqrt {-g x^{2} + f}}{3 \, f^{2} x^{3}}, \frac {4 \, b g^{\frac {3}{2}} p x^{3} \arctan \left (\frac {\sqrt {-g x^{2} + f} \sqrt {g} x}{g x^{2} - f}\right ) + {\left (2 \, {\left (b g p - a g\right )} x^{2} - a f - {\left (2 \, b g p x^{2} + b f p\right )} \log \left (-d g x^{2} + d f\right ) - {\left (2 \, b g x^{2} + b f\right )} \log \left (c\right )\right )} \sqrt {-g x^{2} + f}}{3 \, f^{2} x^{3}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (125) = 250\).

Time = 0.74 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.12 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^4 \sqrt {f-g x^2}} \, dx=-\frac {2}{3} \, b p {\left (\frac {2 \, {\left (3 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )} \sqrt {-g} g \log \left (-d g x^{2} + d f\right )}{{\left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )}^{3}} + \frac {\sqrt {-g} g \log \left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2}\right )}{f^{2}} - \frac {\sqrt {-g} g \log \left ({\left | g x^{2} - f \right |}\right )}{f^{2}} + \frac {2 \, \sqrt {-g} g}{{\left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )} f}\right )} - \frac {4 \, {\left (3 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )} b \sqrt {-g} g \log \left (c\right )}{3 \, {\left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )}^{3}} - \frac {4 \, {\left (3 \, {\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )} a \sqrt {-g} g}{3 \, {\left ({\left (\sqrt {-g} x - \sqrt {-g x^{2} + f}\right )}^{2} - f\right )}^{3}} \] Input:

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

Output:

-2/3*b*p*(2*(3*(sqrt(-g)*x - sqrt(-g*x^2 + f))^2 - f)*sqrt(-g)*g*log(-d*g* 
x^2 + d*f)/((sqrt(-g)*x - sqrt(-g*x^2 + f))^2 - f)^3 + sqrt(-g)*g*log((sqr 
t(-g)*x - sqrt(-g*x^2 + f))^2)/f^2 - sqrt(-g)*g*log(abs(g*x^2 - f))/f^2 + 
2*sqrt(-g)*g/(((sqrt(-g)*x - sqrt(-g*x^2 + f))^2 - f)*f)) - 4/3*(3*(sqrt(- 
g)*x - sqrt(-g*x^2 + f))^2 - f)*b*sqrt(-g)*g*log(c)/((sqrt(-g)*x - sqrt(-g 
*x^2 + f))^2 - f)^3 - 4/3*(3*(sqrt(-g)*x - sqrt(-g*x^2 + f))^2 - f)*a*sqrt 
(-g)*g/((sqrt(-g)*x - sqrt(-g*x^2 + f))^2 - f)^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.44 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^4 \sqrt {f-g x^2}} \, dx=\frac {-4 \sqrt {g}\, \mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right ) b g p \,x^{3}-\sqrt {-g \,x^{2}+f}\, \mathrm {log}\left (\frac {f^{p} d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b f -2 \sqrt {-g \,x^{2}+f}\, \mathrm {log}\left (\frac {f^{p} d^{p} \left (-\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p} c}{\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )^{2}+1\right )^{2 p}}\right ) b g \,x^{2}-\sqrt {-g \,x^{2}+f}\, a f -2 \sqrt {-g \,x^{2}+f}\, a g \,x^{2}+2 \sqrt {-g \,x^{2}+f}\, b g p \,x^{2}}{3 f^{2} x^{3}} \] Input:

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

Output:

( - 4*sqrt(g)*asin((sqrt(g)*x)/sqrt(f))*b*g*p*x**3 - sqrt(f - g*x**2)*log( 
(f**p*d**p*( - tan(asin((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p)*c)/(tan(asi 
n((sqrt(g)*x)/sqrt(f))/2)**2 + 1)**(2*p))*b*f - 2*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*x**2 - sqrt(f - g*x**2)*a*f - 2 
*sqrt(f - g*x**2)*a*g*x**2 + 2*sqrt(f - g*x**2)*b*g*p*x**2)/(3*f**2*x**3)