\(\int \frac {a+b \log (c (d f-d g x^2)^p)}{x^3 \sqrt {f-g x^2}} \, dx\) [707]

Optimal result

Integrand size = 36, antiderivative size = 185 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}} \, dx=\frac {b g p \text {arctanh}\left (\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )}{f^{3/2}}-\frac {\sqrt {f-g x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{2 f x^2}-\frac {g \text {arctanh}\left (\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right ) \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{2 f^{3/2}}-\frac {b g p \operatorname {PolyLog}\left (2,-\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )}{2 f^{3/2}}+\frac {b g p \operatorname {PolyLog}\left (2,\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )}{2 f^{3/2}} \] Output:

b*g*p*arctanh((-g*x^2+f)^(1/2)/f^(1/2))/f^(3/2)-1/2*(-g*x^2+f)^(1/2)*(a+b* 
ln(c*(-d*g*x^2+d*f)^p))/f/x^2-1/2*g*arctanh((-g*x^2+f)^(1/2)/f^(1/2))*(a+b 
*ln(c*(-d*g*x^2+d*f)^p))/f^(3/2)-1/2*b*g*p*polylog(2,-(-g*x^2+f)^(1/2)/f^( 
1/2))/f^(3/2)+1/2*b*g*p*polylog(2,(-g*x^2+f)^(1/2)/f^(1/2))/f^(3/2)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.75 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}} \, dx=\frac {-2 a \sqrt {f} \sqrt {f-g x^2}-2 b \sqrt {f} \sqrt {f-g x^2} \log \left (c \left (d \left (f-g x^2\right )\right )^p\right )-2 b g p x^2 \log \left (\sqrt {f}-\sqrt {f-g x^2}\right )+2 b g p x^2 \log \left (\sqrt {f}+\sqrt {f-g x^2}\right )+a g x^2 \log \left (1-\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )+b g x^2 \log \left (c \left (d \left (f-g x^2\right )\right )^p\right ) \log \left (1-\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )-a g x^2 \log \left (1+\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )-b g x^2 \log \left (c \left (d \left (f-g x^2\right )\right )^p\right ) \log \left (1+\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )-2 b g p x^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )+2 b g p x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )}{4 f^{3/2} x^2} \] Input:

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

Output:

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

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}}dx\)

Input:

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

\[\int \frac {a +b \ln \left (c \left (-d g \,x^{2}+d f \right )^{p}\right )}{x^{3} \sqrt {-g \,x^{2}+f}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}} \, dx=\int { \frac {b \log \left ({\left (-d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {-g x^{2} + f} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(-d*g*x^2+d*f)^p))/x^3/(-g*x^2+f)^(1/2),x, algorithm= 
"fricas")
 

Output:

integral(-(sqrt(-g*x^2 + f)*b*log((-d*g*x^2 + d*f)^p*c) + sqrt(-g*x^2 + f) 
*a)/(g*x^5 - f*x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d f - d g x^{2}\right )^{p} \right )}}{x^{3} \sqrt {f - g x^{2}}}\, dx \] Input:

integrate((a+b*ln(c*(-d*g*x**2+d*f)**p))/x**3/(-g*x**2+f)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*f - d*g*x**2)**p))/(x**3*sqrt(f - g*x**2)), x)
 

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}} \, dx=\int { \frac {b \log \left ({\left (-d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {-g x^{2} + f} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(-d*g*x^2+d*f)^p))/x^3/(-g*x^2+f)^(1/2),x, algorithm= 
"maxima")
 

Output:

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

Giac [F]

\[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}} \, dx=\int { \frac {b \log \left ({\left (-d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {-g x^{2} + f} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(-d*g*x^2+d*f)^p))/x^3/(-g*x^2+f)^(1/2),x, algorithm= 
"giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,f-d\,g\,x^2\right )}^p\right )}{x^3\,\sqrt {f-g\,x^2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x^3 \sqrt {f-g x^2}} \, dx=\frac {-\sqrt {-g \,x^{2}+f}\, a f +\sqrt {f}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {g}\, x}{\sqrt {f}}\right )}{2}\right )\right ) a g \,x^{2}+2 \left (\int \frac {\mathrm {log}\left (\left (-d g \,x^{2}+d f \right )^{p} c \right )}{\sqrt {-g \,x^{2}+f}\, x^{3}}d x \right ) b \,f^{2} x^{2}}{2 f^{2} x^{2}} \] Input:

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

Output:

( - sqrt(f - g*x**2)*a*f + sqrt(f)*log(tan(asin((sqrt(g)*x)/sqrt(f))/2))*a 
*g*x**2 + 2*int(log((d*f - d*g*x**2)**p*c)/(sqrt(f - g*x**2)*x**3),x)*b*f* 
*2*x**2)/(2*f**2*x**2)