\(\int \frac {a+b \log (c (d f-d g x^2)^p)}{x \sqrt {f-g x^2}} \, dx\) [706]

Optimal result

Integrand size = 36, antiderivative size = 105 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x \sqrt {f-g x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right ) \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f}}-\frac {b p \operatorname {PolyLog}\left (2,-\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {b p \operatorname {PolyLog}\left (2,\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )}{\sqrt {f}} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{x \sqrt {f-g x^2}} \, dx=\frac {\left (a+b \log \left (c \left (d \left (f-g x^2\right )\right )^p\right )\right ) \left (\log \left (1-\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )-\log \left (1+\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )\right )-2 b p \operatorname {PolyLog}\left (2,-\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )+2 b p \operatorname {PolyLog}\left (2,\frac {\sqrt {f-g x^2}}{\sqrt {f}}\right )}{2 \sqrt {f}} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 2929

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

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^3 - f*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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