\(\int \frac {a+b \log (c (d f-d g x^2)^p)}{\sqrt {f-g x^2}} \, dx\) [710]

Optimal result

Integrand size = 33, antiderivative size = 242 \[ \int \frac {a+b \log \left (c \left (d f-d g x^2\right )^p\right )}{\sqrt {f-g x^2}} \, dx=\frac {\sqrt {-\frac {f}{g}+x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right ) \log \left (x+\sqrt {-\frac {f}{g}+x^2}\right )}{\sqrt {f-g x^2}}+\frac {b p \sqrt {-\frac {f}{g}+x^2} \log ^2\left (x+\sqrt {-\frac {f}{g}+x^2}\right )}{\sqrt {f-g x^2}}-\frac {2 b p \sqrt {-\frac {f}{g}+x^2} \log \left (x+\sqrt {-\frac {f}{g}+x^2}\right ) \log \left (1-\frac {g \left (x+\sqrt {-\frac {f}{g}+x^2}\right )^2}{f}\right )}{\sqrt {f-g x^2}}-\frac {b p \sqrt {-\frac {f}{g}+x^2} \operatorname {PolyLog}\left (2,\frac {g \left (x+\sqrt {-\frac {f}{g}+x^2}\right )^2}{f}\right )}{\sqrt {f-g x^2}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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.

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

\(\Big \downarrow \) 2923

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2923

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + 
 (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Unintegrable[(f + g*x^s)^r*(a + b*Log 
[c*(d + e*x^n)^p])^q, x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, r, s}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

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^2 - f), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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