\(\int \frac {x^2 (a+b \log (c (d f-d g x^2)^p))}{\sqrt {f-g x^2}} \, dx\) [709]

Optimal result

Integrand size = 36, antiderivative size = 613 \[ \int \frac {x^2 \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{\sqrt {f-g x^2}} \, dx=\frac {b f^2 p \sqrt {-\frac {f}{g}+x^2}}{8 g^2 \sqrt {f-g x^2} \left (x+\sqrt {-\frac {f}{g}+x^2}\right )^2}+\frac {a \sqrt {-\frac {f}{g}+x^2} \left (x+\sqrt {-\frac {f}{g}+x^2}\right )^2}{8 \sqrt {f-g x^2}}-\frac {b p \sqrt {-\frac {f}{g}+x^2} \left (x+\sqrt {-\frac {f}{g}+x^2}\right )^2}{8 \sqrt {f-g x^2}}+\frac {b \sqrt {-\frac {f}{g}+x^2} \left (x+\sqrt {-\frac {f}{g}+x^2}\right )^2 \log \left (c \left (d f-d g x^2\right )^p\right )}{8 \sqrt {f-g x^2}}-\frac {f^2 \sqrt {-\frac {f}{g}+x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right )}{8 g^2 \sqrt {f-g x^2} \left (x+\sqrt {-\frac {f}{g}+x^2}\right )^2}-\frac {b f p \sqrt {-\frac {f}{g}+x^2} \log \left (x+\sqrt {-\frac {f}{g}+x^2}\right )}{2 g \sqrt {f-g x^2}}+\frac {f \sqrt {-\frac {f}{g}+x^2} \left (a+b \log \left (c \left (d f-d g x^2\right )^p\right )\right ) \log \left (x+\sqrt {-\frac {f}{g}+x^2}\right )}{2 g \sqrt {f-g x^2}}+\frac {b f p \sqrt {-\frac {f}{g}+x^2} \log ^2\left (x+\sqrt {-\frac {f}{g}+x^2}\right )}{2 g \sqrt {f-g x^2}}-\frac {b f p \sqrt {-\frac {f}{g}+x^2} \log \left (x+\sqrt {-\frac {f}{g}+x^2}\right ) \log \left (1-\frac {g \left (x+\sqrt {-\frac {f}{g}+x^2}\right )^2}{f}\right )}{g \sqrt {f-g x^2}}-\frac {b f p \sqrt {-\frac {f}{g}+x^2} \operatorname {PolyLog}\left (2,\frac {g \left (x+\sqrt {-\frac {f}{g}+x^2}\right )^2}{f}\right )}{2 g \sqrt {f-g x^2}} \] Output:

1/8*b*f^2*p*(-f/g+x^2)^(1/2)/g^2/(-g*x^2+f)^(1/2)/(x+(-f/g+x^2)^(1/2))^2+1 
/8*a*(-f/g+x^2)^(1/2)*(x+(-f/g+x^2)^(1/2))^2/(-g*x^2+f)^(1/2)-1/8*b*p*(-f/ 
g+x^2)^(1/2)*(x+(-f/g+x^2)^(1/2))^2/(-g*x^2+f)^(1/2)+1/8*b*(-f/g+x^2)^(1/2 
)*(x+(-f/g+x^2)^(1/2))^2*ln(c*(-d*g*x^2+d*f)^p)/(-g*x^2+f)^(1/2)-1/8*f^2*( 
-f/g+x^2)^(1/2)*(a+b*ln(c*(-d*g*x^2+d*f)^p))/g^2/(-g*x^2+f)^(1/2)/(x+(-f/g 
+x^2)^(1/2))^2-1/2*b*f*p*(-f/g+x^2)^(1/2)*ln(x+(-f/g+x^2)^(1/2))/g/(-g*x^2 
+f)^(1/2)+1/2*f*(-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/(-g*x^2+f)^(1/2)+1/2*b*f*p*(-f/g+x^2)^(1/2)*ln(x+(-f/g+x^2)^( 
1/2))^2/g/(-g*x^2+f)^(1/2)-b*f*p*(-f/g+x^2)^(1/2)*ln(x+(-f/g+x^2)^(1/2))*l 
n(1-g*(x+(-f/g+x^2)^(1/2))^2/f)/g/(-g*x^2+f)^(1/2)-1/2*b*f*p*(-f/g+x^2)^(1 
/2)*polylog(2,g*(x+(-f/g+x^2)^(1/2))^2/f)/g/(-g*x^2+f)^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[(x^2*(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.

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

\(\Big \downarrow \) 2929

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

Input:

Int[(x^2*(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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(x^2*(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*f - sqrt(f - g*x**2)*a*g*x + 2*int((l 
og((d*f - d*g*x**2)**p*c)*x**2)/sqrt(f - g*x**2),x)*b*g**2)/(2*g**2)