\(\int \frac {x^2 (a+b \log (c (d f+d g x^2)^p))}{\sqrt {f+g x^2}} \, dx\) [696]

Optimal result

Integrand size = 34, antiderivative size = 406 \[ \int \frac {x^2 \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{\sqrt {f+g x^2}} \, dx=\frac {b f^2 p}{8 g^{3/2} \left (\sqrt {g} x+\sqrt {f+g x^2}\right )^2}+\frac {a \left (\sqrt {g} x+\sqrt {f+g x^2}\right )^2}{8 g^{3/2}}-\frac {b p \left (\sqrt {g} x+\sqrt {f+g x^2}\right )^2}{8 g^{3/2}}+\frac {b \left (\sqrt {g} x+\sqrt {f+g x^2}\right )^2 \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )}{8 g^{3/2}}-\frac {f^2 \left (a+b \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )\right )}{8 g^{3/2} \left (\sqrt {g} x+\sqrt {f+g x^2}\right )^2}+\frac {b f p \log \left (\sqrt {g} x+\sqrt {f+g x^2}\right )}{2 g^{3/2}}-\frac {f \left (a+b \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )\right ) \log \left (\sqrt {g} x+\sqrt {f+g x^2}\right )}{2 g^{3/2}}-\frac {b f p \log ^2\left (\sqrt {g} x+\sqrt {f+g x^2}\right )}{2 g^{3/2}}+\frac {b f p \log \left (\sqrt {g} x+\sqrt {f+g x^2}\right ) \log \left (1+\frac {\left (\sqrt {g} x+\sqrt {f+g x^2}\right )^2}{f}\right )}{g^{3/2}}+\frac {b f p \operatorname {PolyLog}\left (2,-\frac {\left (\sqrt {g} x+\sqrt {f+g x^2}\right )^2}{f}\right )}{2 g^{3/2}} \] Output:

1/8*b*f^2*p/g^(3/2)/(g^(1/2)*x+(g*x^2+f)^(1/2))^2+1/8*a*(g^(1/2)*x+(g*x^2+ 
f)^(1/2))^2/g^(3/2)-1/8*b*p*(g^(1/2)*x+(g*x^2+f)^(1/2))^2/g^(3/2)+1/8*b*(g 
^(1/2)*x+(g*x^2+f)^(1/2))^2*ln(c*(d*(g*x^2+f))^p)/g^(3/2)-1/8*f^2*(a+b*ln( 
c*(d*(g*x^2+f))^p))/g^(3/2)/(g^(1/2)*x+(g*x^2+f)^(1/2))^2+1/2*b*f*p*ln(g^( 
1/2)*x+(g*x^2+f)^(1/2))/g^(3/2)-1/2*f*(a+b*ln(c*(d*(g*x^2+f))^p))*ln(g^(1/ 
2)*x+(g*x^2+f)^(1/2))/g^(3/2)-1/2*b*f*p*ln(g^(1/2)*x+(g*x^2+f)^(1/2))^2/g^ 
(3/2)+b*f*p*ln(g^(1/2)*x+(g*x^2+f)^(1/2))*ln(1+(g^(1/2)*x+(g*x^2+f)^(1/2)) 
^2/f)/g^(3/2)+1/2*b*f*p*polylog(2,-(g^(1/2)*x+(g*x^2+f)^(1/2))^2/f)/g^(3/2 
)
 

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="f 
ricas")
 

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="m 
axima")
 

Output:

1/2*a*(sqrt(g*x^2 + f)*x/g - f*arcsinh(g*x/sqrt(f*g))/g^(3/2)) + b*integra 
te((p*x^2*log(g*x^2 + f) + (p*log(d) + log(c))*x^2)/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="g 
iac")
 

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

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

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

Output:

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