\(\int \frac {a+b \log (c (d f+d g x^2)^p)}{\sqrt {f+g x^2}} \, dx\) [697]

Optimal result

Integrand size = 31, antiderivative size = 169 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{\sqrt {f+g x^2}} \, dx=\frac {\left (a+b \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )\right ) \log \left (\sqrt {g} x+\sqrt {f+g x^2}\right )}{\sqrt {g}}+\frac {b p \log ^2\left (\sqrt {g} x+\sqrt {f+g x^2}\right )}{\sqrt {g}}-\frac {2 b p \log \left (\sqrt {g} x+\sqrt {f+g x^2}\right ) \log \left (1+\frac {\left (\sqrt {g} x+\sqrt {f+g x^2}\right )^2}{f}\right )}{\sqrt {g}}-\frac {b p \operatorname {PolyLog}\left (2,-\frac {\left (\sqrt {g} x+\sqrt {f+g x^2}\right )^2}{f}\right )}{\sqrt {g}} \] Output:

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

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="maxim 
a")
 

Output:

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

(sqrt(g)*log((sqrt(f + g*x**2) + 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