\(\int \frac {a+b \log (c (d f+d g x^2)^p)}{x^3 \sqrt {f+g x^2}} \, dx\) [694]

Optimal result

Integrand size = 34, antiderivative size = 179 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}} \, dx=-\frac {b g p \text {arctanh}\left (\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{f^{3/2}}-\frac {\sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{2 f x^2}+\frac {g \text {arctanh}\left (\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right ) \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{2 f^{3/2}}+\frac {b g p \operatorname {PolyLog}\left (2,-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{2 f^{3/2}}-\frac {b g p \operatorname {PolyLog}\left (2,\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{2 f^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.73 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}} \, dx=-\frac {2 a \sqrt {f} \sqrt {f+g x^2}+2 b \sqrt {f} \sqrt {f+g x^2} \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )-2 b g p x^2 \log \left (\sqrt {f}-\sqrt {f+g x^2}\right )+2 b g p x^2 \log \left (\sqrt {f}+\sqrt {f+g x^2}\right )+a g x^2 \log \left (1-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )+b g x^2 \log \left (c \left (d \left (f+g x^2\right )\right )^p\right ) \log \left (1-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )-a g x^2 \log \left (1+\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )-b g x^2 \log \left (c \left (d \left (f+g x^2\right )\right )^p\right ) \log \left (1+\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )-2 b g p x^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )+2 b g p x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{4 f^{3/2} x^2} \] Input:

Integrate[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^3*Sqrt[f + g*x^2]),x]
 

Output:

-1/4*(2*a*Sqrt[f]*Sqrt[f + g*x^2] + 2*b*Sqrt[f]*Sqrt[f + g*x^2]*Log[c*(d*( 
f + g*x^2))^p] - 2*b*g*p*x^2*Log[Sqrt[f] - Sqrt[f + g*x^2]] + 2*b*g*p*x^2* 
Log[Sqrt[f] + Sqrt[f + g*x^2]] + a*g*x^2*Log[1 - Sqrt[f + g*x^2]/Sqrt[f]] 
+ b*g*x^2*Log[c*(d*(f + g*x^2))^p]*Log[1 - Sqrt[f + g*x^2]/Sqrt[f]] - a*g* 
x^2*Log[1 + Sqrt[f + g*x^2]/Sqrt[f]] - b*g*x^2*Log[c*(d*(f + g*x^2))^p]*Lo 
g[1 + Sqrt[f + g*x^2]/Sqrt[f]] - 2*b*g*p*x^2*PolyLog[2, -(Sqrt[f + g*x^2]/ 
Sqrt[f])] + 2*b*g*p*x^2*PolyLog[2, Sqrt[f + g*x^2]/Sqrt[f]])/(f^(3/2)*x^2)
 

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}}dx\)

Input:

Int[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^3*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]

\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+d f \right )^{p}\right )}{x^{3} \sqrt {g \,x^{2}+f}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + f} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^3/(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^5 + f*x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d f + d g x^{2}\right )^{p} \right )}}{x^{3} \sqrt {f + g x^{2}}}\, dx \] Input:

integrate((a+b*ln(c*(d*g*x**2+d*f)**p))/x**3/(g*x**2+f)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*f + d*g*x**2)**p))/(x**3*sqrt(f + g*x**2)), x)
 

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + f} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^3/(g*x^2+f)^(1/2),x, algorithm="m 
axima")
 

Output:

1/2*a*(g*arcsinh(f/(sqrt(f*g)*abs(x)))/f^(3/2) - sqrt(g*x^2 + f)/(f*x^2)) 
+ b*integrate((p*log(g*x^2 + f) + p*log(d) + log(c))/(sqrt(g*x^2 + f)*x^3) 
, x)
 

Giac [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + f} x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^3/(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^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,g\,x^2+d\,f\right )}^p\right )}{x^3\,\sqrt {g\,x^2+f}} \,d x \] Input:

int((a + b*log(c*(d*f + d*g*x^2)^p))/(x^3*(f + g*x^2)^(1/2)),x)
 

Output:

int((a + b*log(c*(d*f + d*g*x^2)^p))/(x^3*(f + g*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^3 \sqrt {f+g x^2}} \, dx=\frac {-\sqrt {g \,x^{2}+f}\, a f -\sqrt {f}\, \mathrm {log}\left (\frac {\sqrt {g \,x^{2}+f}-\sqrt {f}+\sqrt {g}\, x}{\sqrt {f}}\right ) a g \,x^{2}+\sqrt {f}\, \mathrm {log}\left (\frac {\sqrt {g \,x^{2}+f}+\sqrt {f}+\sqrt {g}\, x}{\sqrt {f}}\right ) a g \,x^{2}+2 \left (\int \frac {\mathrm {log}\left (\left (d g \,x^{2}+d f \right )^{p} c \right )}{\sqrt {g \,x^{2}+f}\, x^{3}}d x \right ) b \,f^{2} x^{2}}{2 f^{2} x^{2}} \] Input:

int((a+b*log(c*(d*g*x^2+d*f)^p))/x^3/(g*x^2+f)^(1/2),x)
 

Output:

( - sqrt(f + g*x**2)*a*f - sqrt(f)*log((sqrt(f + g*x**2) - sqrt(f) + sqrt( 
g)*x)/sqrt(f))*a*g*x**2 + sqrt(f)*log((sqrt(f + g*x**2) + sqrt(f) + sqrt(g 
)*x)/sqrt(f))*a*g*x**2 + 2*int(log((d*f + d*g*x**2)**p*c)/(sqrt(f + g*x**2 
)*x**3),x)*b*f**2*x**2)/(2*f**2*x**2)