\(\int \frac {a+b \log (c (d f+d g x^2)^p)}{x^5 \sqrt {f+g x^2}} \, dx\) [695]

Optimal result

Integrand size = 34, antiderivative size = 251 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}} \, dx=-\frac {b g p \sqrt {f+g x^2}}{4 f^2 x^2}+\frac {b g^2 p \text {arctanh}\left (\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{f^{5/2}}-\frac {\sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{4 f x^4}+\frac {3 g \sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{8 f^2 x^2}-\frac {3 g^2 \text {arctanh}\left (\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right ) \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{8 f^{5/2}}-\frac {3 b g^2 p \operatorname {PolyLog}\left (2,-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{8 f^{5/2}}+\frac {3 b g^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{8 f^{5/2}} \] Output:

-1/4*b*g*p*(g*x^2+f)^(1/2)/f^2/x^2+b*g^2*p*arctanh((g*x^2+f)^(1/2)/f^(1/2) 
)/f^(5/2)-1/4*(g*x^2+f)^(1/2)*(a+b*ln(c*(d*g*x^2+d*f)^p))/f/x^4+3/8*g*(g*x 
^2+f)^(1/2)*(a+b*ln(c*(d*g*x^2+d*f)^p))/f^2/x^2-3/8*g^2*arctanh((g*x^2+f)^ 
(1/2)/f^(1/2))*(a+b*ln(c*(d*g*x^2+d*f)^p))/f^(5/2)-3/8*b*g^2*p*polylog(2,- 
(g*x^2+f)^(1/2)/f^(1/2))/f^(5/2)+3/8*b*g^2*p*polylog(2,(g*x^2+f)^(1/2)/f^( 
1/2))/f^(5/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.64 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}} \, dx=\frac {-4 a f^{3/2} \sqrt {f+g x^2}+6 a \sqrt {f} g x^2 \sqrt {f+g x^2}-4 b \sqrt {f} g p x^2 \sqrt {f+g x^2}-4 b f^{3/2} \sqrt {f+g x^2} \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )+6 b \sqrt {f} g x^2 \sqrt {f+g x^2} \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )-8 b g^2 p x^4 \log \left (\sqrt {f}-\sqrt {f+g x^2}\right )+8 b g^2 p x^4 \log \left (\sqrt {f}+\sqrt {f+g x^2}\right )+3 a g^2 x^4 \log \left (1-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )+3 b g^2 x^4 \log \left (c \left (d \left (f+g x^2\right )\right )^p\right ) \log \left (1-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )-3 a g^2 x^4 \log \left (1+\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )-3 b g^2 x^4 \log \left (c \left (d \left (f+g x^2\right )\right )^p\right ) \log \left (1+\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )-6 b g^2 p x^4 \operatorname {PolyLog}\left (2,-\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )+6 b g^2 p x^4 \operatorname {PolyLog}\left (2,\frac {\sqrt {f+g x^2}}{\sqrt {f}}\right )}{16 f^{5/2} x^4} \] Input:

Integrate[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^5*Sqrt[f + g*x^2]),x]
 

Output:

(-4*a*f^(3/2)*Sqrt[f + g*x^2] + 6*a*Sqrt[f]*g*x^2*Sqrt[f + g*x^2] - 4*b*Sq 
rt[f]*g*p*x^2*Sqrt[f + g*x^2] - 4*b*f^(3/2)*Sqrt[f + g*x^2]*Log[c*(d*(f + 
g*x^2))^p] + 6*b*Sqrt[f]*g*x^2*Sqrt[f + g*x^2]*Log[c*(d*(f + g*x^2))^p] - 
8*b*g^2*p*x^4*Log[Sqrt[f] - Sqrt[f + g*x^2]] + 8*b*g^2*p*x^4*Log[Sqrt[f] + 
 Sqrt[f + g*x^2]] + 3*a*g^2*x^4*Log[1 - Sqrt[f + g*x^2]/Sqrt[f]] + 3*b*g^2 
*x^4*Log[c*(d*(f + g*x^2))^p]*Log[1 - Sqrt[f + g*x^2]/Sqrt[f]] - 3*a*g^2*x 
^4*Log[1 + Sqrt[f + g*x^2]/Sqrt[f]] - 3*b*g^2*x^4*Log[c*(d*(f + g*x^2))^p] 
*Log[1 + Sqrt[f + g*x^2]/Sqrt[f]] - 6*b*g^2*p*x^4*PolyLog[2, -(Sqrt[f + g* 
x^2]/Sqrt[f])] + 6*b*g^2*p*x^4*PolyLog[2, Sqrt[f + g*x^2]/Sqrt[f]])/(16*f^ 
(5/2)*x^4)
 

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}}dx\)

Input:

Int[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^5*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]

\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+d f \right )^{p}\right )}{x^{5} \sqrt {g \,x^{2}+f}}d x\]

Input:

int((a+b*ln(c*(d*g*x^2+d*f)^p))/x^5/(g*x^2+f)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*g*x^2+d*f)^p))/x^5/(g*x^2+f)^(1/2),x)
 

Fricas [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + f} x^{5}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^5/(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^7 + f*x^5), x)
 

Sympy [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d f + d g x^{2}\right )^{p} \right )}}{x^{5} \sqrt {f + g x^{2}}}\, dx \] Input:

integrate((a+b*ln(c*(d*g*x**2+d*f)**p))/x**5/(g*x**2+f)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*f + d*g*x**2)**p))/(x**5*sqrt(f + g*x**2)), x)
 

Maxima [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + f} x^{5}} \,d x } \] Input:

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^5/(g*x^2+f)^(1/2),x, algorithm="m 
axima")
 

Output:

-1/8*a*(3*g^2*arcsinh(f/(sqrt(f*g)*abs(x)))/f^(5/2) - 3*sqrt(g*x^2 + f)*g/ 
(f^2*x^2) + 2*sqrt(g*x^2 + f)/(f*x^4)) + b*integrate((p*log(g*x^2 + f) + p 
*log(d) + log(c))/(sqrt(g*x^2 + f)*x^5), x)
 

Giac [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}} \, dx=\int { \frac {b \log \left ({\left (d g x^{2} + d f\right )}^{p} c\right ) + a}{\sqrt {g x^{2} + f} x^{5}} \,d x } \] Input:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,g\,x^2+d\,f\right )}^p\right )}{x^5\,\sqrt {g\,x^2+f}} \,d x \] Input:

int((a + b*log(c*(d*f + d*g*x^2)^p))/(x^5*(f + g*x^2)^(1/2)),x)
 

Output:

int((a + b*log(c*(d*f + d*g*x^2)^p))/(x^5*(f + g*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^5 \sqrt {f+g x^2}} \, dx=\frac {-2 \sqrt {g \,x^{2}+f}\, a \,f^{2}+3 \sqrt {g \,x^{2}+f}\, a f g \,x^{2}+3 \sqrt {f}\, \mathrm {log}\left (\frac {\sqrt {g \,x^{2}+f}-\sqrt {f}+\sqrt {g}\, x}{\sqrt {f}}\right ) a \,g^{2} x^{4}-3 \sqrt {f}\, \mathrm {log}\left (\frac {\sqrt {g \,x^{2}+f}+\sqrt {f}+\sqrt {g}\, x}{\sqrt {f}}\right ) a \,g^{2} x^{4}+8 \left (\int \frac {\mathrm {log}\left (\left (d g \,x^{2}+d f \right )^{p} c \right )}{\sqrt {g \,x^{2}+f}\, x^{5}}d x \right ) b \,f^{3} x^{4}}{8 f^{3} x^{4}} \] Input:

int((a+b*log(c*(d*g*x^2+d*f)^p))/x^5/(g*x^2+f)^(1/2),x)
 

Output:

( - 2*sqrt(f + g*x**2)*a*f**2 + 3*sqrt(f + g*x**2)*a*f*g*x**2 + 3*sqrt(f)* 
log((sqrt(f + g*x**2) - sqrt(f) + sqrt(g)*x)/sqrt(f))*a*g**2*x**4 - 3*sqrt 
(f)*log((sqrt(f + g*x**2) + sqrt(f) + sqrt(g)*x)/sqrt(f))*a*g**2*x**4 + 8* 
int(log((d*f + d*g*x**2)**p*c)/(sqrt(f + g*x**2)*x**5),x)*b*f**3*x**4)/(8* 
f**3*x**4)