\(\int \frac {a+b \log (c (d f+d g x^2)^p)}{x^4 \sqrt {f+g x^2}} \, dx\) [699]

Optimal result

Integrand size = 34, antiderivative size = 139 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^4 \sqrt {f+g x^2}} \, dx=-\frac {2 b g p \sqrt {f+g x^2}}{3 f^2 x}-\frac {4 b g^{3/2} p \text {arctanh}\left (\frac {\sqrt {g} x}{\sqrt {f+g x^2}}\right )}{3 f^2}-\frac {\sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{3 f x^3}+\frac {2 g \sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{3 f^2 x} \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^4 \sqrt {f+g x^2}} \, dx=-\frac {\sqrt {f+g x^2} \left (a f-2 a g x^2+2 b g p x^2\right )+b \left (f-2 g x^2\right ) \sqrt {f+g x^2} \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )+4 b g^{3/2} p x^3 \log \left (g x+\sqrt {g} \sqrt {f+g x^2}\right )}{3 f^2 x^3} \] Input:

Integrate[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^4*Sqrt[f + g*x^2]),x]
 

Output:

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

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.

\(\displaystyle \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^4 \sqrt {f+g x^2}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^4 \sqrt {f+g x^2}}dx\)

Input:

Int[(a + b*Log[c*(d*f + d*g*x^2)^p])/(x^4*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]

\[\int \frac {a +b \ln \left (c \left (d g \,x^{2}+d f \right )^{p}\right )}{x^{4} \sqrt {g \,x^{2}+f}}d x\]

Input:

int((a+b*ln(c*(d*g*x^2+d*f)^p))/x^4/(g*x^2+f)^(1/2),x)
 

Output:

int((a+b*ln(c*(d*g*x^2+d*f)^p))/x^4/(g*x^2+f)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.65 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^4 \sqrt {f+g x^2}} \, dx=\left [\frac {2 \, b g^{\frac {3}{2}} p x^{3} \log \left (-2 \, g x^{2} + 2 \, \sqrt {g x^{2} + f} \sqrt {g} x - f\right ) - {\left (2 \, {\left (b g p - a g\right )} x^{2} + a f - {\left (2 \, b g p x^{2} - b f p\right )} \log \left (d g x^{2} + d f\right ) - {\left (2 \, b g x^{2} - b f\right )} \log \left (c\right )\right )} \sqrt {g x^{2} + f}}{3 \, f^{2} x^{3}}, \frac {4 \, b \sqrt {-g} g p x^{3} \arctan \left (\frac {\sqrt {-g} x}{\sqrt {g x^{2} + f}}\right ) - {\left (2 \, {\left (b g p - a g\right )} x^{2} + a f - {\left (2 \, b g p x^{2} - b f p\right )} \log \left (d g x^{2} + d f\right ) - {\left (2 \, b g x^{2} - b f\right )} \log \left (c\right )\right )} \sqrt {g x^{2} + f}}{3 \, f^{2} x^{3}}\right ] \] Input:

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^4/(g*x^2+f)^(1/2),x, algorithm="f 
ricas")
 

Output:

[1/3*(2*b*g^(3/2)*p*x^3*log(-2*g*x^2 + 2*sqrt(g*x^2 + f)*sqrt(g)*x - f) - 
(2*(b*g*p - a*g)*x^2 + a*f - (2*b*g*p*x^2 - b*f*p)*log(d*g*x^2 + d*f) - (2 
*b*g*x^2 - b*f)*log(c))*sqrt(g*x^2 + f))/(f^2*x^3), 1/3*(4*b*sqrt(-g)*g*p* 
x^3*arctan(sqrt(-g)*x/sqrt(g*x^2 + f)) - (2*(b*g*p - a*g)*x^2 + a*f - (2*b 
*g*p*x^2 - b*f*p)*log(d*g*x^2 + d*f) - (2*b*g*x^2 - b*f)*log(c))*sqrt(g*x^ 
2 + f))/(f^2*x^3)]
 

Sympy [F]

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

integrate((a+b*ln(c*(d*g*x**2+d*f)**p))/x**4/(g*x**2+f)**(1/2),x)
 

Output:

Integral((a + b*log(c*(d*f + d*g*x**2)**p))/(x**4*sqrt(f + g*x**2)), x)
 

Maxima [F]

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

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^4/(g*x^2+f)^(1/2),x, algorithm="m 
axima")
 

Output:

-1/3*(12*g^3*p*integrate(1/3*x^2/((f^2*g*x^2 + f^3)*sqrt(g*x^2 + f)), x) - 
 (2*(g^2*p*log(d) - 3*g^2*p + g^2*log(c))*x^4 - f^2*p*log(d) + (f*g*p*log( 
d) - 2*f*g*p + f*g*log(c))*x^2 - f^2*log(c) + (2*g^2*p*x^4 + f*g*p*x^2 - f 
^2*p)*log(g*x^2 + f))/(sqrt(g*x^2 + f)*f^2*x^3))*b + 1/3*a*(2*sqrt(g*x^2 + 
 f)*g/(f^2*x) - sqrt(g*x^2 + f)/(f*x^3))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (119) = 238\).

Time = 0.70 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.89 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^4 \sqrt {f+g x^2}} \, dx=\frac {2}{3} \, b p {\left (\frac {2 \, {\left (3 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )} g^{\frac {3}{2}} \log \left (d g x^{2} + d f\right )}{{\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{3}} + \frac {g^{\frac {3}{2}} \log \left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2}\right )}{f^{2}} - \frac {g^{\frac {3}{2}} \log \left ({\left | g x^{2} + f \right |}\right )}{f^{2}} + \frac {2 \, g^{\frac {3}{2}}}{{\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )} f}\right )} + \frac {4 \, {\left (3 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )} b g^{\frac {3}{2}} \log \left (c\right )}{3 \, {\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{3}} + \frac {4 \, {\left (3 \, {\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )} a g^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {g} x - \sqrt {g x^{2} + f}\right )}^{2} - f\right )}^{3}} \] Input:

integrate((a+b*log(c*(d*g*x^2+d*f)^p))/x^4/(g*x^2+f)^(1/2),x, algorithm="g 
iac")
 

Output:

2/3*b*p*(2*(3*(sqrt(g)*x - sqrt(g*x^2 + f))^2 - f)*g^(3/2)*log(d*g*x^2 + d 
*f)/((sqrt(g)*x - sqrt(g*x^2 + f))^2 - f)^3 + g^(3/2)*log((sqrt(g)*x - sqr 
t(g*x^2 + f))^2)/f^2 - g^(3/2)*log(abs(g*x^2 + f))/f^2 + 2*g^(3/2)/(((sqrt 
(g)*x - sqrt(g*x^2 + f))^2 - f)*f)) + 4/3*(3*(sqrt(g)*x - sqrt(g*x^2 + f)) 
^2 - f)*b*g^(3/2)*log(c)/((sqrt(g)*x - sqrt(g*x^2 + f))^2 - f)^3 + 4/3*(3* 
(sqrt(g)*x - sqrt(g*x^2 + f))^2 - f)*a*g^(3/2)/((sqrt(g)*x - sqrt(g*x^2 + 
f))^2 - f)^3
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^4 \sqrt {f+g x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,g\,x^2+d\,f\right )}^p\right )}{x^4\,\sqrt {g\,x^2+f}} \,d x \] Input:

int((a + b*log(c*(d*f + d*g*x^2)^p))/(x^4*(f + g*x^2)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int((a + b*log(c*(d*f + d*g*x^2)^p))/(x^4*(f + g*x^2)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.32 \[ \int \frac {a+b \log \left (c \left (d f+d g x^2\right )^p\right )}{x^4 \sqrt {f+g x^2}} \, dx=\frac {-3 \sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b f +6 \sqrt {g \,x^{2}+f}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b g \,x^{2}-3 \sqrt {g \,x^{2}+f}\, a f +6 \sqrt {g \,x^{2}+f}\, a g \,x^{2}-6 \sqrt {g \,x^{2}+f}\, b g p \,x^{2}-12 \sqrt {g}\, \mathrm {log}\left (\frac {2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}}{f}\right ) b g p \,x^{3}+6 \sqrt {g}\, \mathrm {log}\left (\frac {d^{p} \left (2 \sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +2 f +2 g \,x^{2}\right )^{2 p} c}{\left (\sqrt {g \,x^{2}+f}+\sqrt {g}\, x \right )^{2 p} 2^{2 p}}\right ) b g \,x^{3}-6 \sqrt {g}\, a g \,x^{3}+2 \sqrt {g}\, b g p \,x^{3}}{9 f^{2} x^{3}} \] Input:

int((a+b*log(c*(d*g*x^2+d*f)^p))/x^4/(g*x^2+f)^(1/2),x)
 

Output:

( - 3*sqrt(f + g*x**2)*log((d**p*(2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 2*g 
*x**2)**(2*p)*c)/((sqrt(f + g*x**2) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*f + 6 
*sqrt(f + g*x**2)*log((d**p*(2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 2*g*x**2 
)**(2*p)*c)/((sqrt(f + g*x**2) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*g*x**2 - 3 
*sqrt(f + g*x**2)*a*f + 6*sqrt(f + g*x**2)*a*g*x**2 - 6*sqrt(f + g*x**2)*b 
*g*p*x**2 - 12*sqrt(g)*log((2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 2*g*x**2) 
/f)*b*g*p*x**3 + 6*sqrt(g)*log((d**p*(2*sqrt(g)*sqrt(f + g*x**2)*x + 2*f + 
 2*g*x**2)**(2*p)*c)/((sqrt(f + g*x**2) + sqrt(g)*x)**(2*p)*2**(2*p)))*b*g 
*x**3 - 6*sqrt(g)*a*g*x**3 + 2*sqrt(g)*b*g*p*x**3)/(9*f**2*x**3)