\(\int \frac {x^5 (a+b \log (c (4 d-d g x^2)^p))}{\sqrt {4-g x^2}} \, dx\) [677]

Optimal result

Integrand size = 36, antiderivative size = 177 \[ \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {32 b p \sqrt {4-g x^2}}{g^3}-\frac {16 b p \left (4-g x^2\right )^{3/2}}{9 g^3}+\frac {2 b p \left (4-g x^2\right )^{5/2}}{25 g^3}-\frac {16 \sqrt {4-g x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{g^3}+\frac {8 \left (4-g x^2\right )^{3/2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{3 g^3}-\frac {\left (4-g x^2\right )^{5/2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{5 g^3} \] Output:

32*b*p*(-g*x^2+4)^(1/2)/g^3-16/9*b*p*(-g*x^2+4)^(3/2)/g^3+2/25*b*p*(-g*x^2 
+4)^(5/2)/g^3-16*(-g*x^2+4)^(1/2)*(a+b*ln(c*(-d*g*x^2+4*d)^p))/g^3+8/3*(-g 
*x^2+4)^(3/2)*(a+b*ln(c*(-d*g*x^2+4*d)^p))/g^3-1/5*(-g*x^2+4)^(5/2)*(a+b*l 
n(c*(-d*g*x^2+4*d)^p))/g^3
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.53 \[ \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {\sqrt {4-g x^2} \left (-15 a \left (128+16 g x^2+3 g^2 x^4\right )+2 b p \left (2944+128 g x^2+9 g^2 x^4\right )-15 b \left (128+16 g x^2+3 g^2 x^4\right ) \log \left (c \left (d \left (4-g x^2\right )\right )^p\right )\right )}{225 g^3} \] Input:

Integrate[(x^5*(a + b*Log[c*(4*d - d*g*x^2)^p]))/Sqrt[4 - g*x^2],x]
 

Output:

(Sqrt[4 - g*x^2]*(-15*a*(128 + 16*g*x^2 + 3*g^2*x^4) + 2*b*p*(2944 + 128*g 
*x^2 + 9*g^2*x^4) - 15*b*(128 + 16*g*x^2 + 3*g^2*x^4)*Log[c*(d*(4 - g*x^2) 
)^p]))/(225*g^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.

\(\displaystyle \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}}dx\)

Input:

Int[(x^5*(a + b*Log[c*(4*d - d*g*x^2)^p]))/Sqrt[4 - 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]

\[\int \frac {x^{5} \left (a +b \ln \left (c \left (-d g \,x^{2}+4 d \right )^{p}\right )\right )}{\sqrt {-g \,x^{2}+4}}d x\]

Input:

int(x^5*(a+b*ln(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
 

Output:

int(x^5*(a+b*ln(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67 \[ \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {{\left (9 \, {\left (2 \, b g^{2} p - 5 \, a g^{2}\right )} x^{4} + 16 \, {\left (16 \, b g p - 15 \, a g\right )} x^{2} + 5888 \, b p - 15 \, {\left (3 \, b g^{2} p x^{4} + 16 \, b g p x^{2} + 128 \, b p\right )} \log \left (-d g x^{2} + 4 \, d\right ) - 15 \, {\left (3 \, b g^{2} x^{4} + 16 \, b g x^{2} + 128 \, b\right )} \log \left (c\right ) - 1920 \, a\right )} \sqrt {-g x^{2} + 4}}{225 \, g^{3}} \] Input:

integrate(x^5*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= 
"fricas")
 

Output:

1/225*(9*(2*b*g^2*p - 5*a*g^2)*x^4 + 16*(16*b*g*p - 15*a*g)*x^2 + 5888*b*p 
 - 15*(3*b*g^2*p*x^4 + 16*b*g*p*x^2 + 128*b*p)*log(-d*g*x^2 + 4*d) - 15*(3 
*b*g^2*x^4 + 16*b*g*x^2 + 128*b)*log(c) - 1920*a)*sqrt(-g*x^2 + 4)/g^3
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\text {Timed out} \] Input:

integrate(x**5*(a+b*ln(c*(-d*g*x**2+4*d)**p))/(-g*x**2+4)**(1/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.97 \[ \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=-\frac {1}{15} \, {\left (\frac {3 \, \sqrt {-g x^{2} + 4} x^{4}}{g} + \frac {16 \, \sqrt {-g x^{2} + 4} x^{2}}{g^{2}} + \frac {128 \, \sqrt {-g x^{2} + 4}}{g^{3}}\right )} b \log \left ({\left (-d g x^{2} + 4 \, d\right )}^{p} c\right ) - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-g x^{2} + 4} x^{4}}{g} + \frac {16 \, \sqrt {-g x^{2} + 4} x^{2}}{g^{2}} + \frac {128 \, \sqrt {-g x^{2} + 4}}{g^{3}}\right )} a + \frac {2 \, {\left (9 \, {\left (-g x^{2} + 4\right )}^{\frac {5}{2}} - 200 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 3600 \, \sqrt {-g x^{2} + 4}\right )} b p}{225 \, g^{3}} \] Input:

integrate(x^5*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= 
"maxima")
 

Output:

-1/15*(3*sqrt(-g*x^2 + 4)*x^4/g + 16*sqrt(-g*x^2 + 4)*x^2/g^2 + 128*sqrt(- 
g*x^2 + 4)/g^3)*b*log((-d*g*x^2 + 4*d)^p*c) - 1/15*(3*sqrt(-g*x^2 + 4)*x^4 
/g + 16*sqrt(-g*x^2 + 4)*x^2/g^2 + 128*sqrt(-g*x^2 + 4)/g^3)*a + 2/225*(9* 
(-g*x^2 + 4)^(5/2) - 200*(-g*x^2 + 4)^(3/2) + 3600*sqrt(-g*x^2 + 4))*b*p/g 
^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.30 \[ \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=-\frac {b p {\left (\frac {15 \, {\left (3 \, {\left (g x^{2} - 4\right )}^{2} \sqrt {-g x^{2} + 4} - 40 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 240 \, \sqrt {-g x^{2} + 4}\right )} \log \left (-d g x^{2} + 4 \, d\right )}{g^{2}} - \frac {2 \, {\left (9 \, {\left (g x^{2} - 4\right )}^{2} \sqrt {-g x^{2} + 4} - 200 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 3600 \, \sqrt {-g x^{2} + 4}\right )}}{g^{2}}\right )} + \frac {15 \, {\left (3 \, {\left (g x^{2} - 4\right )}^{2} \sqrt {-g x^{2} + 4} - 40 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 240 \, \sqrt {-g x^{2} + 4}\right )} b \log \left (c\right )}{g^{2}} + \frac {15 \, {\left (3 \, {\left (g x^{2} - 4\right )}^{2} \sqrt {-g x^{2} + 4} - 40 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 240 \, \sqrt {-g x^{2} + 4}\right )} a}{g^{2}}}{225 \, g} \] Input:

integrate(x^5*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= 
"giac")
 

Output:

-1/225*(b*p*(15*(3*(g*x^2 - 4)^2*sqrt(-g*x^2 + 4) - 40*(-g*x^2 + 4)^(3/2) 
+ 240*sqrt(-g*x^2 + 4))*log(-d*g*x^2 + 4*d)/g^2 - 2*(9*(g*x^2 - 4)^2*sqrt( 
-g*x^2 + 4) - 200*(-g*x^2 + 4)^(3/2) + 3600*sqrt(-g*x^2 + 4))/g^2) + 15*(3 
*(g*x^2 - 4)^2*sqrt(-g*x^2 + 4) - 40*(-g*x^2 + 4)^(3/2) + 240*sqrt(-g*x^2 
+ 4))*b*log(c)/g^2 + 15*(3*(g*x^2 - 4)^2*sqrt(-g*x^2 + 4) - 40*(-g*x^2 + 4 
)^(3/2) + 240*sqrt(-g*x^2 + 4))*a/g^2)/g
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\int \frac {x^5\,\left (a+b\,\ln \left (c\,{\left (4\,d-d\,g\,x^2\right )}^p\right )\right )}{\sqrt {4-g\,x^2}} \,d x \] Input:

int((x^5*(a + b*log(c*(4*d - d*g*x^2)^p)))/(4 - g*x^2)^(1/2),x)
 

Output:

int((x^5*(a + b*log(c*(4*d - d*g*x^2)^p)))/(4 - g*x^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.86 \[ \int \frac {x^5 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {\sqrt {-g \,x^{2}+4}\, \left (-45 \,\mathrm {log}\left (\frac {d^{p} \left (-g \,x^{2}+4\right )^{p} 4^{p} c}{2^{2 p}}\right ) b \,g^{2} x^{4}-240 \,\mathrm {log}\left (\frac {d^{p} \left (-g \,x^{2}+4\right )^{p} 4^{p} c}{2^{2 p}}\right ) b g \,x^{2}-1920 \,\mathrm {log}\left (\frac {d^{p} \left (-g \,x^{2}+4\right )^{p} 4^{p} c}{2^{2 p}}\right ) b -45 a \,g^{2} x^{4}-240 a g \,x^{2}-1920 a +18 b \,g^{2} p \,x^{4}+256 b g p \,x^{2}+5888 b p \right )}{225 g^{3}} \] Input:

int(x^5*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
 

Output:

(sqrt( - g*x**2 + 4)*( - 45*log((d**p*( - g*x**2 + 4)**p*4**p*c)/2**(2*p)) 
*b*g**2*x**4 - 240*log((d**p*( - g*x**2 + 4)**p*4**p*c)/2**(2*p))*b*g*x**2 
 - 1920*log((d**p*( - g*x**2 + 4)**p*4**p*c)/2**(2*p))*b - 45*a*g**2*x**4 
- 240*a*g*x**2 - 1920*a + 18*b*g**2*p*x**4 + 256*b*g*p*x**2 + 5888*b*p))/( 
225*g**3)