\(\int \frac {x^7 (a+b \log (c (4 d-d g x^2)^p))}{\sqrt {4-g x^2}} \, dx\) [676]

Optimal result

Integrand size = 36, antiderivative size = 235 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {128 b p \sqrt {4-g x^2}}{g^4}-\frac {32 b p \left (4-g x^2\right )^{3/2}}{3 g^4}+\frac {24 b p \left (4-g x^2\right )^{5/2}}{25 g^4}-\frac {2 b p \left (4-g x^2\right )^{7/2}}{49 g^4}-\frac {64 \sqrt {4-g x^2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{g^4}+\frac {16 \left (4-g x^2\right )^{3/2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{g^4}-\frac {12 \left (4-g x^2\right )^{5/2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{5 g^4}+\frac {\left (4-g x^2\right )^{7/2} \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{7 g^4} \] Output:

128*b*p*(-g*x^2+4)^(1/2)/g^4-32/3*b*p*(-g*x^2+4)^(3/2)/g^4+24/25*b*p*(-g*x 
^2+4)^(5/2)/g^4-2/49*b*p*(-g*x^2+4)^(7/2)/g^4-64*(-g*x^2+4)^(1/2)*(a+b*ln( 
c*(-d*g*x^2+4*d)^p))/g^4+16*(-g*x^2+4)^(3/2)*(a+b*ln(c*(-d*g*x^2+4*d)^p))/ 
g^4-12/5*(-g*x^2+4)^(5/2)*(a+b*ln(c*(-d*g*x^2+4*d)^p))/g^4+1/7*(-g*x^2+4)^ 
(7/2)*(a+b*ln(c*(-d*g*x^2+4*d)^p))/g^4
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.50 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {\sqrt {4-g x^2} \left (-105 a \left (1024+128 g x^2+24 g^2 x^4+5 g^3 x^6\right )+2 b p \left (180224+9088 g x^2+864 g^2 x^4+75 g^3 x^6\right )-105 b \left (1024+128 g x^2+24 g^2 x^4+5 g^3 x^6\right ) \log \left (c \left (d \left (4-g x^2\right )\right )^p\right )\right )}{3675 g^4} \] Input:

Integrate[(x^7*(a + b*Log[c*(4*d - d*g*x^2)^p]))/Sqrt[4 - g*x^2],x]
 

Output:

(Sqrt[4 - g*x^2]*(-105*a*(1024 + 128*g*x^2 + 24*g^2*x^4 + 5*g^3*x^6) + 2*b 
*p*(180224 + 9088*g*x^2 + 864*g^2*x^4 + 75*g^3*x^6) - 105*b*(1024 + 128*g* 
x^2 + 24*g^2*x^4 + 5*g^3*x^6)*Log[c*(d*(4 - g*x^2))^p]))/(3675*g^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.

\(\displaystyle \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx\)

\(\Big \downarrow \) 2929

\(\displaystyle \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}}dx\)

Input:

Int[(x^7*(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]

\[\int \frac {x^{7} \left (a +b \ln \left (c \left (-d g \,x^{2}+4 d \right )^{p}\right )\right )}{\sqrt {-g \,x^{2}+4}}d x\]

Input:

int(x^7*(a+b*ln(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
 

Output:

int(x^7*(a+b*ln(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.66 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {{\left (75 \, {\left (2 \, b g^{3} p - 7 \, a g^{3}\right )} x^{6} + 72 \, {\left (24 \, b g^{2} p - 35 \, a g^{2}\right )} x^{4} + 128 \, {\left (142 \, b g p - 105 \, a g\right )} x^{2} + 360448 \, b p - 105 \, {\left (5 \, b g^{3} p x^{6} + 24 \, b g^{2} p x^{4} + 128 \, b g p x^{2} + 1024 \, b p\right )} \log \left (-d g x^{2} + 4 \, d\right ) - 105 \, {\left (5 \, b g^{3} x^{6} + 24 \, b g^{2} x^{4} + 128 \, b g x^{2} + 1024 \, b\right )} \log \left (c\right ) - 107520 \, a\right )} \sqrt {-g x^{2} + 4}}{3675 \, g^{4}} \] Input:

integrate(x^7*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= 
"fricas")
 

Output:

1/3675*(75*(2*b*g^3*p - 7*a*g^3)*x^6 + 72*(24*b*g^2*p - 35*a*g^2)*x^4 + 12 
8*(142*b*g*p - 105*a*g)*x^2 + 360448*b*p - 105*(5*b*g^3*p*x^6 + 24*b*g^2*p 
*x^4 + 128*b*g*p*x^2 + 1024*b*p)*log(-d*g*x^2 + 4*d) - 105*(5*b*g^3*x^6 + 
24*b*g^2*x^4 + 128*b*g*x^2 + 1024*b)*log(c) - 107520*a)*sqrt(-g*x^2 + 4)/g 
^4
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\text {Timed out} \] Input:

integrate(x**7*(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 = 0.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.93 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=-\frac {1}{35} \, {\left (\frac {5 \, \sqrt {-g x^{2} + 4} x^{6}}{g} + \frac {24 \, \sqrt {-g x^{2} + 4} x^{4}}{g^{2}} + \frac {128 \, \sqrt {-g x^{2} + 4} x^{2}}{g^{3}} + \frac {1024 \, \sqrt {-g x^{2} + 4}}{g^{4}}\right )} b \log \left ({\left (-d g x^{2} + 4 \, d\right )}^{p} c\right ) - \frac {1}{35} \, {\left (\frac {5 \, \sqrt {-g x^{2} + 4} x^{6}}{g} + \frac {24 \, \sqrt {-g x^{2} + 4} x^{4}}{g^{2}} + \frac {128 \, \sqrt {-g x^{2} + 4} x^{2}}{g^{3}} + \frac {1024 \, \sqrt {-g x^{2} + 4}}{g^{4}}\right )} a - \frac {2 \, {\left (75 \, {\left (-g x^{2} + 4\right )}^{\frac {7}{2}} - 1764 \, {\left (-g x^{2} + 4\right )}^{\frac {5}{2}} + 19600 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} - 235200 \, \sqrt {-g x^{2} + 4}\right )} b p}{3675 \, g^{4}} \] Input:

integrate(x^7*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= 
"maxima")
 

Output:

-1/35*(5*sqrt(-g*x^2 + 4)*x^6/g + 24*sqrt(-g*x^2 + 4)*x^4/g^2 + 128*sqrt(- 
g*x^2 + 4)*x^2/g^3 + 1024*sqrt(-g*x^2 + 4)/g^4)*b*log((-d*g*x^2 + 4*d)^p*c 
) - 1/35*(5*sqrt(-g*x^2 + 4)*x^6/g + 24*sqrt(-g*x^2 + 4)*x^4/g^2 + 128*sqr 
t(-g*x^2 + 4)*x^2/g^3 + 1024*sqrt(-g*x^2 + 4)/g^4)*a - 2/3675*(75*(-g*x^2 
+ 4)^(7/2) - 1764*(-g*x^2 + 4)^(5/2) + 19600*(-g*x^2 + 4)^(3/2) - 235200*s 
qrt(-g*x^2 + 4))*b*p/g^4
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.34 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=-\frac {b p {\left (\frac {105 \, {\left (5 \, {\left (g x^{2} - 4\right )}^{3} \sqrt {-g x^{2} + 4} + 84 \, {\left (g x^{2} - 4\right )}^{2} \sqrt {-g x^{2} + 4} - 560 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 2240 \, \sqrt {-g x^{2} + 4}\right )} \log \left (-d g x^{2} + 4 \, d\right )}{g^{3}} - \frac {2 \, {\left (75 \, {\left (g x^{2} - 4\right )}^{3} \sqrt {-g x^{2} + 4} + 1764 \, {\left (g x^{2} - 4\right )}^{2} \sqrt {-g x^{2} + 4} - 19600 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 235200 \, \sqrt {-g x^{2} + 4}\right )}}{g^{3}}\right )} + \frac {105 \, {\left (5 \, {\left (g x^{2} - 4\right )}^{3} \sqrt {-g x^{2} + 4} + 84 \, {\left (g x^{2} - 4\right )}^{2} \sqrt {-g x^{2} + 4} - 560 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 2240 \, \sqrt {-g x^{2} + 4}\right )} b \log \left (c\right )}{g^{3}} + \frac {105 \, {\left (5 \, {\left (g x^{2} - 4\right )}^{3} \sqrt {-g x^{2} + 4} + 84 \, {\left (g x^{2} - 4\right )}^{2} \sqrt {-g x^{2} + 4} - 560 \, {\left (-g x^{2} + 4\right )}^{\frac {3}{2}} + 2240 \, \sqrt {-g x^{2} + 4}\right )} a}{g^{3}}}{3675 \, g} \] Input:

integrate(x^7*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x, algorithm= 
"giac")
 

Output:

-1/3675*(b*p*(105*(5*(g*x^2 - 4)^3*sqrt(-g*x^2 + 4) + 84*(g*x^2 - 4)^2*sqr 
t(-g*x^2 + 4) - 560*(-g*x^2 + 4)^(3/2) + 2240*sqrt(-g*x^2 + 4))*log(-d*g*x 
^2 + 4*d)/g^3 - 2*(75*(g*x^2 - 4)^3*sqrt(-g*x^2 + 4) + 1764*(g*x^2 - 4)^2* 
sqrt(-g*x^2 + 4) - 19600*(-g*x^2 + 4)^(3/2) + 235200*sqrt(-g*x^2 + 4))/g^3 
) + 105*(5*(g*x^2 - 4)^3*sqrt(-g*x^2 + 4) + 84*(g*x^2 - 4)^2*sqrt(-g*x^2 + 
 4) - 560*(-g*x^2 + 4)^(3/2) + 2240*sqrt(-g*x^2 + 4))*b*log(c)/g^3 + 105*( 
5*(g*x^2 - 4)^3*sqrt(-g*x^2 + 4) + 84*(g*x^2 - 4)^2*sqrt(-g*x^2 + 4) - 560 
*(-g*x^2 + 4)^(3/2) + 2240*sqrt(-g*x^2 + 4))*a/g^3)/g
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\int \frac {x^7\,\left (a+b\,\ln \left (c\,{\left (4\,d-d\,g\,x^2\right )}^p\right )\right )}{\sqrt {4-g\,x^2}} \,d x \] Input:

int((x^7*(a + b*log(c*(4*d - d*g*x^2)^p)))/(4 - g*x^2)^(1/2),x)
 

Output:

int((x^7*(a + b*log(c*(4*d - d*g*x^2)^p)))/(4 - g*x^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.88 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d-d g x^2\right )^p\right )\right )}{\sqrt {4-g x^2}} \, dx=\frac {\sqrt {-g \,x^{2}+4}\, \left (-525 \,\mathrm {log}\left (\frac {d^{p} \left (-g \,x^{2}+4\right )^{p} 4^{p} c}{2^{2 p}}\right ) b \,g^{3} x^{6}-2520 \,\mathrm {log}\left (\frac {d^{p} \left (-g \,x^{2}+4\right )^{p} 4^{p} c}{2^{2 p}}\right ) b \,g^{2} x^{4}-13440 \,\mathrm {log}\left (\frac {d^{p} \left (-g \,x^{2}+4\right )^{p} 4^{p} c}{2^{2 p}}\right ) b g \,x^{2}-107520 \,\mathrm {log}\left (\frac {d^{p} \left (-g \,x^{2}+4\right )^{p} 4^{p} c}{2^{2 p}}\right ) b -525 a \,g^{3} x^{6}-2520 a \,g^{2} x^{4}-13440 a g \,x^{2}-107520 a +150 b \,g^{3} p \,x^{6}+1728 b \,g^{2} p \,x^{4}+18176 b g p \,x^{2}+360448 b p \right )}{3675 g^{4}} \] Input:

int(x^7*(a+b*log(c*(-d*g*x^2+4*d)^p))/(-g*x^2+4)^(1/2),x)
 

Output:

(sqrt( - g*x**2 + 4)*( - 525*log((d**p*( - g*x**2 + 4)**p*4**p*c)/2**(2*p) 
)*b*g**3*x**6 - 2520*log((d**p*( - g*x**2 + 4)**p*4**p*c)/2**(2*p))*b*g**2 
*x**4 - 13440*log((d**p*( - g*x**2 + 4)**p*4**p*c)/2**(2*p))*b*g*x**2 - 10 
7520*log((d**p*( - g*x**2 + 4)**p*4**p*c)/2**(2*p))*b - 525*a*g**3*x**6 - 
2520*a*g**2*x**4 - 13440*a*g*x**2 - 107520*a + 150*b*g**3*p*x**6 + 1728*b* 
g**2*p*x**4 + 18176*b*g*p*x**2 + 360448*b*p))/(3675*g**4)