\(\int \frac {x^7 (a+b \log (c (4 d+d g x^2)^p))}{\sqrt {4+g x^2}} \, dx\) [663]

Optimal result

Integrand size = 34, antiderivative size = 223 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{\sqrt {4+g x^2}} \, dx=\frac {128 b p \sqrt {4+g x^2}}{g^4}-\frac {32 b p \left (4+g x^2\right )^{3/2}}{3 g^4}+\frac {24 b p \left (4+g x^2\right )^{5/2}}{25 g^4}-\frac {2 b p \left (4+g x^2\right )^{7/2}}{49 g^4}-\frac {64 \sqrt {4+g x^2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{g^4}+\frac {16 \left (4+g x^2\right )^{3/2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{g^4}-\frac {12 \left (4+g x^2\right )^{5/2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{5 g^4}+\frac {\left (4+g x^2\right )^{7/2} \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{7 g^4} \] 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 = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.52 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{\sqrt {4+g x^2}} \, dx=\frac {\sqrt {4+g x^2} \left (2 b p \left (180224-9088 g x^2+864 g^2 x^4-75 g^3 x^6\right )+105 a \left (-1024+128 g x^2-24 g^2 x^4+5 g^3 x^6\right )+105 b \left (-1024+128 g x^2-24 g^2 x^4+5 g^3 x^6\right ) \log \left (c \left (d \left (4+g x^2\right )\right )^p\right )\right )}{3675 g^4} \] 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]*(2*b*p*(180224 - 9088*g*x^2 + 864*g^2*x^4 - 75*g^3*x^6) + 
 105*a*(-1024 + 128*g*x^2 - 24*g^2*x^4 + 5*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.

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

\(\Big \downarrow \) 2929

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

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 = 0.10 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.69 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{\sqrt {4+g x^2}} \, dx=-\frac {{\left (75 \, {\left (2 \, b g^{3} p - 7 \, a g^{3}\right )} x^{6} - 72 \, {\left (24 \, b g^{2} p - 35 \, a g^{2}\right )} x^{4} + 128 \, {\left (142 \, b g p - 105 \, a g\right )} x^{2} - 360448 \, b p - 105 \, {\left (5 \, b g^{3} p x^{6} - 24 \, b g^{2} p x^{4} + 128 \, b g p x^{2} - 1024 \, b p\right )} \log \left (d g x^{2} + 4 \, d\right ) - 105 \, {\left (5 \, b g^{3} x^{6} - 24 \, b g^{2} x^{4} + 128 \, b g x^{2} - 1024 \, b\right )} \log \left (c\right ) + 107520 \, a\right )} \sqrt {g x^{2} + 4}}{3675 \, g^{4}} \] Input:

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

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 + 1 
28*(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 [A] (verification not implemented)

Time = 15.29 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.49 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{\sqrt {4+g x^2}} \, dx=\begin {cases} \frac {a x^{6} \sqrt {g x^{2} + 4}}{7 g} - \frac {24 a x^{4} \sqrt {g x^{2} + 4}}{35 g^{2}} + \frac {128 a x^{2} \sqrt {g x^{2} + 4}}{35 g^{3}} - \frac {1024 a \sqrt {g x^{2} + 4}}{35 g^{4}} - \frac {2 b p x^{6} \sqrt {g x^{2} + 4}}{49 g} + \frac {b x^{6} \sqrt {g x^{2} + 4} \log {\left (c \left (d g x^{2} + 4 d\right )^{p} \right )}}{7 g} + \frac {576 b p x^{4} \sqrt {g x^{2} + 4}}{1225 g^{2}} - \frac {24 b x^{4} \sqrt {g x^{2} + 4} \log {\left (c \left (d g x^{2} + 4 d\right )^{p} \right )}}{35 g^{2}} - \frac {18176 b p x^{2} \sqrt {g x^{2} + 4}}{3675 g^{3}} + \frac {128 b x^{2} \sqrt {g x^{2} + 4} \log {\left (c \left (d g x^{2} + 4 d\right )^{p} \right )}}{35 g^{3}} + \frac {360448 b p \sqrt {g x^{2} + 4}}{3675 g^{4}} - \frac {1024 b \sqrt {g x^{2} + 4} \log {\left (c \left (d g x^{2} + 4 d\right )^{p} \right )}}{35 g^{4}} & \text {for}\: g \neq 0 \\\frac {x^{8} \left (\frac {a}{2} + \frac {b \log {\left (c \left (4 d\right )^{p} \right )}}{2}\right )}{8} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*x**6*sqrt(g*x**2 + 4)/(7*g) - 24*a*x**4*sqrt(g*x**2 + 4)/(35* 
g**2) + 128*a*x**2*sqrt(g*x**2 + 4)/(35*g**3) - 1024*a*sqrt(g*x**2 + 4)/(3 
5*g**4) - 2*b*p*x**6*sqrt(g*x**2 + 4)/(49*g) + b*x**6*sqrt(g*x**2 + 4)*log 
(c*(d*g*x**2 + 4*d)**p)/(7*g) + 576*b*p*x**4*sqrt(g*x**2 + 4)/(1225*g**2) 
- 24*b*x**4*sqrt(g*x**2 + 4)*log(c*(d*g*x**2 + 4*d)**p)/(35*g**2) - 18176* 
b*p*x**2*sqrt(g*x**2 + 4)/(3675*g**3) + 128*b*x**2*sqrt(g*x**2 + 4)*log(c* 
(d*g*x**2 + 4*d)**p)/(35*g**3) + 360448*b*p*sqrt(g*x**2 + 4)/(3675*g**4) - 
 1024*b*sqrt(g*x**2 + 4)*log(c*(d*g*x**2 + 4*d)**p)/(35*g**4), Ne(g, 0)), 
(x**8*(a/2 + b*log(c*(4*d)**p)/2)/8, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.92 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{\sqrt {4+g x^2}} \, dx=\frac {1}{35} \, {\left (\frac {5 \, \sqrt {g x^{2} + 4} x^{6}}{g} - \frac {24 \, \sqrt {g x^{2} + 4} x^{4}}{g^{2}} + \frac {128 \, \sqrt {g x^{2} + 4} x^{2}}{g^{3}} - \frac {1024 \, \sqrt {g x^{2} + 4}}{g^{4}}\right )} b \log \left ({\left (d g x^{2} + 4 \, d\right )}^{p} c\right ) + \frac {1}{35} \, {\left (\frac {5 \, \sqrt {g x^{2} + 4} x^{6}}{g} - \frac {24 \, \sqrt {g x^{2} + 4} x^{4}}{g^{2}} + \frac {128 \, \sqrt {g x^{2} + 4} x^{2}}{g^{3}} - \frac {1024 \, \sqrt {g x^{2} + 4}}{g^{4}}\right )} a - \frac {2 \, {\left (75 \, {\left (g x^{2} + 4\right )}^{\frac {7}{2}} - 1764 \, {\left (g x^{2} + 4\right )}^{\frac {5}{2}} + 19600 \, {\left (g x^{2} + 4\right )}^{\frac {3}{2}} - 235200 \, \sqrt {g x^{2} + 4}\right )} b p}{3675 \, g^{4}} \] Input:

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

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*sqrt(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*sqrt(g*x^2 + 4 
))*b*p/g^4
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.01 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{\sqrt {4+g x^2}} \, dx=\frac {b p {\left (\frac {105 \, {\left (5 \, {\left (g x^{2} + 4\right )}^{\frac {7}{2}} - 84 \, {\left (g x^{2} + 4\right )}^{\frac {5}{2}} + 560 \, {\left (g x^{2} + 4\right )}^{\frac {3}{2}} - 2240 \, \sqrt {g x^{2} + 4}\right )} \log \left (d g x^{2} + 4 \, d\right )}{g^{3}} - \frac {2 \, {\left (75 \, {\left (g x^{2} + 4\right )}^{\frac {7}{2}} - 1764 \, {\left (g x^{2} + 4\right )}^{\frac {5}{2}} + 19600 \, {\left (g x^{2} + 4\right )}^{\frac {3}{2}} - 235200 \, \sqrt {g x^{2} + 4}\right )}}{g^{3}}\right )} + \frac {105 \, {\left (5 \, {\left (g x^{2} + 4\right )}^{\frac {7}{2}} - 84 \, {\left (g x^{2} + 4\right )}^{\frac {5}{2}} + 560 \, {\left (g x^{2} + 4\right )}^{\frac {3}{2}} - 2240 \, \sqrt {g x^{2} + 4}\right )} b \log \left (c\right )}{g^{3}} + \frac {105 \, {\left (5 \, {\left (g x^{2} + 4\right )}^{\frac {7}{2}} - 84 \, {\left (g x^{2} + 4\right )}^{\frac {5}{2}} + 560 \, {\left (g x^{2} + 4\right )}^{\frac {3}{2}} - 2240 \, \sqrt {g x^{2} + 4}\right )} a}{g^{3}}}{3675 \, g} \] Input:

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

Output:

1/3675*(b*p*(105*(5*(g*x^2 + 4)^(7/2) - 84*(g*x^2 + 4)^(5/2) + 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)^(7/2) - 1764*(g*x^2 + 4)^(5/2) + 19600*(g*x^2 + 4)^(3/2) - 235200*sqrt 
(g*x^2 + 4))/g^3) + 105*(5*(g*x^2 + 4)^(7/2) - 84*(g*x^2 + 4)^(5/2) + 560* 
(g*x^2 + 4)^(3/2) - 2240*sqrt(g*x^2 + 4))*b*log(c)/g^3 + 105*(5*(g*x^2 + 4 
)^(7/2) - 84*(g*x^2 + 4)^(5/2) + 560*(g*x^2 + 4)^(3/2) - 2240*sqrt(g*x^2 + 
 4))*a/g^3)/g
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.43 \[ \int \frac {x^7 \left (a+b \log \left (c \left (4 d+d g x^2\right )^p\right )\right )}{\sqrt {4+g x^2}} \, dx=\frac {\sqrt {g \,x^{2}+4}\, \left (525 \,\mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (2 \sqrt {g \,x^{2}+4}+2 \sqrt {g}\, x \right )^{2 p}}\right ) b \,g^{3} x^{6}-2520 \,\mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (2 \sqrt {g \,x^{2}+4}+2 \sqrt {g}\, x \right )^{2 p}}\right ) b \,g^{2} x^{4}+13440 \,\mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (2 \sqrt {g \,x^{2}+4}+2 \sqrt {g}\, x \right )^{2 p}}\right ) b g \,x^{2}-107520 \,\mathrm {log}\left (\frac {d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+4}\, x +g \,x^{2}+4\right )^{2 p} 4^{p} c}{\left (2 \sqrt {g \,x^{2}+4}+2 \sqrt {g}\, x \right )^{2 p}}\right ) b +525 a \,g^{3} x^{6}-2520 a \,g^{2} x^{4}+13440 a g \,x^{2}-107520 a -150 b \,g^{3} p \,x^{6}+1728 b \,g^{2} p \,x^{4}-18176 b g p \,x^{2}+360448 b p \right )}{3675 g^{4}} \] 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*(sqrt(g)*sqrt(g*x**2 + 4)*x + g*x**2 + 4) 
**(2*p)*4**p*c)/(2*sqrt(g*x**2 + 4) + 2*sqrt(g)*x)**(2*p))*b*g**3*x**6 - 2 
520*log((d**p*(sqrt(g)*sqrt(g*x**2 + 4)*x + g*x**2 + 4)**(2*p)*4**p*c)/(2* 
sqrt(g*x**2 + 4) + 2*sqrt(g)*x)**(2*p))*b*g**2*x**4 + 13440*log((d**p*(sqr 
t(g)*sqrt(g*x**2 + 4)*x + g*x**2 + 4)**(2*p)*4**p*c)/(2*sqrt(g*x**2 + 4) + 
 2*sqrt(g)*x)**(2*p))*b*g*x**2 - 107520*log((d**p*(sqrt(g)*sqrt(g*x**2 + 4 
)*x + g*x**2 + 4)**(2*p)*4**p*c)/(2*sqrt(g*x**2 + 4) + 2*sqrt(g)*x)**(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))/(367 
5*g**4)