\(\int \frac {x^7 (a+b \log (c (d f+d g x^2)^p))}{\sqrt {f+g x^2}} \, dx\) [689]

Optimal result

Integrand size = 34, antiderivative size = 236 \[ \int \frac {x^7 \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{\sqrt {f+g x^2}} \, dx=\frac {2 b f^3 p \sqrt {f+g x^2}}{g^4}-\frac {2 b f^2 p \left (f+g x^2\right )^{3/2}}{3 g^4}+\frac {6 b f p \left (f+g x^2\right )^{5/2}}{25 g^4}-\frac {2 b p \left (f+g x^2\right )^{7/2}}{49 g^4}-\frac {f^3 \sqrt {f+g x^2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{g^4}+\frac {f^2 \left (f+g x^2\right )^{3/2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{g^4}-\frac {3 f \left (f+g x^2\right )^{5/2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{5 g^4}+\frac {\left (f+g x^2\right )^{7/2} \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{7 g^4} \] Output:

2*b*f^3*p*(g*x^2+f)^(1/2)/g^4-2/3*b*f^2*p*(g*x^2+f)^(3/2)/g^4+6/25*b*f*p*( 
g*x^2+f)^(5/2)/g^4-2/49*b*p*(g*x^2+f)^(7/2)/g^4-f^3*(g*x^2+f)^(1/2)*(a+b*l 
n(c*(d*g*x^2+d*f)^p))/g^4+f^2*(g*x^2+f)^(3/2)*(a+b*ln(c*(d*g*x^2+d*f)^p))/ 
g^4-3/5*f*(g*x^2+f)^(5/2)*(a+b*ln(c*(d*g*x^2+d*f)^p))/g^4+1/7*(g*x^2+f)^(7 
/2)*(a+b*ln(c*(d*g*x^2+d*f)^p))/g^4
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.59 \[ \int \frac {x^7 \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{\sqrt {f+g x^2}} \, dx=-\frac {\sqrt {f+g x^2} \left (105 a \left (16 f^3-8 f^2 g x^2+6 f g^2 x^4-5 g^3 x^6\right )+2 b p \left (-2816 f^3+568 f^2 g x^2-216 f g^2 x^4+75 g^3 x^6\right )+105 b \left (16 f^3-8 f^2 g x^2+6 f g^2 x^4-5 g^3 x^6\right ) \log \left (c \left (d \left (f+g x^2\right )\right )^p\right )\right )}{3675 g^4} \] Input:

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

Output:

-1/3675*(Sqrt[f + g*x^2]*(105*a*(16*f^3 - 8*f^2*g*x^2 + 6*f*g^2*x^4 - 5*g^ 
3*x^6) + 2*b*p*(-2816*f^3 + 568*f^2*g*x^2 - 216*f*g^2*x^4 + 75*g^3*x^6) + 
105*b*(16*f^3 - 8*f^2*g*x^2 + 6*f*g^2*x^4 - 5*g^3*x^6)*Log[c*(d*(f + g*x^2 
))^p]))/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.

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

\(\Big \downarrow \) 2929

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

Input:

Int[(x^7*(a + b*Log[c*(d*f + d*g*x^2)^p]))/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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.77 \[ \int \frac {x^7 \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{\sqrt {f+g x^2}} \, dx=-\frac {{\left (75 \, {\left (2 \, b g^{3} p - 7 \, a g^{3}\right )} x^{6} - 5632 \, b f^{3} p - 18 \, {\left (24 \, b f g^{2} p - 35 \, a f g^{2}\right )} x^{4} + 1680 \, a f^{3} + 8 \, {\left (142 \, b f^{2} g p - 105 \, a f^{2} g\right )} x^{2} - 105 \, {\left (5 \, b g^{3} p x^{6} - 6 \, b f g^{2} p x^{4} + 8 \, b f^{2} g p x^{2} - 16 \, b f^{3} p\right )} \log \left (d g x^{2} + d f\right ) - 105 \, {\left (5 \, b g^{3} x^{6} - 6 \, b f g^{2} x^{4} + 8 \, b f^{2} g x^{2} - 16 \, b f^{3}\right )} \log \left (c\right )\right )} \sqrt {g x^{2} + f}}{3675 \, g^{4}} \] Input:

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

Output:

-1/3675*(75*(2*b*g^3*p - 7*a*g^3)*x^6 - 5632*b*f^3*p - 18*(24*b*f*g^2*p - 
35*a*f*g^2)*x^4 + 1680*a*f^3 + 8*(142*b*f^2*g*p - 105*a*f^2*g)*x^2 - 105*( 
5*b*g^3*p*x^6 - 6*b*f*g^2*p*x^4 + 8*b*f^2*g*p*x^2 - 16*b*f^3*p)*log(d*g*x^ 
2 + d*f) - 105*(5*b*g^3*x^6 - 6*b*f*g^2*x^4 + 8*b*f^2*g*x^2 - 16*b*f^3)*lo 
g(c))*sqrt(g*x^2 + f)/g^4
 

Sympy [A] (verification not implemented)

Time = 16.29 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.53 \[ \int \frac {x^7 \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{\sqrt {f+g x^2}} \, dx=\begin {cases} - \frac {16 a f^{3} \sqrt {f + g x^{2}}}{35 g^{4}} + \frac {8 a f^{2} x^{2} \sqrt {f + g x^{2}}}{35 g^{3}} - \frac {6 a f x^{4} \sqrt {f + g x^{2}}}{35 g^{2}} + \frac {a x^{6} \sqrt {f + g x^{2}}}{7 g} + \frac {5632 b f^{3} p \sqrt {f + g x^{2}}}{3675 g^{4}} - \frac {16 b f^{3} \sqrt {f + g x^{2}} \log {\left (c \left (d f + d g x^{2}\right )^{p} \right )}}{35 g^{4}} - \frac {1136 b f^{2} p x^{2} \sqrt {f + g x^{2}}}{3675 g^{3}} + \frac {8 b f^{2} x^{2} \sqrt {f + g x^{2}} \log {\left (c \left (d f + d g x^{2}\right )^{p} \right )}}{35 g^{3}} + \frac {144 b f p x^{4} \sqrt {f + g x^{2}}}{1225 g^{2}} - \frac {6 b f x^{4} \sqrt {f + g x^{2}} \log {\left (c \left (d f + d g x^{2}\right )^{p} \right )}}{35 g^{2}} - \frac {2 b p x^{6} \sqrt {f + g x^{2}}}{49 g} + \frac {b x^{6} \sqrt {f + g x^{2}} \log {\left (c \left (d f + d g x^{2}\right )^{p} \right )}}{7 g} & \text {for}\: g \neq 0 \\\frac {x^{8} \left (a + b \log {\left (c \left (d f\right )^{p} \right )}\right )}{8 \sqrt {f}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-16*a*f**3*sqrt(f + g*x**2)/(35*g**4) + 8*a*f**2*x**2*sqrt(f + 
g*x**2)/(35*g**3) - 6*a*f*x**4*sqrt(f + g*x**2)/(35*g**2) + a*x**6*sqrt(f 
+ g*x**2)/(7*g) + 5632*b*f**3*p*sqrt(f + g*x**2)/(3675*g**4) - 16*b*f**3*s 
qrt(f + g*x**2)*log(c*(d*f + d*g*x**2)**p)/(35*g**4) - 1136*b*f**2*p*x**2* 
sqrt(f + g*x**2)/(3675*g**3) + 8*b*f**2*x**2*sqrt(f + g*x**2)*log(c*(d*f + 
 d*g*x**2)**p)/(35*g**3) + 144*b*f*p*x**4*sqrt(f + g*x**2)/(1225*g**2) - 6 
*b*f*x**4*sqrt(f + g*x**2)*log(c*(d*f + d*g*x**2)**p)/(35*g**2) - 2*b*p*x* 
*6*sqrt(f + g*x**2)/(49*g) + b*x**6*sqrt(f + g*x**2)*log(c*(d*f + d*g*x**2 
)**p)/(7*g), Ne(g, 0)), (x**8*(a + b*log(c*(d*f)**p))/(8*sqrt(f)), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.96 \[ \int \frac {x^7 \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{\sqrt {f+g x^2}} \, dx=\frac {1}{35} \, {\left (\frac {5 \, \sqrt {g x^{2} + f} x^{6}}{g} - \frac {6 \, \sqrt {g x^{2} + f} f x^{4}}{g^{2}} + \frac {8 \, \sqrt {g x^{2} + f} f^{2} x^{2}}{g^{3}} - \frac {16 \, \sqrt {g x^{2} + f} f^{3}}{g^{4}}\right )} b \log \left ({\left (d g x^{2} + d f\right )}^{p} c\right ) + \frac {1}{35} \, {\left (\frac {5 \, \sqrt {g x^{2} + f} x^{6}}{g} - \frac {6 \, \sqrt {g x^{2} + f} f x^{4}}{g^{2}} + \frac {8 \, \sqrt {g x^{2} + f} f^{2} x^{2}}{g^{3}} - \frac {16 \, \sqrt {g x^{2} + f} f^{3}}{g^{4}}\right )} a - \frac {2 \, {\left (75 \, {\left (g x^{2} + f\right )}^{\frac {7}{2}} - 441 \, {\left (g x^{2} + f\right )}^{\frac {5}{2}} f + 1225 \, {\left (g x^{2} + f\right )}^{\frac {3}{2}} f^{2} - 3675 \, \sqrt {g x^{2} + f} f^{3}\right )} b p}{3675 \, g^{4}} \] Input:

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

Output:

1/35*(5*sqrt(g*x^2 + f)*x^6/g - 6*sqrt(g*x^2 + f)*f*x^4/g^2 + 8*sqrt(g*x^2 
 + f)*f^2*x^2/g^3 - 16*sqrt(g*x^2 + f)*f^3/g^4)*b*log((d*g*x^2 + d*f)^p*c) 
 + 1/35*(5*sqrt(g*x^2 + f)*x^6/g - 6*sqrt(g*x^2 + f)*f*x^4/g^2 + 8*sqrt(g* 
x^2 + f)*f^2*x^2/g^3 - 16*sqrt(g*x^2 + f)*f^3/g^4)*a - 2/3675*(75*(g*x^2 + 
 f)^(7/2) - 441*(g*x^2 + f)^(5/2)*f + 1225*(g*x^2 + f)^(3/2)*f^2 - 3675*sq 
rt(g*x^2 + f)*f^3)*b*p/g^4
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.07 \[ \int \frac {x^7 \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{\sqrt {f+g x^2}} \, dx=\frac {b p {\left (\frac {105 \, {\left (5 \, {\left (g x^{2} + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x^{2} + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x^{2} + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x^{2} + f} f^{3}\right )} \log \left (d g x^{2} + d f\right )}{g^{3}} - \frac {2 \, {\left (75 \, {\left (g x^{2} + f\right )}^{\frac {7}{2}} - 441 \, {\left (g x^{2} + f\right )}^{\frac {5}{2}} f + 1225 \, {\left (g x^{2} + f\right )}^{\frac {3}{2}} f^{2} - 3675 \, \sqrt {g x^{2} + f} f^{3}\right )}}{g^{3}}\right )} + \frac {105 \, {\left (5 \, {\left (g x^{2} + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x^{2} + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x^{2} + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x^{2} + f} f^{3}\right )} b \log \left (c\right )}{g^{3}} + \frac {105 \, {\left (5 \, {\left (g x^{2} + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x^{2} + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x^{2} + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x^{2} + f} f^{3}\right )} a}{g^{3}}}{3675 \, g} \] Input:

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

Output:

1/3675*(b*p*(105*(5*(g*x^2 + f)^(7/2) - 21*(g*x^2 + f)^(5/2)*f + 35*(g*x^2 
 + f)^(3/2)*f^2 - 35*sqrt(g*x^2 + f)*f^3)*log(d*g*x^2 + d*f)/g^3 - 2*(75*( 
g*x^2 + f)^(7/2) - 441*(g*x^2 + f)^(5/2)*f + 1225*(g*x^2 + f)^(3/2)*f^2 - 
3675*sqrt(g*x^2 + f)*f^3)/g^3) + 105*(5*(g*x^2 + f)^(7/2) - 21*(g*x^2 + f) 
^(5/2)*f + 35*(g*x^2 + f)^(3/2)*f^2 - 35*sqrt(g*x^2 + f)*f^3)*b*log(c)/g^3 
 + 105*(5*(g*x^2 + f)^(7/2) - 21*(g*x^2 + f)^(5/2)*f + 35*(g*x^2 + f)^(3/2 
)*f^2 - 35*sqrt(g*x^2 + f)*f^3)*a/g^3)/g
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.47 \[ \int \frac {x^7 \left (a+b \log \left (c \left (d f+d g x^2\right )^p\right )\right )}{\sqrt {f+g x^2}} \, dx=\frac {\sqrt {g \,x^{2}+f}\, \left (-1680 \,\mathrm {log}\left (\frac {f^{p} d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +f +g \,x^{2}\right )^{2 p} c}{\left (\sqrt {f}\, \sqrt {g \,x^{2}+f}+\sqrt {g}\, \sqrt {f}\, x \right )^{2 p}}\right ) b \,f^{3}+840 \,\mathrm {log}\left (\frac {f^{p} d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +f +g \,x^{2}\right )^{2 p} c}{\left (\sqrt {f}\, \sqrt {g \,x^{2}+f}+\sqrt {g}\, \sqrt {f}\, x \right )^{2 p}}\right ) b \,f^{2} g \,x^{2}-630 \,\mathrm {log}\left (\frac {f^{p} d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +f +g \,x^{2}\right )^{2 p} c}{\left (\sqrt {f}\, \sqrt {g \,x^{2}+f}+\sqrt {g}\, \sqrt {f}\, x \right )^{2 p}}\right ) b f \,g^{2} x^{4}+525 \,\mathrm {log}\left (\frac {f^{p} d^{p} \left (\sqrt {g}\, \sqrt {g \,x^{2}+f}\, x +f +g \,x^{2}\right )^{2 p} c}{\left (\sqrt {f}\, \sqrt {g \,x^{2}+f}+\sqrt {g}\, \sqrt {f}\, x \right )^{2 p}}\right ) b \,g^{3} x^{6}-1680 a \,f^{3}+840 a \,f^{2} g \,x^{2}-630 a f \,g^{2} x^{4}+525 a \,g^{3} x^{6}+5632 b \,f^{3} p -1136 b \,f^{2} g p \,x^{2}+432 b f \,g^{2} p \,x^{4}-150 b \,g^{3} p \,x^{6}\right )}{3675 g^{4}} \] Input:

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

Output:

(sqrt(f + g*x**2)*( - 1680*log((f**p*d**p*(sqrt(g)*sqrt(f + g*x**2)*x + f 
+ g*x**2)**(2*p)*c)/(sqrt(f)*sqrt(f + g*x**2) + sqrt(g)*sqrt(f)*x)**(2*p)) 
*b*f**3 + 840*log((f**p*d**p*(sqrt(g)*sqrt(f + g*x**2)*x + f + g*x**2)**(2 
*p)*c)/(sqrt(f)*sqrt(f + g*x**2) + sqrt(g)*sqrt(f)*x)**(2*p))*b*f**2*g*x** 
2 - 630*log((f**p*d**p*(sqrt(g)*sqrt(f + g*x**2)*x + f + g*x**2)**(2*p)*c) 
/(sqrt(f)*sqrt(f + g*x**2) + sqrt(g)*sqrt(f)*x)**(2*p))*b*f*g**2*x**4 + 52 
5*log((f**p*d**p*(sqrt(g)*sqrt(f + g*x**2)*x + f + g*x**2)**(2*p)*c)/(sqrt 
(f)*sqrt(f + g*x**2) + sqrt(g)*sqrt(f)*x)**(2*p))*b*g**3*x**6 - 1680*a*f** 
3 + 840*a*f**2*g*x**2 - 630*a*f*g**2*x**4 + 525*a*g**3*x**6 + 5632*b*f**3* 
p - 1136*b*f**2*g*p*x**2 + 432*b*f*g**2*p*x**4 - 150*b*g**3*p*x**6))/(3675 
*g**4)