Integrand size = 50, antiderivative size = 87 \[ \int \frac {a \left (-3 c+2 \sqrt {2} c+\sqrt {6} a x\right )}{c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2} \, dx=-\sqrt {-3+4 \sqrt {2}} \arctan \left (\frac {\sqrt {3} c-2 \sqrt {2} a x}{\sqrt {-3+4 \sqrt {2}} c}\right )+\frac {1}{2} \sqrt {3} \log \left (c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2\right ) \] Output:
-(-3+4*2^(1/2))^(1/2)*arctan((c*3^(1/2)-2*2^(1/2)*a*x)/(-3+4*2^(1/2))^(1/2 )/c)+1/2*3^(1/2)*ln(c^2-3^(1/2)*a*c*x+2^(1/2)*a^2*x^2)
Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09 \[ \int \frac {a \left (-3 c+2 \sqrt {2} c+\sqrt {6} a x\right )}{c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2} \, dx=a \left (\frac {\sqrt {-3+4 \sqrt {2}} \arctan \left (\frac {-\sqrt {3} c+2 \sqrt {2} a x}{\sqrt {-3+4 \sqrt {2}} c}\right )}{a}+\frac {\sqrt {3} \log \left (c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2\right )}{2 a}\right ) \] Input:
Integrate[(a*(-3*c + 2*Sqrt[2]*c + Sqrt[6]*a*x))/(c^2 - Sqrt[3]*a*c*x + Sq rt[2]*a^2*x^2),x]
Output:
a*((Sqrt[-3 + 4*Sqrt[2]]*ArcTan[(-(Sqrt[3]*c) + 2*Sqrt[2]*a*x)/(Sqrt[-3 + 4*Sqrt[2]]*c)])/a + (Sqrt[3]*Log[c^2 - Sqrt[3]*a*c*x + Sqrt[2]*a^2*x^2])/( 2*a))
Time = 0.34 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.28, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {27, 25, 1142, 25, 27, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {a \left (\sqrt {6} a x+2 \sqrt {2} c-3 c\right )}{\sqrt {2} a^2 x^2-\sqrt {3} a c x+c^2} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a \int -\frac {\left (3-2 \sqrt {2}\right ) c-\sqrt {6} a x}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a \int \frac {\left (3-2 \sqrt {2}\right ) c-\sqrt {6} a x}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -a \left (\frac {1}{2} \left (3-4 \sqrt {2}\right ) c \int \frac {1}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx-\frac {\sqrt {3} \int -\frac {a \left (\sqrt {3} c-2 \sqrt {2} a x\right )}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx}{2 a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a \left (\frac {1}{2} \left (3-4 \sqrt {2}\right ) c \int \frac {1}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx+\frac {\sqrt {3} \int \frac {a \left (\sqrt {3} c-2 \sqrt {2} a x\right )}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx}{2 a}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -a \left (\frac {1}{2} \left (3-4 \sqrt {2}\right ) c \int \frac {1}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} c-2 \sqrt {2} a x}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -a \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} c-2 \sqrt {2} a x}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx-\left (3-4 \sqrt {2}\right ) c \int \frac {1}{\left (3-4 \sqrt {2}\right ) a^2 c^2-\left (2 \sqrt {2} a^2 x-\sqrt {3} a c\right )^2}d\left (2 \sqrt {2} a^2 x-\sqrt {3} a c\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -a \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} c-2 \sqrt {2} a x}{c^2-\sqrt {3} a x c+\sqrt {2} a^2 x^2}dx+\frac {\left (3-4 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {2} a^2 x-\sqrt {3} a c}{\sqrt {4 \sqrt {2}-3} a c}\right )}{\sqrt {4 \sqrt {2}-3} a}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -a \left (\frac {\left (3-4 \sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {2} a^2 x-\sqrt {3} a c}{\sqrt {4 \sqrt {2}-3} a c}\right )}{\sqrt {4 \sqrt {2}-3} a}-\frac {\sqrt {3} \log \left (\sqrt {2} a^2 x^2-\sqrt {3} a c x+c^2\right )}{2 a}\right )\) |
Input:
Int[(a*(-3*c + 2*Sqrt[2]*c + Sqrt[6]*a*x))/(c^2 - Sqrt[3]*a*c*x + Sqrt[2]* a^2*x^2),x]
Output:
-(a*(((3 - 4*Sqrt[2])*ArcTan[(-(Sqrt[3]*a*c) + 2*Sqrt[2]*a^2*x)/(Sqrt[-3 + 4*Sqrt[2]]*a*c)])/(Sqrt[-3 + 4*Sqrt[2]]*a) - (Sqrt[3]*Log[c^2 - Sqrt[3]*a *c*x + Sqrt[2]*a^2*x^2])/(2*a)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 1.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.48
| method | result | size |
| default | \(a \sqrt {2}\, \left (\frac {\sqrt {6}\, \ln \left (\sqrt {3}\, \sqrt {2}\, a c x -2 x^{2} a^{2}-\sqrt {2}\, c^{2}\right )}{4 a}+\frac {2 \left (-\frac {\sqrt {6}\, \sqrt {2}\, \sqrt {3}\, c}{4}-2 \sqrt {2}\, c +3 c \right ) \arctan \left (\frac {\sqrt {3}\, \sqrt {2}\, a c -4 x \,a^{2}}{\sqrt {8 \sqrt {2}\, c^{2} a^{2}-6 a^{2} c^{2}}}\right )}{\sqrt {8 \sqrt {2}\, c^{2} a^{2}-6 a^{2} c^{2}}}\right )\) | \(129\) |
Input:
int(a*(-3*c+2*2^(1/2)*c+6^(1/2)*a*x)/(c^2-3^(1/2)*a*c*x+2^(1/2)*a^2*x^2),x ,method=_RETURNVERBOSE)
Output:
a*2^(1/2)*(1/4*6^(1/2)/a*ln(3^(1/2)*2^(1/2)*a*c*x-2*x^2*a^2-2^(1/2)*c^2)+2 *(-1/4*6^(1/2)*2^(1/2)*3^(1/2)*c-2*2^(1/2)*c+3*c)/(8*2^(1/2)*c^2*a^2-6*a^2 *c^2)^(1/2)*arctan((3^(1/2)*2^(1/2)*a*c-4*x*a^2)/(8*2^(1/2)*c^2*a^2-6*a^2* c^2)^(1/2)))
\[ \int \frac {a \left (-3 c+2 \sqrt {2} c+\sqrt {6} a x\right )}{c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2} \, dx=\int { \frac {{\left (\sqrt {6} a x + 2 \, \sqrt {2} c - 3 \, c\right )} a}{\sqrt {2} a^{2} x^{2} - \sqrt {3} a c x + c^{2}} \,d x } \] Input:
integrate(a*(-3*c+2*2^(1/2)*c+6^(1/2)*a*x)/(c^2-3^(1/2)*a*c*x+2^(1/2)*a^2* x^2),x, algorithm="fricas")
Output:
0
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (76) = 152\).
Time = 0.52 (sec) , antiderivative size = 287, normalized size of antiderivative = 3.30 \[ \int \frac {a \left (-3 c+2 \sqrt {2} c+\sqrt {6} a x\right )}{c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2} \, dx=\left (\frac {\sqrt {3}}{2} - \frac {\sqrt {2} \sqrt {315 - 236 \sqrt {2}}}{2 \left (-8 + 3 \sqrt {2}\right )}\right ) \log {\left (- \frac {24243 \sqrt {3} c}{- 34472 a + 24243 \sqrt {2} a} + \frac {17236 \sqrt {6} c}{- 34472 a + 24243 \sqrt {2} a} + x + \left (\frac {\sqrt {3}}{2} - \frac {\sqrt {2} \sqrt {315 - 236 \sqrt {2}}}{2 \left (-8 + 3 \sqrt {2}\right )}\right ) \left (- \frac {24 \sqrt {2} c}{- 48 a + 41 \sqrt {2} a} + \frac {41 c}{- 48 a + 41 \sqrt {2} a}\right ) \right )} + \left (\frac {\sqrt {3}}{2} + \frac {\sqrt {2} \sqrt {315 - 236 \sqrt {2}}}{2 \left (-8 + 3 \sqrt {2}\right )}\right ) \log {\left (- \frac {24243 \sqrt {3} c}{- 34472 a + 24243 \sqrt {2} a} + \frac {17236 \sqrt {6} c}{- 34472 a + 24243 \sqrt {2} a} + x + \left (\frac {\sqrt {3}}{2} + \frac {\sqrt {2} \sqrt {315 - 236 \sqrt {2}}}{2 \left (-8 + 3 \sqrt {2}\right )}\right ) \left (- \frac {24 \sqrt {2} c}{- 48 a + 41 \sqrt {2} a} + \frac {41 c}{- 48 a + 41 \sqrt {2} a}\right ) \right )} \] Input:
integrate(a*(-3*c+2*2**(1/2)*c+6**(1/2)*a*x)/(c**2-3**(1/2)*a*c*x+2**(1/2) *a**2*x**2),x)
Output:
(sqrt(3)/2 - sqrt(2)*sqrt(315 - 236*sqrt(2))/(2*(-8 + 3*sqrt(2))))*log(-24 243*sqrt(3)*c/(-34472*a + 24243*sqrt(2)*a) + 17236*sqrt(6)*c/(-34472*a + 2 4243*sqrt(2)*a) + x + (sqrt(3)/2 - sqrt(2)*sqrt(315 - 236*sqrt(2))/(2*(-8 + 3*sqrt(2))))*(-24*sqrt(2)*c/(-48*a + 41*sqrt(2)*a) + 41*c/(-48*a + 41*sq rt(2)*a))) + (sqrt(3)/2 + sqrt(2)*sqrt(315 - 236*sqrt(2))/(2*(-8 + 3*sqrt( 2))))*log(-24243*sqrt(3)*c/(-34472*a + 24243*sqrt(2)*a) + 17236*sqrt(6)*c/ (-34472*a + 24243*sqrt(2)*a) + x + (sqrt(3)/2 + sqrt(2)*sqrt(315 - 236*sqr t(2))/(2*(-8 + 3*sqrt(2))))*(-24*sqrt(2)*c/(-48*a + 41*sqrt(2)*a) + 41*c/( -48*a + 41*sqrt(2)*a)))
Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.17 \[ \int \frac {a \left (-3 c+2 \sqrt {2} c+\sqrt {6} a x\right )}{c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2} \, dx=\frac {1}{4} \, a {\left (\frac {2 \, \sqrt {2} {\left (\sqrt {6} \sqrt {3} - 6 \, \sqrt {2} + 8\right )} \arctan \left (\frac {2 \, \sqrt {2} a^{2} x - \sqrt {3} a c}{a c \sqrt {4 \, \sqrt {2} - 3}}\right )}{a \sqrt {4 \, \sqrt {2} - 3}} + \frac {\sqrt {6} \sqrt {2} \log \left (\sqrt {2} a^{2} x^{2} - \sqrt {3} a c x + c^{2}\right )}{a}\right )} \] Input:
integrate(a*(-3*c+2*2^(1/2)*c+6^(1/2)*a*x)/(c^2-3^(1/2)*a*c*x+2^(1/2)*a^2* x^2),x, algorithm="maxima")
Output:
1/4*a*(2*sqrt(2)*(sqrt(6)*sqrt(3) - 6*sqrt(2) + 8)*arctan((2*sqrt(2)*a^2*x - sqrt(3)*a*c)/(a*c*sqrt(4*sqrt(2) - 3)))/(a*sqrt(4*sqrt(2) - 3)) + sqrt( 6)*sqrt(2)*log(sqrt(2)*a^2*x^2 - sqrt(3)*a*c*x + c^2)/a)
Time = 0.14 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97 \[ \int \frac {a \left (-3 c+2 \sqrt {2} c+\sqrt {6} a x\right )}{c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2} \, dx=-\frac {1}{2} \, a {\left (\frac {2 \, {\left (3 \, \sqrt {2} - 8\right )} \arctan \left (\frac {4 \, a x - \sqrt {6} c}{c \sqrt {8 \, \sqrt {2} - 6}}\right )}{a \sqrt {8 \, \sqrt {2} - 6}} - \frac {\sqrt {3} \log \left (4 \, a^{2} x^{2} - 2 \, \sqrt {6} a c x + 2 \, \sqrt {2} c^{2}\right )}{a}\right )} \] Input:
integrate(a*(-3*c+2*2^(1/2)*c+6^(1/2)*a*x)/(c^2-3^(1/2)*a*c*x+2^(1/2)*a^2* x^2),x, algorithm="giac")
Output:
-1/2*a*(2*(3*sqrt(2) - 8)*arctan((4*a*x - sqrt(6)*c)/(c*sqrt(8*sqrt(2) - 6 )))/(a*sqrt(8*sqrt(2) - 6)) - sqrt(3)*log(4*a^2*x^2 - 2*sqrt(6)*a*c*x + 2* sqrt(2)*c^2)/a)
Time = 10.67 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.33 \[ \int \frac {a \left (-3 c+2 \sqrt {2} c+\sqrt {6} a x\right )}{c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2} \, dx=\ln \left (\sqrt {6}\,c-4\,a\,x+\sqrt {2}\,c\,\sqrt {3-4\,\sqrt {2}}\right )\,\left (\frac {\sqrt {3}}{2}-\frac {3\,\sqrt {2}}{2\,\sqrt {6-8\,\sqrt {2}}}+\frac {4}{\sqrt {6-8\,\sqrt {2}}}\right )+\ln \left (4\,a\,x-\sqrt {6}\,c+\sqrt {2}\,c\,\sqrt {3-4\,\sqrt {2}}\right )\,\left (\frac {3\,\sqrt {2}}{2\,\sqrt {6-8\,\sqrt {2}}}+\frac {\sqrt {3}}{2}-\frac {4}{\sqrt {6-8\,\sqrt {2}}}\right ) \] Input:
int((a*(2*2^(1/2)*c - 3*c + 6^(1/2)*a*x))/(c^2 + 2^(1/2)*a^2*x^2 - 3^(1/2) *a*c*x),x)
Output:
log(6^(1/2)*c - 4*a*x + 2^(1/2)*c*(3 - 4*2^(1/2))^(1/2))*(3^(1/2)/2 - (3*2 ^(1/2))/(2*(6 - 8*2^(1/2))^(1/2)) + 4/(6 - 8*2^(1/2))^(1/2)) + log(4*a*x - 6^(1/2)*c + 2^(1/2)*c*(3 - 4*2^(1/2))^(1/2))*((3*2^(1/2))/(2*(6 - 8*2^(1/ 2))^(1/2)) + 3^(1/2)/2 - 4/(6 - 8*2^(1/2))^(1/2))
\[ \int \frac {a \left (-3 c+2 \sqrt {2} c+\sqrt {6} a x\right )}{c^2-\sqrt {3} a c x+\sqrt {2} a^2 x^2} \, dx=2 \sqrt {6}\, \left (\int \frac {x^{5}}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{6} c^{2}-3 \sqrt {6}\, \left (\int \frac {x^{3}}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{4} c^{4}+3 \sqrt {6}\, \left (\int \frac {x}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{2} c^{6}-3 \sqrt {3}\, \left (\int \frac {x^{5}}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{6} c^{2}-\frac {7 \sqrt {3}\, \left (\int \frac {x^{3}}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{4} c^{4}}{2}-\frac {3 \sqrt {3}\, \left (\int \frac {x}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{2} c^{6}}{2}+\frac {\sqrt {3}\, \mathrm {log}\left (4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}\right )}{8}-4 \sqrt {2}\, \left (\int \frac {x^{4}}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{5} c^{3}+2 \sqrt {2}\, \left (\int \frac {1}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a \,c^{7}+8 \left (\int \frac {x^{6}}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{7} c -18 \left (\int \frac {x^{4}}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{5} c^{3}+5 \left (\int \frac {x^{2}}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a^{3} c^{5}-3 \left (\int \frac {1}{4 a^{8} x^{8}-12 a^{6} c^{2} x^{6}+5 a^{4} c^{4} x^{4}-6 a^{2} c^{6} x^{2}+c^{8}}d x \right ) a \,c^{7} \] Input:
int(a*(-3*c+2*2^(1/2)*c+6^(1/2)*a*x)/(c^2-3^(1/2)*a*c*x+2^(1/2)*a^2*x^2),x )
Output:
(16*sqrt(6)*int(x**5/(4*a**8*x**8 - 12*a**6*c**2*x**6 + 5*a**4*c**4*x**4 - 6*a**2*c**6*x**2 + c**8),x)*a**6*c**2 - 24*sqrt(6)*int(x**3/(4*a**8*x**8 - 12*a**6*c**2*x**6 + 5*a**4*c**4*x**4 - 6*a**2*c**6*x**2 + c**8),x)*a**4* c**4 + 24*sqrt(6)*int(x/(4*a**8*x**8 - 12*a**6*c**2*x**6 + 5*a**4*c**4*x** 4 - 6*a**2*c**6*x**2 + c**8),x)*a**2*c**6 - 24*sqrt(3)*int(x**5/(4*a**8*x* *8 - 12*a**6*c**2*x**6 + 5*a**4*c**4*x**4 - 6*a**2*c**6*x**2 + c**8),x)*a* *6*c**2 - 28*sqrt(3)*int(x**3/(4*a**8*x**8 - 12*a**6*c**2*x**6 + 5*a**4*c* *4*x**4 - 6*a**2*c**6*x**2 + c**8),x)*a**4*c**4 - 12*sqrt(3)*int(x/(4*a**8 *x**8 - 12*a**6*c**2*x**6 + 5*a**4*c**4*x**4 - 6*a**2*c**6*x**2 + c**8),x) *a**2*c**6 + sqrt(3)*log(4*a**8*x**8 - 12*a**6*c**2*x**6 + 5*a**4*c**4*x** 4 - 6*a**2*c**6*x**2 + c**8) - 32*sqrt(2)*int(x**4/(4*a**8*x**8 - 12*a**6* c**2*x**6 + 5*a**4*c**4*x**4 - 6*a**2*c**6*x**2 + c**8),x)*a**5*c**3 + 16* sqrt(2)*int(1/(4*a**8*x**8 - 12*a**6*c**2*x**6 + 5*a**4*c**4*x**4 - 6*a**2 *c**6*x**2 + c**8),x)*a*c**7 + 64*int(x**6/(4*a**8*x**8 - 12*a**6*c**2*x** 6 + 5*a**4*c**4*x**4 - 6*a**2*c**6*x**2 + c**8),x)*a**7*c - 144*int(x**4/( 4*a**8*x**8 - 12*a**6*c**2*x**6 + 5*a**4*c**4*x**4 - 6*a**2*c**6*x**2 + c* *8),x)*a**5*c**3 + 40*int(x**2/(4*a**8*x**8 - 12*a**6*c**2*x**6 + 5*a**4*c **4*x**4 - 6*a**2*c**6*x**2 + c**8),x)*a**3*c**5 - 24*int(1/(4*a**8*x**8 - 12*a**6*c**2*x**6 + 5*a**4*c**4*x**4 - 6*a**2*c**6*x**2 + c**8),x)*a*c**7 )/8