Integrand size = 19, antiderivative size = 82 \[ \int \frac {x^5}{a-b x^2+c x^4} \, dx=\frac {x^2}{2 c}+\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b-2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}+\frac {b \log \left (a-b x^2+c x^4\right )}{4 c^2} \] Output:
1/2*x^2/c+1/2*(-2*a*c+b^2)*arctanh((-2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/c^2/(- 4*a*c+b^2)^(1/2)+1/4*b*ln(c*x^4-b*x^2+a)/c^2
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int \frac {x^5}{a-b x^2+c x^4} \, dx=\frac {2 c x^2+\frac {2 \left (b^2-2 a c\right ) \arctan \left (\frac {-b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+b \log \left (a-b x^2+c x^4\right )}{4 c^2} \] Input:
Integrate[x^5/(a - b*x^2 + c*x^4),x]
Output:
(2*c*x^2 + (2*(b^2 - 2*a*c)*ArcTan[(-b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqr t[-b^2 + 4*a*c] + b*Log[a - b*x^2 + c*x^4])/(4*c^2)
Time = 0.43 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1434, 1143, 2009}
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.
| \(\displaystyle \int \frac {x^5}{a-b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{c x^4-b x^2+a}dx^2\) |
| \(\Big \downarrow \) 1143 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {1}{c}-\frac {a-b x^2}{c \left (c x^4-b x^2+a\right )}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b-2 c x^2}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {b \log \left (a-b x^2+c x^4\right )}{2 c^2}+\frac {x^2}{c}\right )\) |
Input:
Int[x^5/(a - b*x^2 + c*x^4),x]
Output:
(x^2/c + ((b^2 - 2*a*c)*ArcTanh[(b - 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqr t[b^2 - 4*a*c]) + (b*Log[a - b*x^2 + c*x^4])/(2*c^2))/2
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {x^{2}}{2 c}+\frac {\frac {b \ln \left (c \,x^{4}-b \,x^{2}+a \right )}{2 c}+\frac {2 \left (-a +\frac {b^{2}}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}-b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 c}\) | \(86\) |
| risch | \(\frac {x^{2}}{2 c}+\frac {\ln \left (\left (8 a^{2} c^{2}-6 a \,b^{2} c +b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{c \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (8 a^{2} c^{2}-6 a \,b^{2} c +b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{4 c^{2} \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (8 a^{2} c^{2}-6 a \,b^{2} c +b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{4 c^{2} \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (8 a^{2} c^{2}-6 a \,b^{2} c +b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{c \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (8 a^{2} c^{2}-6 a \,b^{2} c +b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{4 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (8 a^{2} c^{2}-6 a \,b^{2} c +b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{2}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{4 c^{2} \left (4 a c -b^{2}\right )}\) | \(667\) |
Input:
int(x^5/(c*x^4-b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/2*x^2/c+1/2/c*(1/2*b/c*ln(c*x^4-b*x^2+a)+2*(-a+1/2*b^2/c)/(4*a*c-b^2)^(1 /2)*arctan((2*c*x^2-b)/(4*a*c-b^2)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.16 \[ \int \frac {x^5}{a-b x^2+c x^4} \, dx=\left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} - {\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} - 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} - b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} - b x^{2} + a}\right ) + {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{4} - b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} - 2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} - b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{4} - b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \] Input:
integrate(x^5/(c*x^4-b*x^2+a),x, algorithm="fricas")
Output:
[1/4*(2*(b^2*c - 4*a*c^2)*x^2 - (b^2 - 2*a*c)*sqrt(b^2 - 4*a*c)*log((2*c^2 *x^4 - 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 - b)*sqrt(b^2 - 4*a*c))/(c*x^4 - b*x^2 + a)) + (b^3 - 4*a*b*c)*log(c*x^4 - b*x^2 + a))/(b^2*c^2 - 4*a*c^3) , 1/4*(2*(b^2*c - 4*a*c^2)*x^2 - 2*(b^2 - 2*a*c)*sqrt(-b^2 + 4*a*c)*arctan (-(2*c*x^2 - b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (b^3 - 4*a*b*c)*log(c* x^4 - b*x^2 + a))/(b^2*c^2 - 4*a*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (71) = 142\).
Time = 1.04 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.79 \[ \int \frac {x^5}{a-b x^2+c x^4} \, dx=\left (\frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {a b - 8 a c^{2} \left (\frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (\frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (\frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {a b - 8 a c^{2} \left (\frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (\frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{4 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac {x^{2}}{2 c} \] Input:
integrate(x**5/(c*x**4-b*x**2+a),x)
Output:
(b/(4*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2*(4*a*c - b**2)))* log(x**2 + (a*b - 8*a*c**2*(b/(4*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2 )/(4*c**2*(4*a*c - b**2))) + 2*b**2*c*(b/(4*c**2) - sqrt(-4*a*c + b**2)*(2 *a*c - b**2)/(4*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + (b/(4*c**2) + sqr t(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2*(4*a*c - b**2)))*log(x**2 + (a*b - 8*a*c**2*(b/(4*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c**2*(4*a*c - b**2))) + 2*b**2*c*(b/(4*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(4*c **2*(4*a*c - b**2))))/(2*a*c - b**2)) + x**2/(2*c)
Exception generated. \[ \int \frac {x^5}{a-b x^2+c x^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^5/(c*x^4-b*x^2+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95 \[ \int \frac {x^5}{a-b x^2+c x^4} \, dx=\frac {x^{2}}{2 \, c} + \frac {b \log \left (c x^{4} - b x^{2} + a\right )}{4 \, c^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{2} - b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \] Input:
integrate(x^5/(c*x^4-b*x^2+a),x, algorithm="giac")
Output:
1/2*x^2/c + 1/4*b*log(c*x^4 - b*x^2 + a)/c^2 + 1/2*(b^2 - 2*a*c)*arctan((2 *c*x^2 - b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^2)
Time = 0.52 (sec) , antiderivative size = 656, normalized size of antiderivative = 8.00 \[ \int \frac {x^5}{a-b x^2+c x^4} \, dx=\frac {x^2}{2\,c}-\frac {\ln \left (c\,x^4-b\,x^2+a\right )\,\left (2\,b^3-8\,a\,b\,c\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {\mathrm {atan}\left (\frac {2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (\frac {\frac {\left (8\,a\,b+\frac {8\,a\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )\,\left (2\,a\,c-b^2\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}+\frac {a\,\left (2\,b^3-8\,a\,b\,c\right )\,\left (2\,a\,c-b^2\right )}{\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}+x^2\,\left (\frac {\frac {\left (2\,a\,c-b^2\right )\,\left (\frac {4\,a\,c^3-6\,b^2\,c^2}{c^2}-\frac {4\,b\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}-\frac {b\,\left (2\,b^3-8\,a\,b\,c\right )\,\left (2\,a\,c-b^2\right )}{2\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}+\frac {b\,\left (\frac {\left (2\,b^3-8\,a\,b\,c\right )\,\left (\frac {4\,a\,c^3-6\,b^2\,c^2}{c^2}-\frac {4\,b\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {b^3-a\,b\,c}{c^2}+\frac {b\,{\left (2\,a\,c-b^2\right )}^2}{2\,c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )+\frac {b\,\left (\frac {a\,b^2}{c^2}+\frac {\left (2\,b^3-8\,a\,b\,c\right )\,\left (8\,a\,b+\frac {8\,a\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {a\,{\left (2\,a\,c-b^2\right )}^2}{c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )}{4\,a^2\,c^2-4\,a\,b^2\,c+b^4}\right )\,\left (2\,a\,c-b^2\right )}{2\,c^2\,\sqrt {4\,a\,c-b^2}} \] Input:
int(x^5/(a - b*x^2 + c*x^4),x)
Output:
x^2/(2*c) - (log(a - b*x^2 + c*x^4)*(2*b^3 - 8*a*b*c))/(2*(16*a*c^3 - 4*b^ 2*c^2)) - (atan((2*c^2*(4*a*c - b^2)*((((8*a*b + (8*a*c^2*(2*b^3 - 8*a*b*c ))/(16*a*c^3 - 4*b^2*c^2))*(2*a*c - b^2))/(8*c^2*(4*a*c - b^2)^(1/2)) + (a *(2*b^3 - 8*a*b*c)*(2*a*c - b^2))/((4*a*c - b^2)^(1/2)*(16*a*c^3 - 4*b^2*c ^2)))/a + x^2*((((2*a*c - b^2)*((4*a*c^3 - 6*b^2*c^2)/c^2 - (4*b*c^2*(2*b^ 3 - 8*a*b*c))/(16*a*c^3 - 4*b^2*c^2)))/(8*c^2*(4*a*c - b^2)^(1/2)) - (b*(2 *b^3 - 8*a*b*c)*(2*a*c - b^2))/(2*(4*a*c - b^2)^(1/2)*(16*a*c^3 - 4*b^2*c^ 2)))/a + (b*(((2*b^3 - 8*a*b*c)*((4*a*c^3 - 6*b^2*c^2)/c^2 - (4*b*c^2*(2*b ^3 - 8*a*b*c))/(16*a*c^3 - 4*b^2*c^2)))/(2*(16*a*c^3 - 4*b^2*c^2)) - (b^3 - a*b*c)/c^2 + (b*(2*a*c - b^2)^2)/(2*c^2*(4*a*c - b^2))))/(2*a*(4*a*c - b ^2)^(1/2))) + (b*((a*b^2)/c^2 + ((2*b^3 - 8*a*b*c)*(8*a*b + (8*a*c^2*(2*b^ 3 - 8*a*b*c))/(16*a*c^3 - 4*b^2*c^2)))/(2*(16*a*c^3 - 4*b^2*c^2)) - (a*(2* a*c - b^2)^2)/(c^2*(4*a*c - b^2))))/(2*a*(4*a*c - b^2)^(1/2))))/(b^4 + 4*a ^2*c^2 - 4*a*b^2*c))*(2*a*c - b^2))/(2*c^2*(4*a*c - b^2)^(1/2))
Time = 0.23 (sec) , antiderivative size = 360, normalized size of antiderivative = 4.39 \[ \int \frac {x^5}{a-b x^2+c x^4} \, dx=\frac {4 \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a c -2 \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b^{2}+4 \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) a c -2 \sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}+b}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}-b}}\right ) b^{2}+4 \,\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a b c -\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b^{3}+4 \,\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a b c -\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}+b}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b^{3}+8 a \,c^{2} x^{2}-2 b^{2} c \,x^{2}}{4 c^{2} \left (4 a c -b^{2}\right )} \] Input:
int(x^5/(c*x^4-b*x^2+a),x)
Output:
(4*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sq rt(c)*sqrt(a) + b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) - b))*a*c - 2*sqr t(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)* sqrt(a) + b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) - b))*b**2 + 4*sqrt(2*s qrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt( a) + b) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) - b))*a*c - 2*sqrt(2*sqrt(c) *sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) + b ) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) - b))*b**2 + 4*log( - sqrt(2*sqrt( c)*sqrt(a) + b)*x + sqrt(a) + sqrt(c)*x**2)*a*b*c - log( - sqrt(2*sqrt(c)* sqrt(a) + b)*x + sqrt(a) + sqrt(c)*x**2)*b**3 + 4*log(sqrt(2*sqrt(c)*sqrt( a) + b)*x + sqrt(a) + sqrt(c)*x**2)*a*b*c - log(sqrt(2*sqrt(c)*sqrt(a) + b )*x + sqrt(a) + sqrt(c)*x**2)*b**3 + 8*a*c**2*x**2 - 2*b**2*c*x**2)/(4*c** 2*(4*a*c - b**2))