Integrand size = 20, antiderivative size = 89 \[ \int \frac {1}{x^3 \left (a+b+2 a x^2+a x^4\right )} \, dx=-\frac {1}{2 (a+b) x^2}+\frac {\sqrt {a} (a-b) \arctan \left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {b} (a+b)^2}-\frac {2 a \log (x)}{(a+b)^2}+\frac {a \log \left (a+b+2 a x^2+a x^4\right )}{2 (a+b)^2} \] Output:
-1/2/(a+b)/x^2+1/2*a^(1/2)*(a-b)*arctan(a^(1/2)*(x^2+1)/b^(1/2))/b^(1/2)/( a+b)^2-2*a*ln(x)/(a+b)^2+1/2*a*ln(a*x^4+2*a*x^2+a+b)/(a+b)^2
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.83 \[ \int \frac {1}{x^3 \left (a+b+2 a x^2+a x^4\right )} \, dx=-\frac {1}{2 (a+b) x^2}-\frac {2 a \log (x)}{(a+b)^2}+\frac {\left (-i a^2+2 a^{3/2} \sqrt {b}+i a b\right ) \log \left (\sqrt {a}-i \sqrt {b}+\sqrt {a} x^2\right )}{4 \sqrt {a} \sqrt {b} (a+b)^2}+\frac {\left (i a^2+2 a^{3/2} \sqrt {b}-i a b\right ) \log \left (\sqrt {a}+i \sqrt {b}+\sqrt {a} x^2\right )}{4 \sqrt {a} \sqrt {b} (a+b)^2} \] Input:
Integrate[1/(x^3*(a + b + 2*a*x^2 + a*x^4)),x]
Output:
-1/2*1/((a + b)*x^2) - (2*a*Log[x])/(a + b)^2 + (((-I)*a^2 + 2*a^(3/2)*Sqr t[b] + I*a*b)*Log[Sqrt[a] - I*Sqrt[b] + Sqrt[a]*x^2])/(4*Sqrt[a]*Sqrt[b]*( a + b)^2) + ((I*a^2 + 2*a^(3/2)*Sqrt[b] - I*a*b)*Log[Sqrt[a] + I*Sqrt[b] + Sqrt[a]*x^2])/(4*Sqrt[a]*Sqrt[b]*(a + b)^2)
Time = 0.50 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1434, 1145, 25, 27, 1200, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a x^4+2 a x^2+a+b\right )} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (a x^4+2 a x^2+a+b\right )}dx^2\) |
| \(\Big \downarrow \) 1145 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {a \left (x^2+2\right )}{x^2 \left (a x^4+2 a x^2+a+b\right )}dx^2}{a+b}-\frac {1}{x^2 (a+b)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {a \left (x^2+2\right )}{x^2 \left (a x^4+2 a x^2+a+b\right )}dx^2}{a+b}-\frac {1}{x^2 (a+b)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a \int \frac {x^2+2}{x^2 \left (a x^4+2 a x^2+a+b\right )}dx^2}{a+b}-\frac {1}{x^2 (a+b)}\right )\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a \int \left (\frac {-2 a x^2-3 a+b}{(a+b) \left (a x^4+2 a x^2+a+b\right )}+\frac {2}{(a+b) x^2}\right )dx^2}{a+b}-\frac {1}{x^2 (a+b)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a \left (-\frac {(a-b) \arctan \left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} (a+b)}+\frac {2 \log \left (x^2\right )}{a+b}-\frac {\log \left (a x^4+2 a x^2+a+b\right )}{a+b}\right )}{a+b}-\frac {1}{x^2 (a+b)}\right )\) |
Input:
Int[1/(x^3*(a + b + 2*a*x^2 + a*x^4)),x]
Output:
(-(1/((a + b)*x^2)) - (a*(-(((a - b)*ArcTan[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/ (Sqrt[a]*Sqrt[b]*(a + b))) + (2*Log[x^2])/(a + b) - Log[a + b + 2*a*x^2 + a*x^4]/(a + b)))/(a + b))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp [1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[m, -1]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
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 = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(\frac {a \left (\ln \left (a \,x^{4}+2 a \,x^{2}+a +b \right )+\frac {\left (a -b \right ) \arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{\sqrt {a b}}\right )}{2 \left (a +b \right )^{2}}-\frac {1}{2 \left (a +b \right ) x^{2}}-\frac {2 a \ln \left (x \right )}{\left (a +b \right )^{2}}\) | \(75\) |
| risch | \(-\frac {1}{2 \left (a +b \right ) x^{2}}-\frac {2 a \ln \left (x \right )}{a^{2}+2 a b +b^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{2} b +2 b^{2} a +b^{3}\right ) \textit {\_Z}^{2}-4 a b \textit {\_Z} +a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{3}+3 a^{2} b +9 b^{2} a +5 b^{3}\right ) \textit {\_R}^{2}+\left (-8 a^{2}-8 a b \right ) \textit {\_R} +4 a \right ) x^{2}+\left (-a^{3}-3 a^{2} b -3 b^{2} a -b^{3}\right ) \textit {\_R}^{2}+\left (-7 a^{2}-6 a b +b^{2}\right ) \textit {\_R} +8 a \right )\right )}{4}\) | \(158\) |
Input:
int(1/x^3/(a*x^4+2*a*x^2+a+b),x,method=_RETURNVERBOSE)
Output:
1/2/(a+b)^2*a*(ln(a*x^4+2*a*x^2+a+b)+(a-b)/(a*b)^(1/2)*arctan(1/2*(2*a*x^2 +2*a)/(a*b)^(1/2)))-1/2/(a+b)/x^2-2*a*ln(x)/(a+b)^2
Time = 0.09 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.31 \[ \int \frac {1}{x^3 \left (a+b+2 a x^2+a x^4\right )} \, dx=\left [-\frac {{\left (a - b\right )} x^{2} \sqrt {-\frac {a}{b}} \log \left (\frac {a x^{4} + 2 \, a x^{2} - 2 \, {\left (b x^{2} + b\right )} \sqrt {-\frac {a}{b}} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right ) - 2 \, a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 8 \, a x^{2} \log \left (x\right ) + 2 \, a + 2 \, b}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} x^{2}}, \frac {{\left (a - b\right )} x^{2} \sqrt {\frac {a}{b}} \arctan \left ({\left (x^{2} + 1\right )} \sqrt {\frac {a}{b}}\right ) + a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) - 4 \, a x^{2} \log \left (x\right ) - a - b}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} x^{2}}\right ] \] Input:
integrate(1/x^3/(a*x^4+2*a*x^2+a+b),x, algorithm="fricas")
Output:
[-1/4*((a - b)*x^2*sqrt(-a/b)*log((a*x^4 + 2*a*x^2 - 2*(b*x^2 + b)*sqrt(-a /b) + a - b)/(a*x^4 + 2*a*x^2 + a + b)) - 2*a*x^2*log(a*x^4 + 2*a*x^2 + a + b) + 8*a*x^2*log(x) + 2*a + 2*b)/((a^2 + 2*a*b + b^2)*x^2), 1/2*((a - b) *x^2*sqrt(a/b)*arctan((x^2 + 1)*sqrt(a/b)) + a*x^2*log(a*x^4 + 2*a*x^2 + a + b) - 4*a*x^2*log(x) - a - b)/((a^2 + 2*a*b + b^2)*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (83) = 166\).
Time = 12.80 (sec) , antiderivative size = 386, normalized size of antiderivative = 4.34 \[ \int \frac {1}{x^3 \left (a+b+2 a x^2+a x^4\right )} \, dx=- \frac {2 a \log {\left (x \right )}}{\left (a + b\right )^{2}} + \left (\frac {a}{2 \left (a + b\right )^{2}} - \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) \log {\left (x^{2} + \frac {4 a^{2} b \left (\frac {a}{2 \left (a + b\right )^{2}} - \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac {a}{2 \left (a + b\right )^{2}} - \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) - 3 a b + 4 b^{3} \left (\frac {a}{2 \left (a + b\right )^{2}} - \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right )}{a^{2} - a b} \right )} + \left (\frac {a}{2 \left (a + b\right )^{2}} + \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) \log {\left (x^{2} + \frac {4 a^{2} b \left (\frac {a}{2 \left (a + b\right )^{2}} + \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac {a}{2 \left (a + b\right )^{2}} + \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right ) - 3 a b + 4 b^{3} \left (\frac {a}{2 \left (a + b\right )^{2}} + \frac {\sqrt {- a b} \left (a - b\right )}{4 b \left (a^{2} + 2 a b + b^{2}\right )}\right )}{a^{2} - a b} \right )} - \frac {1}{x^{2} \cdot \left (2 a + 2 b\right )} \] Input:
integrate(1/x**3/(a*x**4+2*a*x**2+a+b),x)
Output:
-2*a*log(x)/(a + b)**2 + (a/(2*(a + b)**2) - sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2)))*log(x**2 + (4*a**2*b*(a/(2*(a + b)**2) - sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2))) + a**2 + 8*a*b**2*(a/(2*(a + b)**2) - sq rt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2))) - 3*a*b + 4*b**3*(a/(2*(a + b)**2) - sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2))))/(a**2 - a*b)) + (a/(2*(a + b)**2) + sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2)))*log(x* *2 + (4*a**2*b*(a/(2*(a + b)**2) + sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2))) + a**2 + 8*a*b**2*(a/(2*(a + b)**2) + sqrt(-a*b)*(a - b)/(4*b*(a* *2 + 2*a*b + b**2))) - 3*a*b + 4*b**3*(a/(2*(a + b)**2) + sqrt(-a*b)*(a - b)/(4*b*(a**2 + 2*a*b + b**2))))/(a**2 - a*b)) - 1/(x**2*(2*a + 2*b))
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^3 \left (a+b+2 a x^2+a x^4\right )} \, dx=\frac {a \log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {a \log \left (x^{2}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {{\left (a^{2} - a b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b}} - \frac {1}{2 \, {\left (a + b\right )} x^{2}} \] Input:
integrate(1/x^3/(a*x^4+2*a*x^2+a+b),x, algorithm="maxima")
Output:
1/2*a*log(a*x^4 + 2*a*x^2 + a + b)/(a^2 + 2*a*b + b^2) - a*log(x^2)/(a^2 + 2*a*b + b^2) + 1/2*(a^2 - a*b)*arctan((a*x^2 + a)/sqrt(a*b))/((a^2 + 2*a* b + b^2)*sqrt(a*b)) - 1/2/((a + b)*x^2)
Time = 0.15 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^3 \left (a+b+2 a x^2+a x^4\right )} \, dx=\frac {a \log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {a \log \left (x^{2}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {{\left (a^{2} - a b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b}} + \frac {2 \, a x^{2} - a - b}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} x^{2}} \] Input:
integrate(1/x^3/(a*x^4+2*a*x^2+a+b),x, algorithm="giac")
Output:
1/2*a*log(a*x^4 + 2*a*x^2 + a + b)/(a^2 + 2*a*b + b^2) - a*log(x^2)/(a^2 + 2*a*b + b^2) + 1/2*(a^2 - a*b)*arctan((a*x^2 + a)/sqrt(a*b))/((a^2 + 2*a* b + b^2)*sqrt(a*b)) + 1/2*(2*a*x^2 - a - b)/((a^2 + 2*a*b + b^2)*x^2)
Time = 21.91 (sec) , antiderivative size = 3313, normalized size of antiderivative = 37.22 \[ \int \frac {1}{x^3 \left (a+b+2 a x^2+a x^4\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^3*(a + b + 2*a*x^2 + a*x^4)),x)
Output:
(8*a*b*log(((2*a^5)/(a + b)^3 - (a/(2*(a + b)^2) - (-(a*(a - b)^2)/(b*(a + b)^4))^(1/2)/4)*((12*a^5*x^2)/(a + b)^2 - (a/(2*(a + b)^2) - (-(a*(a - b) ^2)/(b*(a + b)^4))^(1/2)/4)*((8*a^4*(3*a - b))/(a + b) + 16*a^4*(a/(2*(a + b)^2) - (-(a*(a - b)^2)/(b*(a + b)^4))^(1/2)/4)*(a + b + a*x^2 - 5*b*x^2) + (4*a^4*x^2*(7*a + 5*b))/(a + b)) + (a^4*(15*a - b))/(a + b)^2) + (a^5*x ^2)/(a + b)^3)*((2*a^5)/(a + b)^3 - (a/(2*(a + b)^2) + (-(a*(a - b)^2)/(b* (a + b)^4))^(1/2)/4)*((12*a^5*x^2)/(a + b)^2 - (a/(2*(a + b)^2) + (-(a*(a - b)^2)/(b*(a + b)^4))^(1/2)/4)*((8*a^4*(3*a - b))/(a + b) + 16*a^4*(a/(2* (a + b)^2) + (-(a*(a - b)^2)/(b*(a + b)^4))^(1/2)/4)*(a + b + a*x^2 - 5*b* x^2) + (4*a^4*x^2*(7*a + 5*b))/(a + b)) + (a^4*(15*a - b))/(a + b)^2) + (a ^5*x^2)/(a + b)^3)))/(32*a*b^2 + 16*a^2*b + 16*b^3) - (2*a*log(x))/(2*a*b + a^2 + b^2) - 1/(2*x^2*(a + b)) + (a^(1/2)*atan(((13*a^2 - 34*a*b + b^2)* ((8*a*b*((14*a^5*b + 15*a^6 - a^4*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) - ( 8*a*b*((40*a^6*b + 24*a^7 - 8*a^4*b^3 + 8*a^5*b^2)/(3*a*b^2 + 3*a^2*b + a^ 3 + b^3) + (8*a*b*(64*a^7*b + 16*a^8 + 16*a^4*b^4 + 64*a^5*b^3 + 96*a^6*b^ 2))/((32*a*b^2 + 16*a^2*b + 16*b^3)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))))/(32 *a*b^2 + 16*a^2*b + 16*b^3)))/(32*a*b^2 + 16*a^2*b + 16*b^3) - (2*a^5)/(3* a*b^2 + 3*a^2*b + a^3 + b^3) + (a^(1/2)*((a^(1/2)*(a - b)*((40*a^6*b + 24* a^7 - 8*a^4*b^3 + 8*a^5*b^2)/(3*a*b^2 + 3*a^2*b + a^3 + b^3) + (8*a*b*(64* a^7*b + 16*a^8 + 16*a^4*b^4 + 64*a^5*b^3 + 96*a^6*b^2))/((32*a*b^2 + 16...
Time = 0.17 (sec) , antiderivative size = 375, normalized size of antiderivative = 4.21 \[ \int \frac {1}{x^3 \left (a+b+2 a x^2+a x^4\right )} \, dx=\frac {-\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}-2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right ) a \,x^{2}+\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}-2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right ) b \,x^{2}-\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}+2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right ) a \,x^{2}+\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}+2 \sqrt {a}\, x}{\sqrt {\sqrt {a}\, \sqrt {a +b}+a}\, \sqrt {2}}\right ) b \,x^{2}+\mathrm {log}\left (-\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}\, x +\sqrt {a +b}+\sqrt {a}\, x^{2}\right ) a b \,x^{2}+\mathrm {log}\left (\sqrt {\sqrt {a}\, \sqrt {a +b}-a}\, \sqrt {2}\, x +\sqrt {a +b}+\sqrt {a}\, x^{2}\right ) a b \,x^{2}-4 \,\mathrm {log}\left (x \right ) a b \,x^{2}-a b -b^{2}}{2 b \,x^{2} \left (a^{2}+2 a b +b^{2}\right )} \] Input:
int(1/x^3/(a*x^4+2*a*x^2+a+b),x)
Output:
( - sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(sqrt(a)*sqrt(a + b) - a)*atan((sqrt (sqrt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*a*x**2 + sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(sqrt(a)*sqrt(a + b) - a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2) - 2*sqrt(a)*x)/(sqr t(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*b*x**2 - sqrt(sqrt(a)*sqrt(a + b) + a )*sqrt(sqrt(a)*sqrt(a + b) - a)*atan((sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2 ) + 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt(2)))*a*x**2 + sqrt(sq rt(a)*sqrt(a + b) + a)*sqrt(sqrt(a)*sqrt(a + b) - a)*atan((sqrt(sqrt(a)*sq rt(a + b) - a)*sqrt(2) + 2*sqrt(a)*x)/(sqrt(sqrt(a)*sqrt(a + b) + a)*sqrt( 2)))*b*x**2 + log( - sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*a*b*x**2 + log(sqrt(sqrt(a)*sqrt(a + b) - a)*sqrt(2)*x + sqrt(a + b) + sqrt(a)*x**2)*a*b*x**2 - 4*log(x)*a*b*x**2 - a*b - b**2)/(2* b*x**2*(a**2 + 2*a*b + b**2))