Integrand size = 50, antiderivative size = 93 \[ \int \frac {1}{a^4+4 a^3 b (s+x)^2+6 a^2 b^2 (s+x)^4+4 a b^3 (s+x)^6+b^4 (s+x)^8} \, dx=\frac {s+x}{6 a \left (a+b (s+x)^2\right )^3}+\frac {5 (s+x)}{24 a^2 \left (a+b (s+x)^2\right )^2}+\frac {5 (s+x)}{16 a^3 \left (a+b (s+x)^2\right )}+\frac {5 \arctan \left (\frac {\sqrt {b} (s+x)}{\sqrt {a}}\right )}{16 a^{7/2} \sqrt {b}} \] Output:
1/6*(s+x)/a/(a+b*(s+x)^2)^3+5/24*(s+x)/a^2/(a+b*(s+x)^2)^2+5/16*(s+x)/a^3/ (a+b*(s+x)^2)+5/16*arctan(b^(1/2)*(s+x)/a^(1/2))/a^(7/2)/b^(1/2)
Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.82 \[ \int \frac {1}{a^4+4 a^3 b (s+x)^2+6 a^2 b^2 (s+x)^4+4 a b^3 (s+x)^6+b^4 (s+x)^8} \, dx=\frac {(s+x) \left (33 a^2+40 a b (s+x)^2+15 b^2 (s+x)^4\right )}{48 a^3 \left (a+b (s+x)^2\right )^3}+\frac {5 \arctan \left (\frac {\sqrt {b} (s+x)}{\sqrt {a}}\right )}{16 a^{7/2} \sqrt {b}} \] Input:
Integrate[(a^4 + 4*a^3*b*(s + x)^2 + 6*a^2*b^2*(s + x)^4 + 4*a*b^3*(s + x) ^6 + b^4*(s + x)^8)^(-1),x]
Output:
((s + x)*(33*a^2 + 40*a*b*(s + x)^2 + 15*b^2*(s + x)^4))/(48*a^3*(a + b*(s + x)^2)^3) + (5*ArcTan[(Sqrt[b]*(s + x))/Sqrt[a]])/(16*a^(7/2)*Sqrt[b])
Leaf count is larger than twice the leaf count of optimal. \(311\) vs. \(2(93)=186\).
Time = 0.80 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.34, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2462, 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.
\(\displaystyle \int \frac {1}{a^4+4 a^3 b (s+x)^2+6 a^2 b^2 (s+x)^4+4 a b^3 (s+x)^6+b^4 (s+x)^8} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (-\frac {5 b}{8 a^3 \left (2 \sqrt {-a} \sqrt {b}-2 b s-2 b x\right )^2}-\frac {5 b}{8 a^3 \left (2 \sqrt {-a} \sqrt {b}+2 b s+2 b x\right )^2}+\frac {b^2}{a^2 \left (2 \sqrt {-a} \sqrt {b}-2 b s-2 b x\right )^4}+\frac {b^2}{a^2 \left (2 \sqrt {-a} \sqrt {b}+2 b s+2 b x\right )^4}+\frac {b^{3/2}}{(-a)^{5/2} \left (2 \sqrt {-a} \sqrt {b}-2 b s-2 b x\right )^3}+\frac {b^{3/2}}{(-a)^{5/2} \left (2 \sqrt {-a} \sqrt {b}+2 b s+2 b x\right )^3}+\frac {5 \sqrt {b}}{16 (-a)^{7/2} \left (2 \sqrt {-a} \sqrt {b}-2 b s-2 b x\right )}+\frac {5 \sqrt {b}}{16 (-a)^{7/2} \left (2 \sqrt {-a} \sqrt {b}+2 b s+2 b x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{32 a^3 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} s-\sqrt {b} x\right )}+\frac {5}{32 a^3 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} s+\sqrt {b} x\right )}+\frac {1}{48 a^2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} s-\sqrt {b} x\right )^3}-\frac {1}{48 a^2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} s+\sqrt {b} x\right )^3}+\frac {1}{16 (-a)^{5/2} \sqrt {b} \left (\sqrt {-a}-\sqrt {b} s-\sqrt {b} x\right )^2}-\frac {1}{16 (-a)^{5/2} \sqrt {b} \left (\sqrt {-a}+\sqrt {b} s+\sqrt {b} x\right )^2}-\frac {5 \log \left (\sqrt {-a}-\sqrt {b} s-\sqrt {b} x\right )}{32 (-a)^{7/2} \sqrt {b}}+\frac {5 \log \left (\sqrt {-a}+\sqrt {b} s+\sqrt {b} x\right )}{32 (-a)^{7/2} \sqrt {b}}\) |
Input:
Int[(a^4 + 4*a^3*b*(s + x)^2 + 6*a^2*b^2*(s + x)^4 + 4*a*b^3*(s + x)^6 + b ^4*(s + x)^8)^(-1),x]
Output:
1/(48*a^2*Sqrt[b]*(Sqrt[-a] - Sqrt[b]*s - Sqrt[b]*x)^3) + 1/(16*(-a)^(5/2) *Sqrt[b]*(Sqrt[-a] - Sqrt[b]*s - Sqrt[b]*x)^2) - 5/(32*a^3*Sqrt[b]*(Sqrt[- a] - Sqrt[b]*s - Sqrt[b]*x)) - 1/(48*a^2*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*s + S qrt[b]*x)^3) - 1/(16*(-a)^(5/2)*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*s + Sqrt[b]*x) ^2) + 5/(32*a^3*Sqrt[b]*(Sqrt[-a] + Sqrt[b]*s + Sqrt[b]*x)) - (5*Log[Sqrt[ -a] - Sqrt[b]*s - Sqrt[b]*x])/(32*(-a)^(7/2)*Sqrt[b]) + (5*Log[Sqrt[-a] + Sqrt[b]*s + Sqrt[b]*x])/(32*(-a)^(7/2)*Sqrt[b])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.60
method | result | size |
default | \(\frac {2 b s +2 b x}{12 a b \left (b \,s^{2}+2 b s x +b \,x^{2}+a \right )^{3}}+\frac {\frac {5 \left (2 b s +2 b x \right )}{48 a b \left (b \,s^{2}+2 b s x +b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {3 \left (2 b s +2 b x \right )}{16 a b \left (b \,s^{2}+2 b s x +b \,x^{2}+a \right )}+\frac {3 \arctan \left (\frac {2 b s +2 b x}{2 \sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{6 a}}{a}\) | \(149\) |
risch | \(\frac {\frac {5 b^{2} x^{5}}{16 a^{3}}+\frac {25 b^{2} s \,x^{4}}{16 a^{3}}+\frac {5 b \left (15 b \,s^{2}+4 a \right ) x^{3}}{24 a^{3}}+\frac {5 b s \left (5 b \,s^{2}+4 a \right ) x^{2}}{8 a^{3}}+\frac {\left (25 b^{2} s^{4}+40 a b \,s^{2}+11 a^{2}\right ) x}{16 a^{3}}+\frac {s \left (15 b^{2} s^{4}+40 a b \,s^{2}+33 a^{2}\right )}{48 a^{3}}}{b^{3} s^{6}+6 b^{3} s^{5} x +15 b^{3} s^{4} x^{2}+20 b^{3} s^{3} x^{3}+15 b^{3} s^{2} x^{4}+6 b^{3} s \,x^{5}+b^{3} x^{6}+3 a \,b^{2} s^{4}+12 a \,b^{2} s^{3} x +18 a \,b^{2} s^{2} x^{2}+12 a \,b^{2} s \,x^{3}+3 a \,b^{2} x^{4}+3 a^{2} b \,s^{2}+6 a^{2} b s x +3 b \,a^{2} x^{2}+a^{3}}-\frac {5 \ln \left (b s +b x +\sqrt {-a b}\right )}{32 \sqrt {-a b}\, a^{3}}+\frac {5 \ln \left (-b s -b x +\sqrt {-a b}\right )}{32 \sqrt {-a b}\, a^{3}}\) | \(319\) |
Input:
int(1/(a^4+4*a^3*b*(s+x)^2+6*a^2*b^2*(s+x)^4+4*a*b^3*(s+x)^6+b^4*(s+x)^8), x,method=_RETURNVERBOSE)
Output:
1/12*(2*b*s+2*b*x)/a/b/(b*s^2+2*b*s*x+b*x^2+a)^3+5/6/a*(1/8*(2*b*s+2*b*x)/ a/b/(b*s^2+2*b*s*x+b*x^2+a)^2+3/4/a*(1/4*(2*b*s+2*b*x)/a/b/(b*s^2+2*b*s*x+ b*x^2+a)+1/2/a/(a*b)^(1/2)*arctan(1/2*(2*b*s+2*b*x)/(a*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (77) = 154\).
Time = 0.10 (sec) , antiderivative size = 970, normalized size of antiderivative = 10.43 \[ \int \frac {1}{a^4+4 a^3 b (s+x)^2+6 a^2 b^2 (s+x)^4+4 a b^3 (s+x)^6+b^4 (s+x)^8} \, dx =\text {Too large to display} \] Input:
integrate(1/(a^4+4*a^3*b*(s+x)^2+6*a^2*b^2*(s+x)^4+4*a*b^3*(s+x)^6+b^4*(s+ x)^8),x, algorithm="fricas")
Output:
[1/96*(30*a*b^3*s^5 + 150*a*b^3*s*x^4 + 30*a*b^3*x^5 + 80*a^2*b^2*s^3 + 66 *a^3*b*s + 20*(15*a*b^3*s^2 + 4*a^2*b^2)*x^3 + 60*(5*a*b^3*s^3 + 4*a^2*b^2 *s)*x^2 - 15*(b^3*s^6 + 6*b^3*s*x^5 + b^3*x^6 + 3*a*b^2*s^4 + 3*a^2*b*s^2 + 3*(5*b^3*s^2 + a*b^2)*x^4 + 4*(5*b^3*s^3 + 3*a*b^2*s)*x^3 + a^3 + 3*(5*b ^3*s^4 + 6*a*b^2*s^2 + a^2*b)*x^2 + 6*(b^3*s^5 + 2*a*b^2*s^3 + a^2*b*s)*x) *sqrt(-a*b)*log((b*s^2 + 2*b*s*x + b*x^2 - 2*sqrt(-a*b)*(s + x) - a)/(b*s^ 2 + 2*b*s*x + b*x^2 + a)) + 6*(25*a*b^3*s^4 + 40*a^2*b^2*s^2 + 11*a^3*b)*x )/(a^4*b^4*s^6 + 6*a^4*b^4*s*x^5 + a^4*b^4*x^6 + 3*a^5*b^3*s^4 + 3*a^6*b^2 *s^2 + a^7*b + 3*(5*a^4*b^4*s^2 + a^5*b^3)*x^4 + 4*(5*a^4*b^4*s^3 + 3*a^5* b^3*s)*x^3 + 3*(5*a^4*b^4*s^4 + 6*a^5*b^3*s^2 + a^6*b^2)*x^2 + 6*(a^4*b^4* s^5 + 2*a^5*b^3*s^3 + a^6*b^2*s)*x), 1/48*(15*a*b^3*s^5 + 75*a*b^3*s*x^4 + 15*a*b^3*x^5 + 40*a^2*b^2*s^3 + 33*a^3*b*s + 10*(15*a*b^3*s^2 + 4*a^2*b^2 )*x^3 + 30*(5*a*b^3*s^3 + 4*a^2*b^2*s)*x^2 + 15*(b^3*s^6 + 6*b^3*s*x^5 + b ^3*x^6 + 3*a*b^2*s^4 + 3*a^2*b*s^2 + 3*(5*b^3*s^2 + a*b^2)*x^4 + 4*(5*b^3* s^3 + 3*a*b^2*s)*x^3 + a^3 + 3*(5*b^3*s^4 + 6*a*b^2*s^2 + a^2*b)*x^2 + 6*( b^3*s^5 + 2*a*b^2*s^3 + a^2*b*s)*x)*sqrt(a*b)*arctan(sqrt(a*b)*(s + x)/a) + 3*(25*a*b^3*s^4 + 40*a^2*b^2*s^2 + 11*a^3*b)*x)/(a^4*b^4*s^6 + 6*a^4*b^4 *s*x^5 + a^4*b^4*x^6 + 3*a^5*b^3*s^4 + 3*a^6*b^2*s^2 + a^7*b + 3*(5*a^4*b^ 4*s^2 + a^5*b^3)*x^4 + 4*(5*a^4*b^4*s^3 + 3*a^5*b^3*s)*x^3 + 3*(5*a^4*b^4* s^4 + 6*a^5*b^3*s^2 + a^6*b^2)*x^2 + 6*(a^4*b^4*s^5 + 2*a^5*b^3*s^3 + a...
Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (92) = 184\).
Time = 0.93 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.81 \[ \int \frac {1}{a^4+4 a^3 b (s+x)^2+6 a^2 b^2 (s+x)^4+4 a b^3 (s+x)^6+b^4 (s+x)^8} \, dx=- \frac {5 \sqrt {- \frac {1}{a^{7} b}} \log {\left (- a^{4} \sqrt {- \frac {1}{a^{7} b}} + s + x \right )}}{32} + \frac {5 \sqrt {- \frac {1}{a^{7} b}} \log {\left (a^{4} \sqrt {- \frac {1}{a^{7} b}} + s + x \right )}}{32} + \frac {33 a^{2} s + 40 a b s^{3} + 15 b^{2} s^{5} + 75 b^{2} s x^{4} + 15 b^{2} x^{5} + x^{3} \cdot \left (40 a b + 150 b^{2} s^{2}\right ) + x^{2} \cdot \left (120 a b s + 150 b^{2} s^{3}\right ) + x \left (33 a^{2} + 120 a b s^{2} + 75 b^{2} s^{4}\right )}{48 a^{6} + 144 a^{5} b s^{2} + 144 a^{4} b^{2} s^{4} + 48 a^{3} b^{3} s^{6} + 288 a^{3} b^{3} s x^{5} + 48 a^{3} b^{3} x^{6} + x^{4} \cdot \left (144 a^{4} b^{2} + 720 a^{3} b^{3} s^{2}\right ) + x^{3} \cdot \left (576 a^{4} b^{2} s + 960 a^{3} b^{3} s^{3}\right ) + x^{2} \cdot \left (144 a^{5} b + 864 a^{4} b^{2} s^{2} + 720 a^{3} b^{3} s^{4}\right ) + x \left (288 a^{5} b s + 576 a^{4} b^{2} s^{3} + 288 a^{3} b^{3} s^{5}\right )} \] Input:
integrate(1/(a**4+4*a**3*b*(s+x)**2+6*a**2*b**2*(s+x)**4+4*a*b**3*(s+x)**6 +b**4*(s+x)**8),x)
Output:
-5*sqrt(-1/(a**7*b))*log(-a**4*sqrt(-1/(a**7*b)) + s + x)/32 + 5*sqrt(-1/( a**7*b))*log(a**4*sqrt(-1/(a**7*b)) + s + x)/32 + (33*a**2*s + 40*a*b*s**3 + 15*b**2*s**5 + 75*b**2*s*x**4 + 15*b**2*x**5 + x**3*(40*a*b + 150*b**2* s**2) + x**2*(120*a*b*s + 150*b**2*s**3) + x*(33*a**2 + 120*a*b*s**2 + 75* b**2*s**4))/(48*a**6 + 144*a**5*b*s**2 + 144*a**4*b**2*s**4 + 48*a**3*b**3 *s**6 + 288*a**3*b**3*s*x**5 + 48*a**3*b**3*x**6 + x**4*(144*a**4*b**2 + 7 20*a**3*b**3*s**2) + x**3*(576*a**4*b**2*s + 960*a**3*b**3*s**3) + x**2*(1 44*a**5*b + 864*a**4*b**2*s**2 + 720*a**3*b**3*s**4) + x*(288*a**5*b*s + 5 76*a**4*b**2*s**3 + 288*a**3*b**3*s**5))
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (77) = 154\).
Time = 0.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.22 \[ \int \frac {1}{a^4+4 a^3 b (s+x)^2+6 a^2 b^2 (s+x)^4+4 a b^3 (s+x)^6+b^4 (s+x)^8} \, dx=\frac {15 \, b^{2} s^{5} + 75 \, b^{2} s x^{4} + 15 \, b^{2} x^{5} + 40 \, a b s^{3} + 10 \, {\left (15 \, b^{2} s^{2} + 4 \, a b\right )} x^{3} + 33 \, a^{2} s + 30 \, {\left (5 \, b^{2} s^{3} + 4 \, a b s\right )} x^{2} + 3 \, {\left (25 \, b^{2} s^{4} + 40 \, a b s^{2} + 11 \, a^{2}\right )} x}{48 \, {\left (a^{3} b^{3} s^{6} + 6 \, a^{3} b^{3} s x^{5} + a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} s^{4} + 3 \, a^{5} b s^{2} + a^{6} + 3 \, {\left (5 \, a^{3} b^{3} s^{2} + a^{4} b^{2}\right )} x^{4} + 4 \, {\left (5 \, a^{3} b^{3} s^{3} + 3 \, a^{4} b^{2} s\right )} x^{3} + 3 \, {\left (5 \, a^{3} b^{3} s^{4} + 6 \, a^{4} b^{2} s^{2} + a^{5} b\right )} x^{2} + 6 \, {\left (a^{3} b^{3} s^{5} + 2 \, a^{4} b^{2} s^{3} + a^{5} b s\right )} x\right )}} + \frac {5 \, \arctan \left (\frac {b s + b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{3}} \] Input:
integrate(1/(a^4+4*a^3*b*(s+x)^2+6*a^2*b^2*(s+x)^4+4*a*b^3*(s+x)^6+b^4*(s+ x)^8),x, algorithm="maxima")
Output:
1/48*(15*b^2*s^5 + 75*b^2*s*x^4 + 15*b^2*x^5 + 40*a*b*s^3 + 10*(15*b^2*s^2 + 4*a*b)*x^3 + 33*a^2*s + 30*(5*b^2*s^3 + 4*a*b*s)*x^2 + 3*(25*b^2*s^4 + 40*a*b*s^2 + 11*a^2)*x)/(a^3*b^3*s^6 + 6*a^3*b^3*s*x^5 + a^3*b^3*x^6 + 3*a ^4*b^2*s^4 + 3*a^5*b*s^2 + a^6 + 3*(5*a^3*b^3*s^2 + a^4*b^2)*x^4 + 4*(5*a^ 3*b^3*s^3 + 3*a^4*b^2*s)*x^3 + 3*(5*a^3*b^3*s^4 + 6*a^4*b^2*s^2 + a^5*b)*x ^2 + 6*(a^3*b^3*s^5 + 2*a^4*b^2*s^3 + a^5*b*s)*x) + 5/16*arctan((b*s + b*x )/sqrt(a*b))/(sqrt(a*b)*a^3)
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.59 \[ \int \frac {1}{a^4+4 a^3 b (s+x)^2+6 a^2 b^2 (s+x)^4+4 a b^3 (s+x)^6+b^4 (s+x)^8} \, dx=\frac {5 \, \arctan \left (\frac {b s + b x}{\sqrt {a b}}\right )}{16 \, \sqrt {a b} a^{3}} + \frac {15 \, b^{2} s^{5} + 75 \, b^{2} s^{4} x + 150 \, b^{2} s^{3} x^{2} + 150 \, b^{2} s^{2} x^{3} + 75 \, b^{2} s x^{4} + 15 \, b^{2} x^{5} + 40 \, a b s^{3} + 120 \, a b s^{2} x + 120 \, a b s x^{2} + 40 \, a b x^{3} + 33 \, a^{2} s + 33 \, a^{2} x}{48 \, {\left (b s^{2} + 2 \, b s x + b x^{2} + a\right )}^{3} a^{3}} \] Input:
integrate(1/(a^4+4*a^3*b*(s+x)^2+6*a^2*b^2*(s+x)^4+4*a*b^3*(s+x)^6+b^4*(s+ x)^8),x, algorithm="giac")
Output:
5/16*arctan((b*s + b*x)/sqrt(a*b))/(sqrt(a*b)*a^3) + 1/48*(15*b^2*s^5 + 75 *b^2*s^4*x + 150*b^2*s^3*x^2 + 150*b^2*s^2*x^3 + 75*b^2*s*x^4 + 15*b^2*x^5 + 40*a*b*s^3 + 120*a*b*s^2*x + 120*a*b*s*x^2 + 40*a*b*x^3 + 33*a^2*s + 33 *a^2*x)/((b*s^2 + 2*b*s*x + b*x^2 + a)^3*a^3)
Time = 21.79 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98 \[ \int \frac {1}{a^4+4 a^3 b (s+x)^2+6 a^2 b^2 (s+x)^4+4 a b^3 (s+x)^6+b^4 (s+x)^8} \, dx=\frac {\frac {11\,\left (s+x\right )}{16\,a}+\frac {5\,b\,{\left (s+x\right )}^3}{6\,a^2}+\frac {5\,b^2\,{\left (s+x\right )}^5}{16\,a^3}}{a^3+b^3\,{\left (s+x\right )}^6+3\,a^2\,b\,{\left (s+x\right )}^2+3\,a\,b^2\,{\left (s+x\right )}^4}+\frac {5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\left (s+x\right )}{\sqrt {a}}\right )}{16\,a^{7/2}\,\sqrt {b}} \] Input:
int(1/(a^4 + b^4*(s + x)^8 + 4*a^3*b*(s + x)^2 + 4*a*b^3*(s + x)^6 + 6*a^2 *b^2*(s + x)^4),x)
Output:
((11*(s + x))/(16*a) + (5*b*(s + x)^3)/(6*a^2) + (5*b^2*(s + x)^5)/(16*a^3 ))/(a^3 + b^3*(s + x)^6 + 3*a^2*b*(s + x)^2 + 3*a*b^2*(s + x)^4) + (5*atan ((b^(1/2)*(s + x))/a^(1/2)))/(16*a^(7/2)*b^(1/2))
Time = 0.15 (sec) , antiderivative size = 808, normalized size of antiderivative = 8.69 \[ \int \frac {1}{a^4+4 a^3 b (s+x)^2+6 a^2 b^2 (s+x)^4+4 a b^3 (s+x)^6+b^4 (s+x)^8} \, dx =\text {Too large to display} \] Input:
int(1/(a^4+4*a^3*b*(s+x)^2+6*a^2*b^2*(s+x)^4+4*a*b^3*(s+x)^6+b^4*(s+x)^8), x)
Output:
(30*sqrt(b)*sqrt(a)*atan((b*s + b*x)/(sqrt(b)*sqrt(a)))*a**3*s + 90*sqrt(b )*sqrt(a)*atan((b*s + b*x)/(sqrt(b)*sqrt(a)))*a**2*b*s**3 + 180*sqrt(b)*sq rt(a)*atan((b*s + b*x)/(sqrt(b)*sqrt(a)))*a**2*b*s**2*x + 90*sqrt(b)*sqrt( a)*atan((b*s + b*x)/(sqrt(b)*sqrt(a)))*a**2*b*s*x**2 + 90*sqrt(b)*sqrt(a)* atan((b*s + b*x)/(sqrt(b)*sqrt(a)))*a*b**2*s**5 + 360*sqrt(b)*sqrt(a)*atan ((b*s + b*x)/(sqrt(b)*sqrt(a)))*a*b**2*s**4*x + 540*sqrt(b)*sqrt(a)*atan(( b*s + b*x)/(sqrt(b)*sqrt(a)))*a*b**2*s**3*x**2 + 360*sqrt(b)*sqrt(a)*atan( (b*s + b*x)/(sqrt(b)*sqrt(a)))*a*b**2*s**2*x**3 + 90*sqrt(b)*sqrt(a)*atan( (b*s + b*x)/(sqrt(b)*sqrt(a)))*a*b**2*s*x**4 + 30*sqrt(b)*sqrt(a)*atan((b* s + b*x)/(sqrt(b)*sqrt(a)))*b**3*s**7 + 180*sqrt(b)*sqrt(a)*atan((b*s + b* x)/(sqrt(b)*sqrt(a)))*b**3*s**6*x + 450*sqrt(b)*sqrt(a)*atan((b*s + b*x)/( sqrt(b)*sqrt(a)))*b**3*s**5*x**2 + 600*sqrt(b)*sqrt(a)*atan((b*s + b*x)/(s qrt(b)*sqrt(a)))*b**3*s**4*x**3 + 450*sqrt(b)*sqrt(a)*atan((b*s + b*x)/(sq rt(b)*sqrt(a)))*b**3*s**3*x**4 + 180*sqrt(b)*sqrt(a)*atan((b*s + b*x)/(sqr t(b)*sqrt(a)))*b**3*s**2*x**5 + 30*sqrt(b)*sqrt(a)*atan((b*s + b*x)/(sqrt( b)*sqrt(a)))*b**3*s*x**6 - 5*a**4 + 51*a**3*b*s**2 + 36*a**3*b*s*x - 15*a* *3*b*x**2 + 65*a**2*b**2*s**4 + 180*a**2*b**2*s**3*x + 150*a**2*b**2*s**2* x**2 + 20*a**2*b**2*s*x**3 - 15*a**2*b**2*x**4 + 25*a*b**3*s**6 + 120*a*b* *3*s**5*x + 225*a*b**3*s**4*x**2 + 200*a*b**3*s**3*x**3 + 75*a*b**3*s**2*x **4 - 5*a*b**3*x**6)/(96*a**4*b*s*(a**3 + 3*a**2*b*s**2 + 6*a**2*b*s*x ...