Integrand size = 26, antiderivative size = 114 \[ \int \frac {b+a x^4}{\left (-a+x^2\right )^2 \left (2 a+x^2\right )^2} \, dx=\frac {8 a^4 x+5 a b x-2 a^3 x^3+b x^3}{36 a^3 \left (2 a^2-a x^2-x^4\right )}-\frac {\left (4 a^3-11 b\right ) \arctan \left (\frac {x}{\sqrt {2} \sqrt {a}}\right )}{108 \sqrt {2} a^{7/2}}+\frac {\left (-5 a^3+7 b\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{54 a^{7/2}} \] Output:
1/36*(-2*a^3*x^3+8*a^4*x+b*x^3+5*a*b*x)/a^3/(-x^4-a*x^2+2*a^2)-1/216*(4*a^ 3-11*b)*arctan(1/2*x*2^(1/2)/a^(1/2))*2^(1/2)/a^(7/2)+1/54*(-5*a^3+7*b)*ar ctanh(x/a^(1/2))/a^(7/2)
Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {b+a x^4}{\left (-a+x^2\right )^2 \left (2 a+x^2\right )^2} \, dx=\frac {-\frac {12 \sqrt {a} \left (a^3+b\right ) x}{-a+x^2}+\frac {6 \sqrt {a} \left (4 a^3+b\right ) x}{2 a+x^2}+\sqrt {2} \left (-4 a^3+11 b\right ) \arctan \left (\frac {x}{\sqrt {2} \sqrt {a}}\right )-4 \left (5 a^3-7 b\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{216 a^{7/2}} \] Input:
Integrate[(b + a*x^4)/((-a + x^2)^2*(2*a + x^2)^2),x]
Output:
((-12*Sqrt[a]*(a^3 + b)*x)/(-a + x^2) + (6*Sqrt[a]*(4*a^3 + b)*x)/(2*a + x ^2) + Sqrt[2]*(-4*a^3 + 11*b)*ArcTan[x/(Sqrt[2]*Sqrt[a])] - 4*(5*a^3 - 7*b )*ArcTanh[x/Sqrt[a]])/(216*a^(7/2))
Time = 0.48 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.45, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {7293, 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.
| \(\displaystyle \int \frac {a x^4+b}{\left (x^2-a\right )^2 \left (2 a+x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 \left (2 a^3-b\right )}{27 a^3 \left (a-x^2\right )}-\frac {2 \left (2 a^3-b\right )}{27 a^3 \left (2 a+x^2\right )}+\frac {a^3+b}{9 a^2 \left (a-x^2\right )^2}+\frac {4 a^3+b}{9 a^2 \left (2 a+x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (a^3+b\right )}{18 a^3 \left (a-x^2\right )}+\frac {x \left (4 a^3+b\right )}{36 a^3 \left (2 a+x^2\right )}-\frac {\sqrt {2} \left (2 a^3-b\right ) \arctan \left (\frac {x}{\sqrt {2} \sqrt {a}}\right )}{27 a^{7/2}}+\frac {\left (4 a^3+b\right ) \arctan \left (\frac {x}{\sqrt {2} \sqrt {a}}\right )}{36 \sqrt {2} a^{7/2}}-\frac {2 \left (2 a^3-b\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{27 a^{7/2}}+\frac {\left (a^3+b\right ) \text {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{18 a^{7/2}}\) |
Input:
Int[(b + a*x^4)/((-a + x^2)^2*(2*a + x^2)^2),x]
Output:
((a^3 + b)*x)/(18*a^3*(a - x^2)) + ((4*a^3 + b)*x)/(36*a^3*(2*a + x^2)) - (Sqrt[2]*(2*a^3 - b)*ArcTan[x/(Sqrt[2]*Sqrt[a])])/(27*a^(7/2)) + ((4*a^3 + b)*ArcTan[x/(Sqrt[2]*Sqrt[a])])/(36*Sqrt[2]*a^(7/2)) - (2*(2*a^3 - b)*Arc Tanh[x/Sqrt[a]])/(27*a^(7/2)) + ((a^3 + b)*ArcTanh[x/Sqrt[a]])/(18*a^(7/2) )
Time = 0.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(-\frac {\frac {\left (-\frac {3 a^{3}}{2}-\frac {3 b}{2}\right ) x}{-x^{2}+a}+\frac {\left (5 a^{3}-7 b \right ) \operatorname {arctanh}\left (\frac {x}{\sqrt {a}}\right )}{2 \sqrt {a}}}{27 a^{3}}+\frac {\frac {\left (\frac {3 a^{3}}{2}+\frac {3 b}{8}\right ) x}{\frac {x^{2}}{2}+a}-\frac {\left (2 a^{3}-\frac {11 b}{2}\right ) \sqrt {2}\, \arctan \left (\frac {x \sqrt {2}}{2 \sqrt {a}}\right )}{4 \sqrt {a}}}{27 a^{3}}\) | \(101\) |
| risch | \(\frac {-\frac {\left (2 a^{3}-b \right ) x^{3}}{36 a^{3}}+\frac {\left (8 a^{3}+5 b \right ) x}{36 a^{2}}}{\left (x^{2}+2 a \right ) \left (-x^{2}+a \right )}+\frac {\sqrt {2}\, \ln \left (2 \left (-a \right )^{\frac {21}{2}}-a^{10} x \sqrt {2}\right ) a^{3}}{108 \left (-a \right )^{\frac {7}{2}}}-\frac {11 \sqrt {2}\, \ln \left (2 \left (-a \right )^{\frac {21}{2}}-a^{10} x \sqrt {2}\right ) b}{432 \left (-a \right )^{\frac {7}{2}}}-\frac {\sqrt {2}\, \ln \left (2 \left (-a \right )^{\frac {21}{2}}+a^{10} x \sqrt {2}\right ) a^{3}}{108 \left (-a \right )^{\frac {7}{2}}}+\frac {11 \sqrt {2}\, \ln \left (2 \left (-a \right )^{\frac {21}{2}}+a^{10} x \sqrt {2}\right ) b}{432 \left (-a \right )^{\frac {7}{2}}}+\frac {5 \ln \left (\left (40 a^{9}-176 b \,a^{6}+263 a^{3} b^{2}-133 b^{3}\right ) x -40 a^{\frac {19}{2}}+176 b \,a^{\frac {13}{2}}-263 a^{\frac {7}{2}} b^{2}+133 b^{3} \sqrt {a}\right )}{108 \sqrt {a}}-\frac {7 \ln \left (\left (40 a^{9}-176 b \,a^{6}+263 a^{3} b^{2}-133 b^{3}\right ) x -40 a^{\frac {19}{2}}+176 b \,a^{\frac {13}{2}}-263 a^{\frac {7}{2}} b^{2}+133 b^{3} \sqrt {a}\right ) b}{108 a^{\frac {7}{2}}}-\frac {5 \ln \left (\left (-40 a^{9}+176 b \,a^{6}-263 a^{3} b^{2}+133 b^{3}\right ) x -40 a^{\frac {19}{2}}+176 b \,a^{\frac {13}{2}}-263 a^{\frac {7}{2}} b^{2}+133 b^{3} \sqrt {a}\right )}{108 \sqrt {a}}+\frac {7 \ln \left (\left (-40 a^{9}+176 b \,a^{6}-263 a^{3} b^{2}+133 b^{3}\right ) x -40 a^{\frac {19}{2}}+176 b \,a^{\frac {13}{2}}-263 a^{\frac {7}{2}} b^{2}+133 b^{3} \sqrt {a}\right ) b}{108 a^{\frac {7}{2}}}\) | \(419\) |
Input:
int((a*x^4+b)/(x^2-a)^2/(x^2+2*a)^2,x,method=_RETURNVERBOSE)
Output:
-1/27/a^3*((-3/2*a^3-3/2*b)*x/(-x^2+a)+1/2*(5*a^3-7*b)/a^(1/2)*arctanh(x/a ^(1/2)))+1/27/a^3*((3/2*a^3+3/8*b)*x/(1/2*x^2+a)-1/4*(2*a^3-11/2*b)*2^(1/2 )/a^(1/2)*arctan(1/2*x*2^(1/2)/a^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (95) = 190\).
Time = 0.09 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.32 \[ \int \frac {b+a x^4}{\left (-a+x^2\right )^2 \left (2 a+x^2\right )^2} \, dx=\left [\frac {6 \, {\left (2 \, a^{4} - a b\right )} x^{3} + \sqrt {2} {\left (8 \, a^{5} - {\left (4 \, a^{3} - 11 \, b\right )} x^{4} - 22 \, a^{2} b - {\left (4 \, a^{4} - 11 \, a b\right )} x^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} x}{2 \, \sqrt {a}}\right ) + 2 \, {\left (10 \, a^{5} - {\left (5 \, a^{3} - 7 \, b\right )} x^{4} - 14 \, a^{2} b - {\left (5 \, a^{4} - 7 \, a b\right )} x^{2}\right )} \sqrt {a} \log \left (\frac {x^{2} + 2 \, \sqrt {a} x + a}{x^{2} - a}\right ) - 6 \, {\left (8 \, a^{5} + 5 \, a^{2} b\right )} x}{216 \, {\left (a^{4} x^{4} + a^{5} x^{2} - 2 \, a^{6}\right )}}, \frac {12 \, {\left (2 \, a^{4} - a b\right )} x^{3} - \sqrt {2} {\left (8 \, a^{5} - {\left (4 \, a^{3} - 11 \, b\right )} x^{4} - 22 \, a^{2} b - {\left (4 \, a^{4} - 11 \, a b\right )} x^{2}\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} x - x^{2} + 2 \, a}{x^{2} + 2 \, a}\right ) - 8 \, {\left (10 \, a^{5} - {\left (5 \, a^{3} - 7 \, b\right )} x^{4} - 14 \, a^{2} b - {\left (5 \, a^{4} - 7 \, a b\right )} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} x}{a}\right ) - 12 \, {\left (8 \, a^{5} + 5 \, a^{2} b\right )} x}{432 \, {\left (a^{4} x^{4} + a^{5} x^{2} - 2 \, a^{6}\right )}}\right ] \] Input:
integrate((a*x^4+b)/(x^2-a)^2/(x^2+2*a)^2,x, algorithm="fricas")
Output:
[1/216*(6*(2*a^4 - a*b)*x^3 + sqrt(2)*(8*a^5 - (4*a^3 - 11*b)*x^4 - 22*a^2 *b - (4*a^4 - 11*a*b)*x^2)*sqrt(a)*arctan(1/2*sqrt(2)*x/sqrt(a)) + 2*(10*a ^5 - (5*a^3 - 7*b)*x^4 - 14*a^2*b - (5*a^4 - 7*a*b)*x^2)*sqrt(a)*log((x^2 + 2*sqrt(a)*x + a)/(x^2 - a)) - 6*(8*a^5 + 5*a^2*b)*x)/(a^4*x^4 + a^5*x^2 - 2*a^6), 1/432*(12*(2*a^4 - a*b)*x^3 - sqrt(2)*(8*a^5 - (4*a^3 - 11*b)*x^ 4 - 22*a^2*b - (4*a^4 - 11*a*b)*x^2)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*x - x^2 + 2*a)/(x^2 + 2*a)) - 8*(10*a^5 - (5*a^3 - 7*b)*x^4 - 14*a^2*b - (5*a ^4 - 7*a*b)*x^2)*sqrt(-a)*arctan(sqrt(-a)*x/a) - 12*(8*a^5 + 5*a^2*b)*x)/( a^4*x^4 + a^5*x^2 - 2*a^6)]
Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (109) = 218\).
Time = 1.34 (sec) , antiderivative size = 989, normalized size of antiderivative = 8.68 \[ \int \frac {b+a x^4}{\left (-a+x^2\right )^2 \left (2 a+x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((a*x**4+b)/(x**2-a)**2/(x**2+2*a)**2,x)
Output:
-sqrt(2)*sqrt(-1/a**7)*(4*a**3 - 11*b)*log(x + (-4*sqrt(2)*a**14*(-1/a**7) **(3/2)*(4*a**3 - 11*b)**3/27 + 1016*sqrt(2)*a**13*sqrt(-1/a**7)*(4*a**3 - 11*b)/27 + 17*sqrt(2)*a**11*b*(-1/a**7)**(3/2)*(4*a**3 - 11*b)**3/108 - 1 444*sqrt(2)*a**10*b*sqrt(-1/a**7)*(4*a**3 - 11*b)/9 + 2081*sqrt(2)*a**7*b* *2*sqrt(-1/a**7)*(4*a**3 - 11*b)/9 - 12307*sqrt(2)*a**4*b**3*sqrt(-1/a**7) *(4*a**3 - 11*b)/108)/(160*a**12 - 1144*a**9*b + 2988*a**6*b**2 - 3425*a** 3*b**3 + 1463*b**4))/432 + sqrt(2)*sqrt(-1/a**7)*(4*a**3 - 11*b)*log(x + ( 4*sqrt(2)*a**14*(-1/a**7)**(3/2)*(4*a**3 - 11*b)**3/27 - 1016*sqrt(2)*a**1 3*sqrt(-1/a**7)*(4*a**3 - 11*b)/27 - 17*sqrt(2)*a**11*b*(-1/a**7)**(3/2)*( 4*a**3 - 11*b)**3/108 + 1444*sqrt(2)*a**10*b*sqrt(-1/a**7)*(4*a**3 - 11*b) /9 - 2081*sqrt(2)*a**7*b**2*sqrt(-1/a**7)*(4*a**3 - 11*b)/9 + 12307*sqrt(2 )*a**4*b**3*sqrt(-1/a**7)*(4*a**3 - 11*b)/108)/(160*a**12 - 1144*a**9*b + 2988*a**6*b**2 - 3425*a**3*b**3 + 1463*b**4))/432 - (5*a**3 - 7*b)*sqrt(a* *(-7))*log(x + (-128*a**14*(5*a**3 - 7*b)**3*(a**(-7))**(3/2)/27 + 4064*a* *13*(5*a**3 - 7*b)*sqrt(a**(-7))/27 + 136*a**11*b*(5*a**3 - 7*b)**3*(a**(- 7))**(3/2)/27 - 5776*a**10*b*(5*a**3 - 7*b)*sqrt(a**(-7))/9 + 8324*a**7*b* *2*(5*a**3 - 7*b)*sqrt(a**(-7))/9 - 12307*a**4*b**3*(5*a**3 - 7*b)*sqrt(a* *(-7))/27)/(160*a**12 - 1144*a**9*b + 2988*a**6*b**2 - 3425*a**3*b**3 + 14 63*b**4))/108 + (5*a**3 - 7*b)*sqrt(a**(-7))*log(x + (128*a**14*(5*a**3 - 7*b)**3*(a**(-7))**(3/2)/27 - 4064*a**13*(5*a**3 - 7*b)*sqrt(a**(-7))/2...
Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {b+a x^4}{\left (-a+x^2\right )^2 \left (2 a+x^2\right )^2} \, dx=\frac {{\left (2 \, a^{3} - b\right )} x^{3} - {\left (8 \, a^{4} + 5 \, a b\right )} x}{36 \, {\left (a^{3} x^{4} + a^{4} x^{2} - 2 \, a^{5}\right )}} - \frac {\sqrt {2} {\left (4 \, a^{3} - 11 \, b\right )} \arctan \left (\frac {\sqrt {2} x}{2 \, \sqrt {a}}\right )}{216 \, a^{\frac {7}{2}}} + \frac {{\left (5 \, a^{3} - 7 \, b\right )} \log \left (\frac {x - \sqrt {a}}{x + \sqrt {a}}\right )}{108 \, a^{\frac {7}{2}}} \] Input:
integrate((a*x^4+b)/(x^2-a)^2/(x^2+2*a)^2,x, algorithm="maxima")
Output:
1/36*((2*a^3 - b)*x^3 - (8*a^4 + 5*a*b)*x)/(a^3*x^4 + a^4*x^2 - 2*a^5) - 1 /216*sqrt(2)*(4*a^3 - 11*b)*arctan(1/2*sqrt(2)*x/sqrt(a))/a^(7/2) + 1/108* (5*a^3 - 7*b)*log((x - sqrt(a))/(x + sqrt(a)))/a^(7/2)
Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.89 \[ \int \frac {b+a x^4}{\left (-a+x^2\right )^2 \left (2 a+x^2\right )^2} \, dx=-\frac {\sqrt {2} {\left (4 \, a^{3} - 11 \, b\right )} \arctan \left (\frac {\sqrt {2} x}{2 \, \sqrt {a}}\right )}{216 \, a^{\frac {7}{2}}} + \frac {{\left (5 \, a^{3} - 7 \, b\right )} \arctan \left (\frac {x}{\sqrt {-a}}\right )}{54 \, \sqrt {-a} a^{3}} + \frac {2 \, a^{3} x^{3} - 8 \, a^{4} x - b x^{3} - 5 \, a b x}{36 \, {\left (x^{4} + a x^{2} - 2 \, a^{2}\right )} a^{3}} \] Input:
integrate((a*x^4+b)/(x^2-a)^2/(x^2+2*a)^2,x, algorithm="giac")
Output:
-1/216*sqrt(2)*(4*a^3 - 11*b)*arctan(1/2*sqrt(2)*x/sqrt(a))/a^(7/2) + 1/54 *(5*a^3 - 7*b)*arctan(x/sqrt(-a))/(sqrt(-a)*a^3) + 1/36*(2*a^3*x^3 - 8*a^4 *x - b*x^3 - 5*a*b*x)/((x^4 + a*x^2 - 2*a^2)*a^3)
Time = 9.77 (sec) , antiderivative size = 425, normalized size of antiderivative = 3.73 \[ \int \frac {b+a x^4}{\left (-a+x^2\right )^2 \left (2 a+x^2\right )^2} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x\,\sqrt {-a^7}}{324\,\left (\frac {23\,a\,b}{648}-\frac {a^4}{162}-\frac {85\,b^2}{1296\,a^2}+\frac {209\,b^3}{5184\,a^5}\right )}+\frac {85\,\sqrt {2}\,b^2\,x\,\sqrt {-a^7}}{2592\,\left (-\frac {a^{10}}{162}+\frac {23\,a^7\,b}{648}-\frac {85\,a^4\,b^2}{1296}+\frac {209\,a\,b^3}{5184}\right )}-\frac {209\,\sqrt {2}\,b^3\,x\,\sqrt {-a^7}}{10368\,\left (-\frac {a^{13}}{162}+\frac {23\,a^{10}\,b}{648}-\frac {85\,a^7\,b^2}{1296}+\frac {209\,a^4\,b^3}{5184}\right )}+\frac {23\,\sqrt {2}\,b\,x\,\sqrt {-a^7}}{1296\,\left (\frac {85\,a\,b^2}{1296}-\frac {23\,a^4\,b}{648}+\frac {a^7}{162}-\frac {209\,b^3}{5184\,a^2}\right )}\right )\,\left (11\,b-4\,a^3\right )\,\sqrt {-a^7}}{216\,a^7}-\frac {\mathrm {atanh}\left (\frac {263\,b^2\,x}{1296\,\sqrt {a^7}\,\left (\frac {11\,b}{81}-\frac {5\,a^3}{162}-\frac {263\,b^2}{1296\,a^3}+\frac {133\,b^3}{1296\,a^6}\right )}-\frac {133\,b^3\,x}{1296\,\sqrt {a^7}\,\left (\frac {11\,a^3\,b}{81}-\frac {5\,a^6}{162}-\frac {263\,b^2}{1296}+\frac {133\,b^3}{1296\,a^3}\right )}+\frac {5\,a^5\,x}{162\,\sqrt {a^7}\,\left (\frac {11\,b}{81\,a}-\frac {5\,a^2}{162}-\frac {263\,b^2}{1296\,a^4}+\frac {133\,b^3}{1296\,a^7}\right )}-\frac {11\,a^2\,b\,x}{81\,\sqrt {a^7}\,\left (\frac {11\,b}{81\,a}-\frac {5\,a^2}{162}-\frac {263\,b^2}{1296\,a^4}+\frac {133\,b^3}{1296\,a^7}\right )}\right )\,\left (7\,b-5\,a^3\right )}{54\,\sqrt {a^7}}-\frac {\frac {x\,\left (8\,a^3+5\,b\right )}{36\,a^2}+\frac {x^3\,\left (b-2\,a^3\right )}{36\,a^3}}{-2\,a^2+a\,x^2+x^4} \] Input:
int((b + a*x^4)/((a - x^2)^2*(2*a + x^2)^2),x)
Output:
(2^(1/2)*atanh((2^(1/2)*x*(-a^7)^(1/2))/(324*((23*a*b)/648 - a^4/162 - (85 *b^2)/(1296*a^2) + (209*b^3)/(5184*a^5))) + (85*2^(1/2)*b^2*x*(-a^7)^(1/2) )/(2592*((209*a*b^3)/5184 + (23*a^7*b)/648 - a^10/162 - (85*a^4*b^2)/1296) ) - (209*2^(1/2)*b^3*x*(-a^7)^(1/2))/(10368*((23*a^10*b)/648 - a^13/162 + (209*a^4*b^3)/5184 - (85*a^7*b^2)/1296)) + (23*2^(1/2)*b*x*(-a^7)^(1/2))/( 1296*((85*a*b^2)/1296 - (23*a^4*b)/648 + a^7/162 - (209*b^3)/(5184*a^2)))) *(11*b - 4*a^3)*(-a^7)^(1/2))/(216*a^7) - (atanh((263*b^2*x)/(1296*(a^7)^( 1/2)*((11*b)/81 - (5*a^3)/162 - (263*b^2)/(1296*a^3) + (133*b^3)/(1296*a^6 ))) - (133*b^3*x)/(1296*(a^7)^(1/2)*((11*a^3*b)/81 - (5*a^6)/162 - (263*b^ 2)/1296 + (133*b^3)/(1296*a^3))) + (5*a^5*x)/(162*(a^7)^(1/2)*((11*b)/(81* a) - (5*a^2)/162 - (263*b^2)/(1296*a^4) + (133*b^3)/(1296*a^7))) - (11*a^2 *b*x)/(81*(a^7)^(1/2)*((11*b)/(81*a) - (5*a^2)/162 - (263*b^2)/(1296*a^4) + (133*b^3)/(1296*a^7))))*(7*b - 5*a^3))/(54*(a^7)^(1/2)) - ((x*(5*b + 8*a ^3))/(36*a^2) + (x^3*(b - 2*a^3))/(36*a^3))/(a*x^2 - 2*a^2 + x^4)
Time = 0.16 (sec) , antiderivative size = 359, normalized size of antiderivative = 3.15 \[ \int \frac {b+a x^4}{\left (-a+x^2\right )^2 \left (2 a+x^2\right )^2} \, dx=\frac {-8 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {a}\, \sqrt {2}}\right ) a^{5}+4 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {a}\, \sqrt {2}}\right ) a^{4} x^{2}+4 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {a}\, \sqrt {2}}\right ) a^{3} x^{4}+22 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {a}\, \sqrt {2}}\right ) a^{2} b -11 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {a}\, \sqrt {2}}\right ) a b \,x^{2}-11 \sqrt {a}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {a}\, \sqrt {2}}\right ) b \,x^{4}+20 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}-x \right ) a^{5}-10 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}-x \right ) a^{4} x^{2}-10 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}-x \right ) a^{3} x^{4}-28 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}-x \right ) a^{2} b +14 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}-x \right ) a b \,x^{2}+14 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}-x \right ) b \,x^{4}-20 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+x \right ) a^{5}+10 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+x \right ) a^{4} x^{2}+10 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+x \right ) a^{3} x^{4}+28 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+x \right ) a^{2} b -14 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+x \right ) a b \,x^{2}-14 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+x \right ) b \,x^{4}+48 a^{5} x -12 a^{4} x^{3}+30 a^{2} b x +6 a b \,x^{3}}{216 a^{4} \left (-x^{4}-a \,x^{2}+2 a^{2}\right )} \] Input:
int((a*x^4+b)/(x^2-a)^2/(x^2+2*a)^2,x)
Output:
( - 8*sqrt(a)*sqrt(2)*atan(x/(sqrt(a)*sqrt(2)))*a**5 + 4*sqrt(a)*sqrt(2)*a tan(x/(sqrt(a)*sqrt(2)))*a**4*x**2 + 4*sqrt(a)*sqrt(2)*atan(x/(sqrt(a)*sqr t(2)))*a**3*x**4 + 22*sqrt(a)*sqrt(2)*atan(x/(sqrt(a)*sqrt(2)))*a**2*b - 1 1*sqrt(a)*sqrt(2)*atan(x/(sqrt(a)*sqrt(2)))*a*b*x**2 - 11*sqrt(a)*sqrt(2)* atan(x/(sqrt(a)*sqrt(2)))*b*x**4 + 20*sqrt(a)*log(sqrt(a) - x)*a**5 - 10*s qrt(a)*log(sqrt(a) - x)*a**4*x**2 - 10*sqrt(a)*log(sqrt(a) - x)*a**3*x**4 - 28*sqrt(a)*log(sqrt(a) - x)*a**2*b + 14*sqrt(a)*log(sqrt(a) - x)*a*b*x** 2 + 14*sqrt(a)*log(sqrt(a) - x)*b*x**4 - 20*sqrt(a)*log(sqrt(a) + x)*a**5 + 10*sqrt(a)*log(sqrt(a) + x)*a**4*x**2 + 10*sqrt(a)*log(sqrt(a) + x)*a**3 *x**4 + 28*sqrt(a)*log(sqrt(a) + x)*a**2*b - 14*sqrt(a)*log(sqrt(a) + x)*a *b*x**2 - 14*sqrt(a)*log(sqrt(a) + x)*b*x**4 + 48*a**5*x - 12*a**4*x**3 + 30*a**2*b*x + 6*a*b*x**3)/(216*a**4*(2*a**2 - a*x**2 - x**4))