Integrand size = 22, antiderivative size = 238 \[ \int \left (1-b x^2\right ) \left (1+b^2 x^4\right )^{5/2} \, dx=\frac {4 x \left (195-77 b x^2\right ) \sqrt {1+b^2 x^4}}{3003}-\frac {8 x \sqrt {1+b^2 x^4}}{39 \left (1+b x^2\right )}+\frac {10 x \left (117-77 b x^2\right ) \left (1+b^2 x^4\right )^{3/2}}{9009}+\frac {1}{143} x \left (13-11 b x^2\right ) \left (1+b^2 x^4\right )^{5/2}+\frac {8 \left (1+b x^2\right ) \sqrt {\frac {1+b^2 x^4}{\left (1+b x^2\right )^2}} E\left (2 \arctan \left (\sqrt {b} x\right )|\frac {1}{2}\right )}{39 \sqrt {b} \sqrt {1+b^2 x^4}}+\frac {472 \left (1+b x^2\right ) \sqrt {\frac {1+b^2 x^4}{\left (1+b x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {b} x\right ),\frac {1}{2}\right )}{3003 \sqrt {b} \sqrt {1+b^2 x^4}} \] Output:
4/3003*x*(-77*b*x^2+195)*(b^2*x^4+1)^(1/2)-8*x*(b^2*x^4+1)^(1/2)/(39*b*x^2 +39)+10/9009*x*(-77*b*x^2+117)*(b^2*x^4+1)^(3/2)+1/143*x*(-11*b*x^2+13)*(b ^2*x^4+1)^(5/2)+8/39*(b*x^2+1)*((b^2*x^4+1)/(b*x^2+1)^2)^(1/2)*EllipticE(s in(2*arctan(b^(1/2)*x)),1/2*2^(1/2))/b^(1/2)/(b^2*x^4+1)^(1/2)+472/3003*(b *x^2+1)*((b^2*x^4+1)/(b*x^2+1)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/2)*x ),1/2*2^(1/2))/b^(1/2)/(b^2*x^4+1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.79 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.20 \[ \int \left (1-b x^2\right ) \left (1+b^2 x^4\right )^{5/2} \, dx=x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},-b^2 x^4\right )-\frac {1}{3} b x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},-b^2 x^4\right ) \] Input:
Integrate[(1 - b*x^2)*(1 + b^2*x^4)^(5/2),x]
Output:
x*Hypergeometric2F1[-5/2, 1/4, 5/4, -(b^2*x^4)] - (b*x^3*Hypergeometric2F1 [-5/2, 3/4, 7/4, -(b^2*x^4)])/3
Time = 0.65 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1491, 27, 1491, 27, 1491, 27, 1512, 761, 1510}
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.
| \(\displaystyle \int \left (1-b x^2\right ) \left (b^2 x^4+1\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {5}{143} \int 2 \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{3/2}dx+\frac {1}{143} x \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} \int \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{3/2}dx+\frac {1}{143} x \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{5/2}\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {10}{143} \left (\frac {1}{21} \int 2 \left (117-77 b x^2\right ) \sqrt {b^2 x^4+1}dx+\frac {1}{63} x \left (117-77 b x^2\right ) \left (b^2 x^4+1\right )^{3/2}\right )+\frac {1}{143} x \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} \left (\frac {2}{21} \int \left (117-77 b x^2\right ) \sqrt {b^2 x^4+1}dx+\frac {1}{63} x \left (117-77 b x^2\right ) \left (b^2 x^4+1\right )^{3/2}\right )+\frac {1}{143} x \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{5/2}\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {10}{143} \left (\frac {2}{21} \left (\frac {1}{15} \int \frac {6 \left (195-77 b x^2\right )}{\sqrt {b^2 x^4+1}}dx+\frac {1}{5} x \sqrt {b^2 x^4+1} \left (195-77 b x^2\right )\right )+\frac {1}{63} x \left (117-77 b x^2\right ) \left (b^2 x^4+1\right )^{3/2}\right )+\frac {1}{143} x \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10}{143} \left (\frac {2}{21} \left (\frac {2}{5} \int \frac {195-77 b x^2}{\sqrt {b^2 x^4+1}}dx+\frac {1}{5} x \sqrt {b^2 x^4+1} \left (195-77 b x^2\right )\right )+\frac {1}{63} x \left (117-77 b x^2\right ) \left (b^2 x^4+1\right )^{3/2}\right )+\frac {1}{143} x \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{5/2}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {10}{143} \left (\frac {2}{21} \left (\frac {2}{5} \left (118 \int \frac {1}{\sqrt {b^2 x^4+1}}dx+77 \int \frac {1-b x^2}{\sqrt {b^2 x^4+1}}dx\right )+\frac {1}{5} x \sqrt {b^2 x^4+1} \left (195-77 b x^2\right )\right )+\frac {1}{63} x \left (117-77 b x^2\right ) \left (b^2 x^4+1\right )^{3/2}\right )+\frac {1}{143} x \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{5/2}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {10}{143} \left (\frac {2}{21} \left (\frac {2}{5} \left (77 \int \frac {1-b x^2}{\sqrt {b^2 x^4+1}}dx+\frac {59 \left (b x^2+1\right ) \sqrt {\frac {b^2 x^4+1}{\left (b x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {b} x\right ),\frac {1}{2}\right )}{\sqrt {b} \sqrt {b^2 x^4+1}}\right )+\frac {1}{5} x \sqrt {b^2 x^4+1} \left (195-77 b x^2\right )\right )+\frac {1}{63} x \left (117-77 b x^2\right ) \left (b^2 x^4+1\right )^{3/2}\right )+\frac {1}{143} x \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{5/2}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {10}{143} \left (\frac {2}{21} \left (\frac {2}{5} \left (\frac {59 \left (b x^2+1\right ) \sqrt {\frac {b^2 x^4+1}{\left (b x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {b} x\right ),\frac {1}{2}\right )}{\sqrt {b} \sqrt {b^2 x^4+1}}+77 \left (\frac {\left (b x^2+1\right ) \sqrt {\frac {b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt {b} x\right )|\frac {1}{2}\right )}{\sqrt {b} \sqrt {b^2 x^4+1}}-\frac {x \sqrt {b^2 x^4+1}}{b x^2+1}\right )\right )+\frac {1}{5} x \sqrt {b^2 x^4+1} \left (195-77 b x^2\right )\right )+\frac {1}{63} x \left (117-77 b x^2\right ) \left (b^2 x^4+1\right )^{3/2}\right )+\frac {1}{143} x \left (13-11 b x^2\right ) \left (b^2 x^4+1\right )^{5/2}\) |
Input:
Int[(1 - b*x^2)*(1 + b^2*x^4)^(5/2),x]
Output:
(x*(13 - 11*b*x^2)*(1 + b^2*x^4)^(5/2))/143 + (10*((x*(117 - 77*b*x^2)*(1 + b^2*x^4)^(3/2))/63 + (2*((x*(195 - 77*b*x^2)*Sqrt[1 + b^2*x^4])/5 + (2*( 77*(-((x*Sqrt[1 + b^2*x^4])/(1 + b*x^2)) + ((1 + b*x^2)*Sqrt[(1 + b^2*x^4) /(1 + b*x^2)^2]*EllipticE[2*ArcTan[Sqrt[b]*x], 1/2])/(Sqrt[b]*Sqrt[1 + b^2 *x^4])) + (59*(1 + b*x^2)*Sqrt[(1 + b^2*x^4)/(1 + b*x^2)^2]*EllipticF[2*Ar cTan[Sqrt[b]*x], 1/2])/(Sqrt[b]*Sqrt[1 + b^2*x^4])))/5))/21))/143
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*( d*(4*p + 3) + e*(4*p + 1)*x^2)*((a + c*x^4)^p/((4*p + 1)*(4*p + 3))), x] + Simp[2*(p/((4*p + 1)*(4*p + 3))) Int[Simp[2*a*d*(4*p + 3) + (2*a*e*(4*p + 1))*x^2, x]*(a + c*x^4)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c *d^2 + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Time = 1.36 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.16
| method | result | size |
| meijerg | \(x \operatorname {hypergeom}\left (\left [-\frac {5}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -b^{2} x^{4}\right )-\frac {b \,x^{3} \operatorname {hypergeom}\left (\left [-\frac {5}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -b^{2} x^{4}\right )}{3}\) | \(38\) |
| risch | \(-\frac {x \left (693 x^{10} b^{5}-819 b^{4} x^{8}+2156 b^{3} x^{6}-2808 b^{2} x^{4}+2387 b \,x^{2}-4329\right ) \sqrt {b^{2} x^{4}+1}}{9009}+\frac {40 \sqrt {-i b \,x^{2}+1}\, \sqrt {i b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i b}, i\right )}{77 \sqrt {i b}\, \sqrt {b^{2} x^{4}+1}}-\frac {8 i \sqrt {-i b \,x^{2}+1}\, \sqrt {i b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {i b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {i b}, i\right )\right )}{39 \sqrt {i b}\, \sqrt {b^{2} x^{4}+1}}\) | \(175\) |
| elliptic | \(-\frac {b^{5} x^{11} \sqrt {b^{2} x^{4}+1}}{13}+\frac {b^{4} x^{9} \sqrt {b^{2} x^{4}+1}}{11}-\frac {28 b^{3} x^{7} \sqrt {b^{2} x^{4}+1}}{117}+\frac {24 b^{2} x^{5} \sqrt {b^{2} x^{4}+1}}{77}-\frac {31 b \,x^{3} \sqrt {b^{2} x^{4}+1}}{117}+\frac {37 x \sqrt {b^{2} x^{4}+1}}{77}+\frac {40 \sqrt {-i b \,x^{2}+1}\, \sqrt {i b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i b}, i\right )}{77 \sqrt {i b}\, \sqrt {b^{2} x^{4}+1}}-\frac {8 i \sqrt {-i b \,x^{2}+1}\, \sqrt {i b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {i b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {i b}, i\right )\right )}{39 \sqrt {i b}\, \sqrt {b^{2} x^{4}+1}}\) | \(228\) |
| default | \(-b \left (\frac {b^{4} x^{11} \sqrt {b^{2} x^{4}+1}}{13}+\frac {28 b^{2} x^{7} \sqrt {b^{2} x^{4}+1}}{117}+\frac {31 x^{3} \sqrt {b^{2} x^{4}+1}}{117}+\frac {8 i \sqrt {-i b \,x^{2}+1}\, \sqrt {i b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {i b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {i b}, i\right )\right )}{39 \sqrt {i b}\, \sqrt {b^{2} x^{4}+1}\, b}\right )+\frac {b^{4} x^{9} \sqrt {b^{2} x^{4}+1}}{11}+\frac {24 b^{2} x^{5} \sqrt {b^{2} x^{4}+1}}{77}+\frac {37 x \sqrt {b^{2} x^{4}+1}}{77}+\frac {40 \sqrt {-i b \,x^{2}+1}\, \sqrt {i b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i b}, i\right )}{77 \sqrt {i b}\, \sqrt {b^{2} x^{4}+1}}\) | \(234\) |
Input:
int((-b*x^2+1)*(b^2*x^4+1)^(5/2),x,method=_RETURNVERBOSE)
Output:
x*hypergeom([-5/2,1/4],[5/4],-b^2*x^4)-1/3*b*x^3*hypergeom([-5/2,3/4],[7/4 ],-b^2*x^4)
Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.53 \[ \int \left (1-b x^2\right ) \left (1+b^2 x^4\right )^{5/2} \, dx=-\frac {1848 \, b x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 24 \, {\left (195 \, b^{2} + 77 \, b\right )} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (693 \, b^{6} x^{12} - 819 \, b^{5} x^{10} + 2156 \, b^{4} x^{8} - 2808 \, b^{3} x^{6} + 2387 \, b^{2} x^{4} - 4329 \, b x^{2} + 1848\right )} \sqrt {b^{2} x^{4} + 1}}{9009 \, b x} \] Input:
integrate((-b*x^2+1)*(b^2*x^4+1)^(5/2),x, algorithm="fricas")
Output:
-1/9009*(1848*b*x*(-1/b^2)^(3/4)*elliptic_e(arcsin((-1/b^2)^(1/4)/x), -1) - 24*(195*b^2 + 77*b)*x*(-1/b^2)^(3/4)*elliptic_f(arcsin((-1/b^2)^(1/4)/x) , -1) + (693*b^6*x^12 - 819*b^5*x^10 + 2156*b^4*x^8 - 2808*b^3*x^6 + 2387* b^2*x^4 - 4329*b*x^2 + 1848)*sqrt(b^2*x^4 + 1))/(b*x)
Result contains complex when optimal does not.
Time = 2.32 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.95 \[ \int \left (1-b x^2\right ) \left (1+b^2 x^4\right )^{5/2} \, dx=- \frac {b^{5} x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} + \frac {b^{4} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} - \frac {b^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {b^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} - \frac {b x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((-b*x**2+1)*(b**2*x**4+1)**(5/2),x)
Output:
-b**5*x**11*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), b**2*x**4*exp_polar(I *pi))/(4*gamma(15/4)) + b**4*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), b **2*x**4*exp_polar(I*pi))/(4*gamma(13/4)) - b**3*x**7*gamma(7/4)*hyper((-1 /2, 7/4), (11/4,), b**2*x**4*exp_polar(I*pi))/(2*gamma(11/4)) + b**2*x**5* gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b**2*x**4*exp_polar(I*pi))/(2*gamma( 9/4)) - b*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b**2*x**4*exp_polar(I *pi))/(4*gamma(7/4)) + x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b**2*x**4*e xp_polar(I*pi))/(4*gamma(5/4))
\[ \int \left (1-b x^2\right ) \left (1+b^2 x^4\right )^{5/2} \, dx=\int { -{\left (b^{2} x^{4} + 1\right )}^{\frac {5}{2}} {\left (b x^{2} - 1\right )} \,d x } \] Input:
integrate((-b*x^2+1)*(b^2*x^4+1)^(5/2),x, algorithm="maxima")
Output:
-integrate((b^2*x^4 + 1)^(5/2)*(b*x^2 - 1), x)
\[ \int \left (1-b x^2\right ) \left (1+b^2 x^4\right )^{5/2} \, dx=\int { -{\left (b^{2} x^{4} + 1\right )}^{\frac {5}{2}} {\left (b x^{2} - 1\right )} \,d x } \] Input:
integrate((-b*x^2+1)*(b^2*x^4+1)^(5/2),x, algorithm="giac")
Output:
integrate(-(b^2*x^4 + 1)^(5/2)*(b*x^2 - 1), x)
Timed out. \[ \int \left (1-b x^2\right ) \left (1+b^2 x^4\right )^{5/2} \, dx=-\int {\left (b^2\,x^4+1\right )}^{5/2}\,\left (b\,x^2-1\right ) \,d x \] Input:
int(-(b^2*x^4 + 1)^(5/2)*(b*x^2 - 1),x)
Output:
-int((b^2*x^4 + 1)^(5/2)*(b*x^2 - 1), x)
\[ \int \left (1-b x^2\right ) \left (1+b^2 x^4\right )^{5/2} \, dx=-\frac {\sqrt {b^{2} x^{4}+1}\, b^{5} x^{11}}{13}+\frac {\sqrt {b^{2} x^{4}+1}\, b^{4} x^{9}}{11}-\frac {28 \sqrt {b^{2} x^{4}+1}\, b^{3} x^{7}}{117}+\frac {24 \sqrt {b^{2} x^{4}+1}\, b^{2} x^{5}}{77}-\frac {31 \sqrt {b^{2} x^{4}+1}\, b \,x^{3}}{117}+\frac {37 \sqrt {b^{2} x^{4}+1}\, x}{77}+\frac {40 \left (\int \frac {\sqrt {b^{2} x^{4}+1}}{b^{2} x^{4}+1}d x \right )}{77}-\frac {8 \left (\int \frac {\sqrt {b^{2} x^{4}+1}\, x^{2}}{b^{2} x^{4}+1}d x \right ) b}{39} \] Input:
int((-b*x^2+1)*(b^2*x^4+1)^(5/2),x)
Output:
( - 693*sqrt(b**2*x**4 + 1)*b**5*x**11 + 819*sqrt(b**2*x**4 + 1)*b**4*x**9 - 2156*sqrt(b**2*x**4 + 1)*b**3*x**7 + 2808*sqrt(b**2*x**4 + 1)*b**2*x**5 - 2387*sqrt(b**2*x**4 + 1)*b*x**3 + 4329*sqrt(b**2*x**4 + 1)*x + 4680*int (sqrt(b**2*x**4 + 1)/(b**2*x**4 + 1),x) - 1848*int((sqrt(b**2*x**4 + 1)*x* *2)/(b**2*x**4 + 1),x)*b)/9009