Integrand size = 23, antiderivative size = 244 \[ \int \left (1-b x^2\right ) \left (-1-b^2 x^4\right )^{5/2} \, dx=\frac {4 x \left (195-77 b x^2\right ) \sqrt {-1-b^2 x^4}}{3003}-\frac {8 x \sqrt {-1-b^2 x^4}}{39 \left (1+b x^2\right )}-\frac {10 x \left (117-77 b x^2\right ) \left (-1-b^2 x^4\right )^{3/2}}{9009}+\frac {1}{143} x \left (13-11 b x^2\right ) \left (-1-b^2 x^4\right )^{5/2}-\frac {8 \left (1+b x^2\right ) \sqrt {\frac {1+b^2 x^4}{\left (1+b x^2\right )^2}} E\left (2 \arctan \left (\sqrt {b} x\right )|\frac {1}{2}\right )}{39 \sqrt {b} \sqrt {-1-b^2 x^4}}-\frac {472 \left (1+b x^2\right ) \sqrt {\frac {1+b^2 x^4}{\left (1+b x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {b} x\right ),\frac {1}{2}\right )}{3003 \sqrt {b} \sqrt {-1-b^2 x^4}} \] 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)*Ellipti cE(sin(2*arctan(b^(1/2)*x)),1/2*2^(1/2))/b^(1/2)/(-b^2*x^4-1)^(1/2)-472/30 03*(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 = 8.89 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.31 \[ \int \left (1-b x^2\right ) \left (-1-b^2 x^4\right )^{5/2} \, dx=-\frac {\sqrt {-1-b^2 x^4} \left (-3 x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},-b^2 x^4\right )+b x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},-b^2 x^4\right )\right )}{3 \sqrt {1+b^2 x^4}} \] Input:
Integrate[(1 - b*x^2)*(-1 - b^2*x^4)^(5/2),x]
Output:
-1/3*(Sqrt[-1 - b^2*x^4]*(-3*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)]))/Sqrt[1 + b^2*x^ 4]
Time = 0.68 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, 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 |
| \(\displaystyle \frac {1}{143} x \left (13-11 b x^2\right ) \left (-b^2 x^4-1\right )^{5/2}-\frac {10}{143} \int \left (13-11 b x^2\right ) \left (-b^2 x^4-1\right )^{3/2}dx\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {1}{143} x \left (13-11 b x^2\right ) \left (-b^2 x^4-1\right )^{5/2}-\frac {10}{143} \left (\frac {1}{21} \int -2 \left (117-77 b x^2\right ) \sqrt {-b^2 x^4-1}dx+\frac {1}{63} x \left (117-77 b x^2\right ) \left (-b^2 x^4-1\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{143} x \left (13-11 b x^2\right ) \left (-b^2 x^4-1\right )^{5/2}-\frac {10}{143} \left (\frac {1}{63} x \left (117-77 b x^2\right ) \left (-b^2 x^4-1\right )^{3/2}-\frac {2}{21} \int \left (117-77 b x^2\right ) \sqrt {-b^2 x^4-1}dx\right )\) |
| \(\Big \downarrow \) 1491 |
| \(\displaystyle \frac {1}{143} x \left (13-11 b x^2\right ) \left (-b^2 x^4-1\right )^{5/2}-\frac {10}{143} \left (\frac {1}{63} x \left (117-77 b x^2\right ) \left (-b^2 x^4-1\right )^{3/2}-\frac {2}{21} \left (\frac {1}{15} \int -\frac {6 \left (195-77 b x^2\right )}{\sqrt {-b^2 x^4-1}}dx+\frac {1}{5} x \sqrt {-b^2 x^4-1} \left (195-77 b x^2\right )\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{143} x \left (13-11 b x^2\right ) \left (-b^2 x^4-1\right )^{5/2}-\frac {10}{143} \left (\frac {1}{63} x \left (117-77 b x^2\right ) \left (-b^2 x^4-1\right )^{3/2}-\frac {2}{21} \left (\frac {1}{5} x \left (195-77 b x^2\right ) \sqrt {-b^2 x^4-1}-\frac {2}{5} \int \frac {195-77 b x^2}{\sqrt {-b^2 x^4-1}}dx\right )\right )\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {1}{143} x \left (13-11 b x^2\right ) \left (-b^2 x^4-1\right )^{5/2}-\frac {10}{143} \left (\frac {1}{63} x \left (117-77 b x^2\right ) \left (-b^2 x^4-1\right )^{3/2}-\frac {2}{21} \left (\frac {1}{5} x \left (195-77 b x^2\right ) \sqrt {-b^2 x^4-1}-\frac {2}{5} \left (118 \int \frac {1}{\sqrt {-b^2 x^4-1}}dx+77 \int \frac {1-b x^2}{\sqrt {-b^2 x^4-1}}dx\right )\right )\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {1}{143} x \left (13-11 b x^2\right ) \left (-b^2 x^4-1\right )^{5/2}-\frac {10}{143} \left (\frac {1}{63} x \left (117-77 b x^2\right ) \left (-b^2 x^4-1\right )^{3/2}-\frac {2}{21} \left (\frac {1}{5} x \left (195-77 b x^2\right ) \sqrt {-b^2 x^4-1}-\frac {2}{5} \left (77 \int \frac {1-b x^2}{\sqrt {-b^2 x^4-1}}dx+\frac {59 \left (b x^2+1\right ) \sqrt {\frac {b^2 x^4+1}{\left (b x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {b} x\right ),\frac {1}{2}\right )}{\sqrt {b} \sqrt {-b^2 x^4-1}}\right )\right )\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {1}{143} x \left (13-11 b x^2\right ) \left (-b^2 x^4-1\right )^{5/2}-\frac {10}{143} \left (\frac {1}{63} x \left (117-77 b x^2\right ) \left (-b^2 x^4-1\right )^{3/2}-\frac {2}{21} \left (\frac {1}{5} x \left (195-77 b x^2\right ) \sqrt {-b^2 x^4-1}-\frac {2}{5} \left (\frac {59 \left (b x^2+1\right ) \sqrt {\frac {b^2 x^4+1}{\left (b x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {b} x\right ),\frac {1}{2}\right )}{\sqrt {b} \sqrt {-b^2 x^4-1}}+77 \left (\frac {\left (b x^2+1\right ) \sqrt {\frac {b^2 x^4+1}{\left (b x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt {b} x\right )|\frac {1}{2}\right )}{\sqrt {b} \sqrt {-b^2 x^4-1}}+\frac {x \sqrt {-b^2 x^4-1}}{b x^2+1}\right )\right )\right )\right )\) |
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* ArcTan[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 = 2.43 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.37
| method | result | size |
| meijerg | \(\frac {{\left (-\operatorname {signum}\left (b^{2} x^{4}+1\right )\right )}^{\frac {5}{2}} x \operatorname {hypergeom}\left (\left [-\frac {5}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -b^{2} x^{4}\right )}{\operatorname {signum}\left (b^{2} x^{4}+1\right )^{\frac {5}{2}}}-\frac {b {\left (-\operatorname {signum}\left (b^{2} x^{4}+1\right )\right )}^{\frac {5}{2}} x^{3} \operatorname {hypergeom}\left (\left [-\frac {5}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -b^{2} x^{4}\right )}{3 \operatorname {signum}\left (b^{2} x^{4}+1\right )^{\frac {5}{2}}}\) | \(90\) |
| risch | \(\frac {x \left (693 x^{10} b^{5}-819 b^{4} x^{8}+2156 b^{3} x^{6}-2808 b^{2} x^{4}+2387 b \,x^{2}-4329\right ) \left (b^{2} x^{4}+1\right )}{9009 \sqrt {-b^{2} x^{4}-1}}-\frac {40 \sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-i b}, i\right )}{77 \sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}-\frac {8 i \sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-i b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-i b}, i\right )\right )}{39 \sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}\) | \(187\) |
| elliptic | \(-\frac {b^{5} x^{11} \sqrt {-b^{2} x^{4}-1}}{13}+\frac {b^{4} x^{9} \sqrt {-b^{2} x^{4}-1}}{11}-\frac {28 b^{3} x^{7} \sqrt {-b^{2} x^{4}-1}}{117}+\frac {24 b^{2} x^{5} \sqrt {-b^{2} x^{4}-1}}{77}-\frac {31 b \,x^{3} \sqrt {-b^{2} x^{4}-1}}{117}+\frac {37 x \sqrt {-b^{2} x^{4}-1}}{77}-\frac {40 \sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-i b}, i\right )}{77 \sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}-\frac {8 i \sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-i b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-i b}, i\right )\right )}{39 \sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}\) | \(236\) |
| default | \(-b \left (\frac {b^{4} x^{11} \sqrt {-b^{2} x^{4}-1}}{13}+\frac {28 b^{2} x^{7} \sqrt {-b^{2} x^{4}-1}}{117}+\frac {31 x^{3} \sqrt {-b^{2} x^{4}-1}}{117}+\frac {8 i \sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-i b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-i b}, i\right )\right )}{39 \sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}\, b}\right )+\frac {b^{4} x^{9} \sqrt {-b^{2} x^{4}-1}}{11}+\frac {24 b^{2} x^{5} \sqrt {-b^{2} x^{4}-1}}{77}+\frac {37 x \sqrt {-b^{2} x^{4}-1}}{77}-\frac {40 \sqrt {i b \,x^{2}+1}\, \sqrt {-i b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-i b}, i\right )}{77 \sqrt {-i b}\, \sqrt {-b^{2} x^{4}-1}}\) | \(242\) |
Input:
int((-b*x^2+1)*(-b^2*x^4-1)^(5/2),x,method=_RETURNVERBOSE)
Output:
(-signum(b^2*x^4+1))^(5/2)/signum(b^2*x^4+1)^(5/2)*x*hypergeom([-5/2,1/4], [5/4],-b^2*x^4)-1/3*b*(-signum(b^2*x^4+1))^(5/2)/signum(b^2*x^4+1)^(5/2)*x ^3*hypergeom([-5/2,3/4],[7/4],-b^2*x^4)
Time = 0.07 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.57 \[ \int \left (1-b x^2\right ) \left (-1-b^2 x^4\right )^{5/2} \, dx=\frac {24 \, \sqrt {-b^{2}} {\left (195 \, b + 77\right )} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 1848 \, \sqrt {-b^{2}} x \left (-\frac {1}{b^{2}}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {1}{b^{2}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (693 \, b^{6} x^{12} - 819 \, b^{5} x^{10} + 2156 \, b^{4} x^{8} - 2808 \, b^{3} x^{6} + 2387 \, b^{2} x^{4} - 4329 \, b x^{2} + 1848\right )} \sqrt {-b^{2} x^{4} - 1}}{9009 \, b x} \] Input:
integrate((-b*x^2+1)*(-b^2*x^4-1)^(5/2),x, algorithm="fricas")
Output:
1/9009*(24*sqrt(-b^2)*(195*b + 77)*x*(-1/b^2)^(3/4)*elliptic_f(arcsin((-1/ b^2)^(1/4)/x), -1) - 1848*sqrt(-b^2)*x*(-1/b^2)^(3/4)*elliptic_e(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 = 2.56 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.97 \[ \int \left (1-b x^2\right ) \left (-1-b^2 x^4\right )^{5/2} \, dx=- \frac {i b^{5} x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} + \frac {i b^{4} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} - \frac {i b^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {i b^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} - \frac {i b x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {b^{2} x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((-b*x**2+1)*(-b**2*x**4-1)**(5/2),x)
Output:
-I*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)) + I*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)) - I*b**3*x**7*gamma(7/4)*hyp er((-1/2, 7/4), (11/4,), b**2*x**4*exp_polar(I*pi))/(2*gamma(11/4)) + I*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)) - I*b*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b**2*x**4*e xp_polar(I*pi))/(4*gamma(7/4)) + I*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b**2*x**4*exp_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)
\[ \int \left (1-b x^2\right ) \left (-1-b^2 x^4\right )^{5/2} \, dx=\frac {i \left (-693 \sqrt {b^{2} x^{4}+1}\, b^{5} x^{11}+819 \sqrt {b^{2} x^{4}+1}\, b^{4} x^{9}-2156 \sqrt {b^{2} x^{4}+1}\, b^{3} x^{7}+2808 \sqrt {b^{2} x^{4}+1}\, b^{2} x^{5}-2387 \sqrt {b^{2} x^{4}+1}\, b \,x^{3}+4329 \sqrt {b^{2} x^{4}+1}\, x +4680 \left (\int \frac {\sqrt {b^{2} x^{4}+1}}{b^{2} x^{4}+1}d x \right )-1848 \left (\int \frac {\sqrt {b^{2} x^{4}+1}\, x^{2}}{b^{2} x^{4}+1}d x \right ) b \right )}{9009} \] Input:
int((-b*x^2+1)*(-b^2*x^4-1)^(5/2),x)
Output:
(i*( - 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