Integrand size = 22, antiderivative size = 197 \[ \int \frac {1-b x^2}{\left (-1+b^2 x^4\right )^{9/2}} \, dx=-\frac {x \left (1-b x^2\right )}{14 \left (-1+b^2 x^4\right )^{7/2}}+\frac {x \left (13-11 b x^2\right )}{140 \left (-1+b^2 x^4\right )^{5/2}}-\frac {x \left (117-77 b x^2\right )}{840 \left (-1+b^2 x^4\right )^{3/2}}+\frac {x \left (195-77 b x^2\right )}{560 \sqrt {-1+b^2 x^4}}+\frac {11 \sqrt {1-b^2 x^4} E\left (\left .\arcsin \left (\sqrt {b} x\right )\right |-1\right )}{80 \sqrt {b} \sqrt {-1+b^2 x^4}}+\frac {59 \sqrt {1-b^2 x^4} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {b} x\right ),-1\right )}{280 \sqrt {b} \sqrt {-1+b^2 x^4}} \] Output:
-1/14*x*(-b*x^2+1)/(b^2*x^4-1)^(7/2)+1/140*x*(-11*b*x^2+13)/(b^2*x^4-1)^(5 /2)-1/840*x*(-77*b*x^2+117)/(b^2*x^4-1)^(3/2)+1/560*x*(-77*b*x^2+195)/(b^2 *x^4-1)^(1/2)+11/80*(-b^2*x^4+1)^(1/2)*EllipticE(b^(1/2)*x,I)/b^(1/2)/(b^2 *x^4-1)^(1/2)+59/280*(-b^2*x^4+1)^(1/2)*EllipticF(b^(1/2)*x,I)/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 = 10.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.58 \[ \int \frac {1-b x^2}{\left (-1+b^2 x^4\right )^{9/2}} \, dx=\frac {x \left (-1095+2379 b^2 x^4-1989 b^4 x^8+585 b^6 x^{12}-585 \left (1-b^2 x^4\right )^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},b^2 x^4\right )+560 b x^2 \left (1-b^2 x^4\right )^{7/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {9}{2},\frac {7}{4},b^2 x^4\right )\right )}{1680 \left (-1+b^2 x^4\right )^{7/2}} \] Input:
Integrate[(1 - b*x^2)/(-1 + b^2*x^4)^(9/2),x]
Output:
(x*(-1095 + 2379*b^2*x^4 - 1989*b^4*x^8 + 585*b^6*x^12 - 585*(1 - b^2*x^4) ^(7/2)*Hypergeometric2F1[1/4, 1/2, 5/4, b^2*x^4] + 560*b*x^2*(1 - b^2*x^4) ^(7/2)*Hypergeometric2F1[3/4, 9/2, 7/4, b^2*x^4]))/(1680*(-1 + b^2*x^4)^(7 /2))
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.
| \(\displaystyle \int \frac {1-b x^2}{\left (b^2 x^4-1\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {1-b x^2}{\left (b^2 x^4-1\right )^{9/2}}dx\) |
Input:
Int[(1 - b*x^2)/(-1 + b^2*x^4)^(9/2),x]
Output:
$Aborted
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.45
| method | result | size |
| meijerg | \(\frac {{\left (-\operatorname {signum}\left (b^{2} x^{4}-1\right )\right )}^{\frac {9}{2}} x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {9}{2}\right ], \left [\frac {5}{4}\right ], b^{2} x^{4}\right )}{\operatorname {signum}\left (b^{2} x^{4}-1\right )^{\frac {9}{2}}}-\frac {b {\left (-\operatorname {signum}\left (b^{2} x^{4}-1\right )\right )}^{\frac {9}{2}} x^{3} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {9}{2}\right ], \left [\frac {7}{4}\right ], b^{2} x^{4}\right )}{3 \operatorname {signum}\left (b^{2} x^{4}-1\right )^{\frac {9}{2}}}\) | \(88\) |
| elliptic | \(\frac {x \sqrt {b^{2} x^{4}-1}}{160 b^{3} \left (x^{2}-\frac {1}{b}\right )^{3}}-\frac {x \sqrt {b^{2} x^{4}-1}}{30 b^{2} \left (x^{2}-\frac {1}{b}\right )^{2}}+\frac {27 \left (b^{2} x^{2}+b \right ) x}{160 b \sqrt {\left (x^{2}-\frac {1}{b}\right ) \left (b^{2} x^{2}+b \right )}}-\frac {x \sqrt {b^{2} x^{4}-1}}{112 b^{4} \left (x^{2}+\frac {1}{b}\right )^{4}}-\frac {39 x \sqrt {b^{2} x^{4}-1}}{1120 b^{3} \left (x^{2}+\frac {1}{b}\right )^{3}}-\frac {157 x \sqrt {b^{2} x^{4}-1}}{1680 b^{2} \left (x^{2}+\frac {1}{b}\right )^{2}}-\frac {49 \left (b^{2} x^{2}-b \right ) x}{160 b \sqrt {\left (x^{2}+\frac {1}{b}\right ) \left (b^{2} x^{2}-b \right )}}+\frac {39 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-b}, i\right )}{112 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}}+\frac {11 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-b}, i\right )\right )}{80 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}}\) | \(317\) |
| default | \(-b \left (-\frac {x^{3} \sqrt {b^{2} x^{4}-1}}{14 b^{8} \left (x^{4}-\frac {1}{b^{2}}\right )^{4}}+\frac {11 x^{3} \sqrt {b^{2} x^{4}-1}}{140 b^{6} \left (x^{4}-\frac {1}{b^{2}}\right )^{3}}-\frac {11 x^{3} \sqrt {b^{2} x^{4}-1}}{120 b^{4} \left (x^{4}-\frac {1}{b^{2}}\right )^{2}}+\frac {11 x^{3}}{80 \sqrt {\left (x^{4}-\frac {1}{b^{2}}\right ) b^{2}}}-\frac {11 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-b}, i\right )\right )}{80 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}\, b}\right )-\frac {x \sqrt {b^{2} x^{4}-1}}{14 b^{8} \left (x^{4}-\frac {1}{b^{2}}\right )^{4}}+\frac {13 x \sqrt {b^{2} x^{4}-1}}{140 b^{6} \left (x^{4}-\frac {1}{b^{2}}\right )^{3}}-\frac {39 x \sqrt {b^{2} x^{4}-1}}{280 b^{4} \left (x^{4}-\frac {1}{b^{2}}\right )^{2}}+\frac {39 x}{112 \sqrt {\left (x^{4}-\frac {1}{b^{2}}\right ) b^{2}}}+\frac {39 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-b}, i\right )}{112 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}}\) | \(328\) |
Input:
int((-b*x^2+1)/(b^2*x^4-1)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/signum(b^2*x^4-1)^(9/2)*(-signum(b^2*x^4-1))^(9/2)*x*hypergeom([1/4,9/2] ,[5/4],b^2*x^4)-1/3*b/signum(b^2*x^4-1)^(9/2)*(-signum(b^2*x^4-1))^(9/2)*x ^3*hypergeom([3/4,9/2],[7/4],b^2*x^4)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (158) = 316\).
Time = 0.08 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.67 \[ \int \frac {1-b x^2}{\left (-1+b^2 x^4\right )^{9/2}} \, dx=-\frac {231 \, {\left (i \, b^{8} x^{14} + i \, b^{7} x^{12} - 3 i \, b^{6} x^{10} - 3 i \, b^{5} x^{8} + 3 i \, b^{4} x^{6} + 3 i \, b^{3} x^{4} - i \, b^{2} x^{2} - i \, b\right )} \sqrt {b} E(\arcsin \left (\sqrt {b} x\right )\,|\,-1) + 3 \, {\left (-i \, {\left (77 \, b^{8} - 195 \, b^{7}\right )} x^{14} - i \, {\left (77 \, b^{7} - 195 \, b^{6}\right )} x^{12} + 3 i \, {\left (77 \, b^{6} - 195 \, b^{5}\right )} x^{10} + 3 i \, {\left (77 \, b^{5} - 195 \, b^{4}\right )} x^{8} - 3 i \, {\left (77 \, b^{4} - 195 \, b^{3}\right )} x^{6} - 3 i \, {\left (77 \, b^{3} - 195 \, b^{2}\right )} x^{4} + i \, {\left (77 \, b^{2} - 195 \, b\right )} x^{2} + 77 i \, b - 195 i\right )} \sqrt {b} F(\arcsin \left (\sqrt {b} x\right )\,|\,-1) + {\left (231 \, b^{7} x^{13} - 354 \, b^{6} x^{11} - 1201 \, b^{5} x^{9} + 788 \, b^{4} x^{7} + 1921 \, b^{3} x^{5} - 458 \, b^{2} x^{3} - 1095 \, b x\right )} \sqrt {b^{2} x^{4} - 1}}{1680 \, {\left (b^{8} x^{14} + b^{7} x^{12} - 3 \, b^{6} x^{10} - 3 \, b^{5} x^{8} + 3 \, b^{4} x^{6} + 3 \, b^{3} x^{4} - b^{2} x^{2} - b\right )}} \] Input:
integrate((-b*x^2+1)/(b^2*x^4-1)^(9/2),x, algorithm="fricas")
Output:
-1/1680*(231*(I*b^8*x^14 + I*b^7*x^12 - 3*I*b^6*x^10 - 3*I*b^5*x^8 + 3*I*b ^4*x^6 + 3*I*b^3*x^4 - I*b^2*x^2 - I*b)*sqrt(b)*elliptic_e(arcsin(sqrt(b)* x), -1) + 3*(-I*(77*b^8 - 195*b^7)*x^14 - I*(77*b^7 - 195*b^6)*x^12 + 3*I* (77*b^6 - 195*b^5)*x^10 + 3*I*(77*b^5 - 195*b^4)*x^8 - 3*I*(77*b^4 - 195*b ^3)*x^6 - 3*I*(77*b^3 - 195*b^2)*x^4 + I*(77*b^2 - 195*b)*x^2 + 77*I*b - 1 95*I)*sqrt(b)*elliptic_f(arcsin(sqrt(b)*x), -1) + (231*b^7*x^13 - 354*b^6* x^11 - 1201*b^5*x^9 + 788*b^4*x^7 + 1921*b^3*x^5 - 458*b^2*x^3 - 1095*b*x) *sqrt(b^2*x^4 - 1))/(b^8*x^14 + b^7*x^12 - 3*b^6*x^10 - 3*b^5*x^8 + 3*b^4* x^6 + 3*b^3*x^4 - b^2*x^2 - b)
Time = 35.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.30 \[ \int \frac {1-b x^2}{\left (-1+b^2 x^4\right )^{9/2}} \, dx=\frac {i b x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {9}{2} \\ \frac {7}{4} \end {matrix}\middle | {b^{2} x^{4}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{2} \\ \frac {5}{4} \end {matrix}\middle | {b^{2} x^{4}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((-b*x**2+1)/(b**2*x**4-1)**(9/2),x)
Output:
I*b*x**3*gamma(3/4)*hyper((3/4, 9/2), (7/4,), b**2*x**4)/(4*gamma(7/4)) - I*x*gamma(1/4)*hyper((1/4, 9/2), (5/4,), b**2*x**4)/(4*gamma(5/4))
\[ \int \frac {1-b x^2}{\left (-1+b^2 x^4\right )^{9/2}} \, dx=\int { -\frac {b x^{2} - 1}{{\left (b^{2} x^{4} - 1\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((-b*x^2+1)/(b^2*x^4-1)^(9/2),x, algorithm="maxima")
Output:
-integrate((b*x^2 - 1)/(b^2*x^4 - 1)^(9/2), x)
\[ \int \frac {1-b x^2}{\left (-1+b^2 x^4\right )^{9/2}} \, dx=\int { -\frac {b x^{2} - 1}{{\left (b^{2} x^{4} - 1\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((-b*x^2+1)/(b^2*x^4-1)^(9/2),x, algorithm="giac")
Output:
integrate(-(b*x^2 - 1)/(b^2*x^4 - 1)^(9/2), x)
Timed out. \[ \int \frac {1-b x^2}{\left (-1+b^2 x^4\right )^{9/2}} \, dx=-\int \frac {b\,x^2-1}{{\left (b^2\,x^4-1\right )}^{9/2}} \,d x \] Input:
int(-(b*x^2 - 1)/(b^2*x^4 - 1)^(9/2),x)
Output:
-int((b*x^2 - 1)/(b^2*x^4 - 1)^(9/2), x)
\[ \int \frac {1-b x^2}{\left (-1+b^2 x^4\right )^{9/2}} \, dx=-\left (\int \frac {\sqrt {b^{2} x^{4}-1}}{b^{9} x^{18}+b^{8} x^{16}-4 b^{7} x^{14}-4 b^{6} x^{12}+6 b^{5} x^{10}+6 b^{4} x^{8}-4 b^{3} x^{6}-4 b^{2} x^{4}+b \,x^{2}+1}d x \right ) \] Input:
int((-b*x^2+1)/(b^2*x^4-1)^(9/2),x)
Output:
- int(sqrt(b**2*x**4 - 1)/(b**9*x**18 + b**8*x**16 - 4*b**7*x**14 - 4*b** 6*x**12 + 6*b**5*x**10 + 6*b**4*x**8 - 4*b**3*x**6 - 4*b**2*x**4 + b*x**2 + 1),x)