Integrand size = 22, antiderivative size = 145 \[ \int \left (1-b x^2\right ) \left (-1+b^2 x^4\right )^{3/2} \, dx=-\frac {2}{105} x \left (15-7 b x^2\right ) \sqrt {-1+b^2 x^4}+\frac {1}{63} x \left (9-7 b x^2\right ) \left (-1+b^2 x^4\right )^{3/2}-\frac {4 \sqrt {1-b^2 x^4} E\left (\left .\arcsin \left (\sqrt {b} x\right )\right |-1\right )}{15 \sqrt {b} \sqrt {-1+b^2 x^4}}+\frac {88 \sqrt {1-b^2 x^4} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {b} x\right ),-1\right )}{105 \sqrt {b} \sqrt {-1+b^2 x^4}} \] Output:
-2/105*x*(-7*b*x^2+15)*(b^2*x^4-1)^(1/2)+1/63*x*(-7*b*x^2+9)*(b^2*x^4-1)^( 3/2)-4/15*(-b^2*x^4+1)^(1/2)*EllipticE(b^(1/2)*x,I)/b^(1/2)/(b^2*x^4-1)^(1 /2)+88/105*(-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 = 7.65 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.51 \[ \int \left (1-b x^2\right ) \left (-1+b^2 x^4\right )^{3/2} \, dx=\frac {\sqrt {-1+b^2 x^4} \left (-3 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},b^2 x^4\right )+b x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},b^2 x^4\right )\right )}{3 \sqrt {1-b^2 x^4}} \] Input:
Integrate[(1 - b*x^2)*(-1 + b^2*x^4)^(3/2),x]
Output:
(Sqrt[-1 + b^2*x^4]*(-3*x*Hypergeometric2F1[-3/2, 1/4, 5/4, b^2*x^4] + b*x ^3*Hypergeometric2F1[-3/2, 3/4, 7/4, b^2*x^4]))/(3*Sqrt[1 - b^2*x^4])
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.
| \(\displaystyle \int \left (1-b x^2\right ) \left (b^2 x^4-1\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \left (1-b x^2\right ) \left (b^2 x^4-1\right )^{3/2}dx\) |
Input:
Int[(1 - b*x^2)*(-1 + b^2*x^4)^(3/2),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 = 2.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.61
| method | result | size |
| meijerg | \(\frac {\operatorname {signum}\left (b^{2} x^{4}-1\right )^{\frac {3}{2}} x \operatorname {hypergeom}\left (\left [-\frac {3}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], b^{2} x^{4}\right )}{{\left (-\operatorname {signum}\left (b^{2} x^{4}-1\right )\right )}^{\frac {3}{2}}}-\frac {b \operatorname {signum}\left (b^{2} x^{4}-1\right )^{\frac {3}{2}} x^{3} \operatorname {hypergeom}\left (\left [-\frac {3}{2}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], b^{2} x^{4}\right )}{3 {\left (-\operatorname {signum}\left (b^{2} x^{4}-1\right )\right )}^{\frac {3}{2}}}\) | \(88\) |
| risch | \(-\frac {x \left (35 b^{3} x^{6}-45 b^{2} x^{4}-77 b \,x^{2}+135\right ) \sqrt {b^{2} x^{4}-1}}{315}+\frac {4 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-b}, i\right )}{7 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}}-\frac {4 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-b}, i\right )\right )}{15 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}}\) | \(147\) |
| elliptic | \(-\frac {b^{3} x^{7} \sqrt {b^{2} x^{4}-1}}{9}+\frac {b^{2} x^{5} \sqrt {b^{2} x^{4}-1}}{7}+\frac {11 b \,x^{3} \sqrt {b^{2} x^{4}-1}}{45}-\frac {3 x \sqrt {b^{2} x^{4}-1}}{7}+\frac {4 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-b}, i\right )}{7 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}}-\frac {4 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-b}, i\right )\right )}{15 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}}\) | \(178\) |
| default | \(-b \left (\frac {b^{2} x^{7} \sqrt {b^{2} x^{4}-1}}{9}-\frac {11 x^{3} \sqrt {b^{2} x^{4}-1}}{45}+\frac {4 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-b}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-b}, i\right )\right )}{15 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}\, b}\right )+\frac {b^{2} x^{5} \sqrt {b^{2} x^{4}-1}}{7}-\frac {3 x \sqrt {b^{2} x^{4}-1}}{7}+\frac {4 \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-b}, i\right )}{7 \sqrt {-b}\, \sqrt {b^{2} x^{4}-1}}\) | \(184\) |
Input:
int((-b*x^2+1)*(b^2*x^4-1)^(3/2),x,method=_RETURNVERBOSE)
Output:
signum(b^2*x^4-1)^(3/2)/(-signum(b^2*x^4-1))^(3/2)*x*hypergeom([-3/2,1/4], [5/4],b^2*x^4)-1/3*b*signum(b^2*x^4-1)^(3/2)/(-signum(b^2*x^4-1))^(3/2)*x^ 3*hypergeom([-3/2,3/4],[7/4],b^2*x^4)
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.62 \[ \int \left (1-b x^2\right ) \left (-1+b^2 x^4\right )^{3/2} \, dx=-\frac {\frac {12 \, {\left (15 \, b - 7\right )} x F(\arcsin \left (\frac {1}{\sqrt {b} x}\right )\,|\,-1)}{\sqrt {b}} + {\left (35 \, b^{4} x^{8} - 45 \, b^{3} x^{6} - 77 \, b^{2} x^{4} + 135 \, b x^{2} + 84\right )} \sqrt {b^{2} x^{4} - 1} + \frac {84 \, x E(\arcsin \left (\frac {1}{\sqrt {b} x}\right )\,|\,-1)}{\sqrt {b}}}{315 \, b x} \] Input:
integrate((-b*x^2+1)*(b^2*x^4-1)^(3/2),x, algorithm="fricas")
Output:
-1/315*(12*(15*b - 7)*x*elliptic_f(arcsin(1/(sqrt(b)*x)), -1)/sqrt(b) + (3 5*b^4*x^8 - 45*b^3*x^6 - 77*b^2*x^4 + 135*b*x^2 + 84)*sqrt(b^2*x^4 - 1) + 84*x*elliptic_e(arcsin(1/(sqrt(b)*x)), -1)/sqrt(b))/(b*x)
Time = 1.68 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.92 \[ \int \left (1-b x^2\right ) \left (-1+b^2 x^4\right )^{3/2} \, dx=- \frac {i b^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {b^{2} x^{4}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {i b^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {b^{2} x^{4}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {i b x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {b^{2} x^{4}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {i x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {b^{2} x^{4}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((-b*x**2+1)*(b**2*x**4-1)**(3/2),x)
Output:
-I*b**3*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), b**2*x**4)/(4*gamma(11 /4)) + I*b**2*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b**2*x**4)/(4*gam ma(9/4)) + I*b*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b**2*x**4)/(4*ga mma(7/4)) - I*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b**2*x**4)/(4*gamma( 5/4))
\[ \int \left (1-b x^2\right ) \left (-1+b^2 x^4\right )^{3/2} \, dx=\int { -{\left (b^{2} x^{4} - 1\right )}^{\frac {3}{2}} {\left (b x^{2} - 1\right )} \,d x } \] Input:
integrate((-b*x^2+1)*(b^2*x^4-1)^(3/2),x, algorithm="maxima")
Output:
-integrate((b^2*x^4 - 1)^(3/2)*(b*x^2 - 1), x)
\[ \int \left (1-b x^2\right ) \left (-1+b^2 x^4\right )^{3/2} \, dx=\int { -{\left (b^{2} x^{4} - 1\right )}^{\frac {3}{2}} {\left (b x^{2} - 1\right )} \,d x } \] Input:
integrate((-b*x^2+1)*(b^2*x^4-1)^(3/2),x, algorithm="giac")
Output:
integrate(-(b^2*x^4 - 1)^(3/2)*(b*x^2 - 1), x)
Timed out. \[ \int \left (1-b x^2\right ) \left (-1+b^2 x^4\right )^{3/2} \, dx=-\int {\left (b^2\,x^4-1\right )}^{3/2}\,\left (b\,x^2-1\right ) \,d x \] Input:
int(-(b^2*x^4 - 1)^(3/2)*(b*x^2 - 1),x)
Output:
-int((b^2*x^4 - 1)^(3/2)*(b*x^2 - 1), x)
\[ \int \left (1-b x^2\right ) \left (-1+b^2 x^4\right )^{3/2} \, dx=-\frac {\sqrt {b^{2} x^{4}-1}\, b^{3} x^{7}}{9}+\frac {\sqrt {b^{2} x^{4}-1}\, b^{2} x^{5}}{7}+\frac {11 \sqrt {b^{2} x^{4}-1}\, b \,x^{3}}{45}-\frac {3 \sqrt {b^{2} x^{4}-1}\, x}{7}+\frac {4 \left (\int \frac {\sqrt {b^{2} x^{4}-1}}{b^{2} x^{4}-1}d x \right )}{7}-\frac {4 \left (\int \frac {\sqrt {b^{2} x^{4}-1}\, x^{2}}{b^{2} x^{4}-1}d x \right ) b}{15} \] Input:
int((-b*x^2+1)*(b^2*x^4-1)^(3/2),x)
Output:
( - 35*sqrt(b**2*x**4 - 1)*b**3*x**7 + 45*sqrt(b**2*x**4 - 1)*b**2*x**5 + 77*sqrt(b**2*x**4 - 1)*b*x**3 - 135*sqrt(b**2*x**4 - 1)*x + 180*int(sqrt(b **2*x**4 - 1)/(b**2*x**4 - 1),x) - 84*int((sqrt(b**2*x**4 - 1)*x**2)/(b**2 *x**4 - 1),x)*b)/315