Integrand size = 23, antiderivative size = 147 \[ \int \frac {1-b x^2}{\left (1-b^2 x^4\right )^{9/2}} \, dx=\frac {x \left (1-b x^2\right )}{14 \left (1-b^2 x^4\right )^{7/2}}+\frac {x \left (13-11 b x^2\right )}{140 \left (1-b^2 x^4\right )^{5/2}}+\frac {x \left (117-77 b x^2\right )}{840 \left (1-b^2 x^4\right )^{3/2}}+\frac {x \left (195-77 b x^2\right )}{560 \sqrt {1-b^2 x^4}}+\frac {11 E\left (\left .\arcsin \left (\sqrt {b} x\right )\right |-1\right )}{80 \sqrt {b}}+\frac {59 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {b} x\right ),-1\right )}{280 \sqrt {b}} \] 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*EllipticE(b^(1/2)*x,I)/b^(1/2)+59/280*EllipticF(b^( 1/2)*x,I)/b^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.61 \[ \int \frac {1-b x^2}{\left (1-b^2 x^4\right )^{9/2}} \, dx=\frac {x \left (-\frac {3 \left (-365+793 b^2 x^4-663 b^4 x^8+195 b^6 x^{12}\right )}{\left (1-b^2 x^4\right )^{7/2}}+585 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},b^2 x^4\right )-560 b x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {9}{2},\frac {7}{4},b^2 x^4\right )\right )}{1680} \] Input:
Integrate[(1 - b*x^2)/(1 - b^2*x^4)^(9/2),x]
Output:
(x*((-3*(-365 + 793*b^2*x^4 - 663*b^4*x^8 + 195*b^6*x^12))/(1 - b^2*x^4)^( 7/2) + 585*Hypergeometric2F1[1/4, 1/2, 5/4, b^2*x^4] - 560*b*x^2*Hypergeom etric2F1[3/4, 9/2, 7/4, b^2*x^4]))/1680
Time = 0.78 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.76, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {1388, 316, 27, 402, 27, 402, 25, 27, 402, 27, 402, 27, 402, 27, 402, 25, 27, 399, 284, 327, 762}
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.
| \(\displaystyle \int \frac {1-b x^2}{\left (1-b^2 x^4\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {1}{\left (1-b x^2\right )^{7/2} \left (b x^2+1\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\int \frac {b \left (11 b x^2+9\right )}{\left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{9/2}}dx}{10 b}+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \int \frac {11 b x^2+9}{\left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{9/2}}dx+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{10} \left (\frac {\int \frac {2 b \left (90 b x^2+17\right )}{\left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{9/2}}dx}{6 b}+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \int \frac {90 b x^2+17}{\left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{9/2}}dx+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {\int -\frac {b \left (73-749 b x^2\right )}{\sqrt {1-b x^2} \left (b x^2+1\right )^{9/2}}dx}{2 b}+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}-\frac {\int \frac {b \left (73-749 b x^2\right )}{\sqrt {1-b x^2} \left (b x^2+1\right )^{9/2}}dx}{2 b}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}-\frac {1}{2} \int \frac {73-749 b x^2}{\sqrt {1-b x^2} \left (b x^2+1\right )^{9/2}}dx\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {\int -\frac {10 b \left (20-411 b x^2\right )}{\sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}dx}{14 b}-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \int \frac {20-411 b x^2}{\sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}dx-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {\int \frac {3 b \left (431 b x^2+77\right )}{\sqrt {1-b x^2} \left (b x^2+1\right )^{5/2}}dx}{10 b}\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \int \frac {431 b x^2+77}{\sqrt {1-b x^2} \left (b x^2+1\right )^{5/2}}dx\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \left (-\frac {\int -\frac {6 b \left (59 b x^2+136\right )}{\sqrt {1-b x^2} \left (b x^2+1\right )^{3/2}}dx}{6 b}-\frac {59 x \sqrt {1-b x^2}}{\left (b x^2+1\right )^{3/2}}\right )\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \left (\int \frac {59 b x^2+136}{\sqrt {1-b x^2} \left (b x^2+1\right )^{3/2}}dx-\frac {59 x \sqrt {1-b x^2}}{\left (b x^2+1\right )^{3/2}}\right )\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \left (-\frac {\int -\frac {b \left (77 b x^2+195\right )}{\sqrt {1-b x^2} \sqrt {b x^2+1}}dx}{2 b}+\frac {77 x \sqrt {1-b x^2}}{2 \sqrt {b x^2+1}}-\frac {59 x \sqrt {1-b x^2}}{\left (b x^2+1\right )^{3/2}}\right )\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \left (\frac {\int \frac {b \left (77 b x^2+195\right )}{\sqrt {1-b x^2} \sqrt {b x^2+1}}dx}{2 b}+\frac {77 x \sqrt {1-b x^2}}{2 \sqrt {b x^2+1}}-\frac {59 x \sqrt {1-b x^2}}{\left (b x^2+1\right )^{3/2}}\right )\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \left (\frac {1}{2} \int \frac {77 b x^2+195}{\sqrt {1-b x^2} \sqrt {b x^2+1}}dx+\frac {77 x \sqrt {1-b x^2}}{2 \sqrt {b x^2+1}}-\frac {59 x \sqrt {1-b x^2}}{\left (b x^2+1\right )^{3/2}}\right )\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \left (\frac {1}{2} \left (118 \int \frac {1}{\sqrt {1-b x^2} \sqrt {b x^2+1}}dx+77 \int \frac {\sqrt {b x^2+1}}{\sqrt {1-b x^2}}dx\right )+\frac {77 x \sqrt {1-b x^2}}{2 \sqrt {b x^2+1}}-\frac {59 x \sqrt {1-b x^2}}{\left (b x^2+1\right )^{3/2}}\right )\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 284 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \left (\frac {1}{2} \left (118 \int \frac {1}{\sqrt {1-b^2 x^4}}dx+77 \int \frac {\sqrt {b x^2+1}}{\sqrt {1-b x^2}}dx\right )+\frac {77 x \sqrt {1-b x^2}}{2 \sqrt {b x^2+1}}-\frac {59 x \sqrt {1-b x^2}}{\left (b x^2+1\right )^{3/2}}\right )\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \left (\frac {1}{2} \left (118 \int \frac {1}{\sqrt {1-b^2 x^4}}dx+\frac {77 E\left (\left .\arcsin \left (\sqrt {b} x\right )\right |-1\right )}{\sqrt {b}}\right )+\frac {77 x \sqrt {1-b x^2}}{2 \sqrt {b x^2+1}}-\frac {59 x \sqrt {1-b x^2}}{\left (b x^2+1\right )^{3/2}}\right )\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{3} \left (\frac {1}{2} \left (-\frac {5}{7} \left (\frac {431 x \sqrt {1-b x^2}}{10 \left (b x^2+1\right )^{5/2}}-\frac {3}{10} \left (\frac {1}{2} \left (\frac {118 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {b} x\right ),-1\right )}{\sqrt {b}}+\frac {77 E\left (\left .\arcsin \left (\sqrt {b} x\right )\right |-1\right )}{\sqrt {b}}\right )+\frac {77 x \sqrt {1-b x^2}}{2 \sqrt {b x^2+1}}-\frac {59 x \sqrt {1-b x^2}}{\left (b x^2+1\right )^{3/2}}\right )\right )-\frac {411 x \sqrt {1-b x^2}}{7 \left (b x^2+1\right )^{7/2}}\right )+\frac {107 x}{2 \sqrt {1-b x^2} \left (b x^2+1\right )^{7/2}}\right )+\frac {10 x}{3 \left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{7/2}}\right )+\frac {x}{10 \left (1-b x^2\right )^{5/2} \left (b x^2+1\right )^{7/2}}\) |
Input:
Int[(1 - b*x^2)/(1 - b^2*x^4)^(9/2),x]
Output:
x/(10*(1 - b*x^2)^(5/2)*(1 + b*x^2)^(7/2)) + ((10*x)/(3*(1 - b*x^2)^(3/2)* (1 + b*x^2)^(7/2)) + ((107*x)/(2*Sqrt[1 - b*x^2]*(1 + b*x^2)^(7/2)) + ((-4 11*x*Sqrt[1 - b*x^2])/(7*(1 + b*x^2)^(7/2)) - (5*((431*x*Sqrt[1 - b*x^2])/ (10*(1 + b*x^2)^(5/2)) - (3*((-59*x*Sqrt[1 - b*x^2])/(1 + b*x^2)^(3/2) + ( 77*x*Sqrt[1 - b*x^2])/(2*Sqrt[1 + b*x^2]) + ((77*EllipticE[ArcSin[Sqrt[b]* x], -1])/Sqrt[b] + (118*EllipticF[ArcSin[Sqrt[b]*x], -1])/Sqrt[b])/2))/10) )/7)/2)/3)/10
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> I nt[(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0]))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 1.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.24
| method | result | size |
| meijerg | \(x \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {9}{2}\right ], \left [\frac {5}{4}\right ], b^{2} x^{4}\right )-\frac {b \,x^{3} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {9}{2}\right ], \left [\frac {7}{4}\right ], b^{2} x^{4}\right )}{3}\) | \(36\) |
| elliptic | \(-\frac {x \sqrt {-b^{2} x^{4}+1}}{160 b^{3} \left (x^{2}-\frac {1}{b}\right )^{3}}+\frac {x \sqrt {-b^{2} x^{4}+1}}{30 b^{2} \left (x^{2}-\frac {1}{b}\right )^{2}}-\frac {27 \left (-b^{2} x^{2}-b \right ) x}{160 b \sqrt {\left (x^{2}-\frac {1}{b}\right ) \left (-b^{2} x^{2}-b \right )}}+\frac {x \sqrt {-b^{2} x^{4}+1}}{112 b^{4} \left (x^{2}+\frac {1}{b}\right )^{4}}+\frac {39 x \sqrt {-b^{2} x^{4}+1}}{1120 b^{3} \left (x^{2}+\frac {1}{b}\right )^{3}}+\frac {157 x \sqrt {-b^{2} x^{4}+1}}{1680 b^{2} \left (x^{2}+\frac {1}{b}\right )^{2}}+\frac {49 \left (-b^{2} x^{2}+b \right ) x}{160 b \sqrt {\left (x^{2}+\frac {1}{b}\right ) \left (-b^{2} x^{2}+b \right )}}+\frac {39 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )}{112 \sqrt {b}\, \sqrt {-b^{2} x^{4}+1}}-\frac {11 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )-\operatorname {EllipticE}\left (\sqrt {b}\, x , i\right )\right )}{80 \sqrt {b}\, \sqrt {-b^{2} x^{4}+1}}\) | \(318\) |
| default | \(-b \left (\frac {x^{3} \sqrt {-b^{2} x^{4}+1}}{14 b^{8} \left (x^{4}-\frac {1}{b^{2}}\right )^{4}}-\frac {11 x^{3} \sqrt {-b^{2} x^{4}+1}}{140 b^{6} \left (x^{4}-\frac {1}{b^{2}}\right )^{3}}+\frac {11 x^{3} \sqrt {-b^{2} x^{4}+1}}{120 b^{4} \left (x^{4}-\frac {1}{b^{2}}\right )^{2}}+\frac {11 x^{3}}{80 \sqrt {-\left (x^{4}-\frac {1}{b^{2}}\right ) b^{2}}}+\frac {11 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \left (\operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )-\operatorname {EllipticE}\left (\sqrt {b}\, x , i\right )\right )}{80 b^{\frac {3}{2}} \sqrt {-b^{2} x^{4}+1}}\right )+\frac {x \sqrt {-b^{2} x^{4}+1}}{14 b^{8} \left (x^{4}-\frac {1}{b^{2}}\right )^{4}}-\frac {13 x \sqrt {-b^{2} x^{4}+1}}{140 b^{6} \left (x^{4}-\frac {1}{b^{2}}\right )^{3}}+\frac {39 x \sqrt {-b^{2} x^{4}+1}}{280 b^{4} \left (x^{4}-\frac {1}{b^{2}}\right )^{2}}+\frac {39 x}{112 \sqrt {-\left (x^{4}-\frac {1}{b^{2}}\right ) b^{2}}}+\frac {39 \sqrt {-b \,x^{2}+1}\, \sqrt {b \,x^{2}+1}\, \operatorname {EllipticF}\left (\sqrt {b}\, x , i\right )}{112 \sqrt {b}\, \sqrt {-b^{2} x^{4}+1}}\) | \(325\) |
Input:
int((-b*x^2+1)/(-b^2*x^4+1)^(9/2),x,method=_RETURNVERBOSE)
Output:
x*hypergeom([1/4,9/2],[5/4],b^2*x^4)-1/3*b*x^3*hypergeom([3/4,9/2],[7/4],b ^2*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (116) = 232\).
Time = 0.08 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.22 \[ \int \frac {1-b x^2}{\left (1-b^2 x^4\right )^{9/2}} \, dx=\frac {231 \, {\left (b^{8} x^{14} + b^{7} x^{12} - 3 \, b^{6} x^{10} - 3 \, b^{5} x^{8} + 3 \, b^{4} x^{6} + 3 \, b^{3} x^{4} - b^{2} x^{2} - b\right )} \sqrt {b} E(\arcsin \left (\sqrt {b} x\right )\,|\,-1) - 3 \, {\left ({\left (77 \, b^{8} - 195 \, b^{7}\right )} x^{14} + {\left (77 \, b^{7} - 195 \, b^{6}\right )} x^{12} - 3 \, {\left (77 \, b^{6} - 195 \, b^{5}\right )} x^{10} - 3 \, {\left (77 \, b^{5} - 195 \, b^{4}\right )} x^{8} + 3 \, {\left (77 \, b^{4} - 195 \, b^{3}\right )} x^{6} + 3 \, {\left (77 \, b^{3} - 195 \, b^{2}\right )} x^{4} - {\left (77 \, b^{2} - 195 \, b\right )} x^{2} - 77 \, b + 195\right )} \sqrt {b} F(\arcsin \left (\sqrt {b} x\right )\,|\,-1) + {\left (231 \, b^{7} x^{13} - 354 \, b^{6} x^{11} - 1201 \, b^{5} x^{9} + 788 \, b^{4} x^{7} + 1921 \, b^{3} x^{5} - 458 \, b^{2} x^{3} - 1095 \, b x\right )} \sqrt {-b^{2} x^{4} + 1}}{1680 \, {\left (b^{8} x^{14} + b^{7} x^{12} - 3 \, b^{6} x^{10} - 3 \, b^{5} x^{8} + 3 \, b^{4} x^{6} + 3 \, b^{3} x^{4} - b^{2} x^{2} - b\right )}} \] Input:
integrate((-b*x^2+1)/(-b^2*x^4+1)^(9/2),x, algorithm="fricas")
Output:
1/1680*(231*(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)*sqrt(b)*elliptic_e(arcsin(sqrt(b)*x), -1) - 3*((77* b^8 - 195*b^7)*x^14 + (77*b^7 - 195*b^6)*x^12 - 3*(77*b^6 - 195*b^5)*x^10 - 3*(77*b^5 - 195*b^4)*x^8 + 3*(77*b^4 - 195*b^3)*x^6 + 3*(77*b^3 - 195*b^ 2)*x^4 - (77*b^2 - 195*b)*x^2 - 77*b + 195)*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 = 35.46 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.48 \[ \int \frac {1-b x^2}{\left (1-b^2 x^4\right )^{9/2}} \, dx=- \frac {b x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {9}{2} \\ \frac {7}{4} \end {matrix}\middle | {b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{2} \\ \frac {5}{4} \end {matrix}\middle | {b^{2} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((-b*x**2+1)/(-b**2*x**4+1)**(9/2),x)
Output:
-b*x**3*gamma(3/4)*hyper((3/4, 9/2), (7/4,), b**2*x**4*exp_polar(2*I*pi))/ (4*gamma(7/4)) + x*gamma(1/4)*hyper((1/4, 9/2), (5/4,), b**2*x**4*exp_pola r(2*I*pi))/(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. \[ \int \frac {1-b x^2}{\left (1-b^2 x^4\right )^{9/2}} \, dx=-\int \frac {b\,x^2-1}{{\left (1-b^2\,x^4\right )}^{9/2}} \,d x \] Input:
int(-(b*x^2 - 1)/(1 - b^2*x^4)^(9/2),x)
Output:
-int((b*x^2 - 1)/(1 - b^2*x^4)^(9/2), x)
\[ \int \frac {1-b x^2}{\left (1-b^2 x^4\right )^{9/2}} \, dx=\int \frac {\sqrt {-b^{2} x^{4}+1}}{b^{9} x^{18}+b^{8} x^{16}-4 b^{7} x^{14}-4 b^{6} x^{12}+6 b^{5} x^{10}+6 b^{4} x^{8}-4 b^{3} x^{6}-4 b^{2} x^{4}+b \,x^{2}+1}d x \] 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)