Integrand size = 26, antiderivative size = 259 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {x \left (54 d-77 e x^2\right )}{420 d^5 \left (d^2-e^2 x^4\right )^{3/2}}+\frac {x}{14 d^2 \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}}+\frac {11 x}{70 d^3 \left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2}}+\frac {x \left (90 d-77 e x^2\right )}{280 d^7 \sqrt {d^2-e^2 x^4}}+\frac {11 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{40 d^{11/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {13 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{280 d^{11/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
1/420*x*(-77*e*x^2+54*d)/d^5/(-e^2*x^4+d^2)^(3/2)+1/14*x/d^2/(e*x^2+d)^2/( -e^2*x^4+d^2)^(3/2)+11/70*x/d^3/(e*x^2+d)/(-e^2*x^4+d^2)^(3/2)+1/280*x*(-7 7*e*x^2+90*d)/d^7/(-e^2*x^4+d^2)^(1/2)+11/40*(1-e^2*x^4/d^2)^(1/2)*Ellipti cE(e^(1/2)*x/d^(1/2),I)/d^(11/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+13/280*(1-e^ 2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/d^(11/2)/e^(1/2)/(-e^2*x^4 +d^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.95 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {\frac {x \left (570 d^5+503 d^4 e x^2-662 d^3 e^2 x^4-694 d^2 e^3 x^6+192 d e^4 x^8+231 e^5 x^{10}\right )}{\left (d-e x^2\right ) \left (d+e x^2\right )^3}+\frac {3 i e \sqrt {1-\frac {e^2 x^4}{d^2}} \left (77 E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )+13 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\left (-\frac {e}{d}\right )^{3/2}}}{840 d^7 \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[1/((d + e*x^2)^2*(d^2 - e^2*x^4)^(5/2)),x]
Output:
((x*(570*d^5 + 503*d^4*e*x^2 - 662*d^3*e^2*x^4 - 694*d^2*e^3*x^6 + 192*d*e ^4*x^8 + 231*e^5*x^10))/((d - e*x^2)*(d + e*x^2)^3) + ((3*I)*e*Sqrt[1 - (e ^2*x^4)/d^2]*(77*EllipticE[I*ArcSinh[Sqrt[-(e/d)]*x], -1] + 13*EllipticF[I *ArcSinh[Sqrt[-(e/d)]*x], -1]))/(-(e/d))^(3/2))/(840*d^7*Sqrt[d^2 - e^2*x^ 4])
Time = 1.08 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.54, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.731, Rules used = {1396, 316, 27, 402, 27, 402, 27, 402, 27, 402, 27, 402, 27, 399, 289, 329, 327, 765, 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.
| \(\displaystyle \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {1}{\left (d-e x^2\right )^{5/2} \left (e x^2+d\right )^{9/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (9 e x^2+5 d\right )}{\left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{9/2}}dx}{6 d^2 e}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {9 e x^2+5 d}{\left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{9/2}}dx}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int -\frac {2 d e \left (2 d-49 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{9/2}}dx}{2 d^2 e}+\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\int \frac {2 d-49 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{9/2}}dx}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\int \frac {d e \left (255 e x^2+23 d\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{7/2}}dx}{14 d^2 e}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\int \frac {255 e x^2+23 d}{\sqrt {d-e x^2} \left (e x^2+d\right )^{7/2}}dx}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {-\frac {\int -\frac {6 d e \left (116 e x^2+77 d\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{10 d^2 e}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \int \frac {116 e x^2+77 d}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (-\frac {\int -\frac {3 d e \left (13 e x^2+167 d\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (\frac {\int \frac {13 e x^2+167 d}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{2 d}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (\frac {\frac {77 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}-\frac {\int -\frac {2 d e \left (77 e x^2+90 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}}{2 d}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (\frac {\frac {\int \frac {77 e x^2+90 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{d}+\frac {77 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{2 d}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (\frac {\frac {13 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx+77 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{d}+\frac {77 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{2 d}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (\frac {\frac {\frac {13 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+77 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{d}+\frac {77 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{2 d}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (\frac {\frac {\frac {77 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {13 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {77 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{2 d}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (\frac {\frac {\frac {13 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {77 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{2 d}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (\frac {\frac {\frac {13 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {77 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{2 d}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{d \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}-\frac {\frac {51 x \sqrt {d-e x^2}}{14 d \left (d+e x^2\right )^{7/2}}-\frac {\frac {3 \left (\frac {\frac {\frac {13 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{d}+\frac {77 x \sqrt {d-e x^2}}{d \sqrt {d+e x^2}}}{2 d}-\frac {13 x \sqrt {d-e x^2}}{2 d \left (d+e x^2\right )^{3/2}}\right )}{5 d}-\frac {116 x \sqrt {d-e x^2}}{5 d \left (d+e x^2\right )^{5/2}}}{14 d}}{d}}{6 d^2}+\frac {x}{6 d^2 \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[1/((d + e*x^2)^2*(d^2 - e^2*x^4)^(5/2)),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(6*d^2*(d - e*x^2)^(3/2)*(d + e*x^2)^( 7/2)) + ((7*x)/(d*Sqrt[d - e*x^2]*(d + e*x^2)^(7/2)) - ((51*x*Sqrt[d - e*x ^2])/(14*d*(d + e*x^2)^(7/2)) - ((-116*x*Sqrt[d - e*x^2])/(5*d*(d + e*x^2) ^(5/2)) + (3*((-13*x*Sqrt[d - e*x^2])/(2*d*(d + e*x^2)^(3/2)) + ((77*x*Sqr t[d - e*x^2])/(d*Sqrt[d + e*x^2]) + ((77*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*E llipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]) + (13*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e] *x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]))/d)/(2*d)))/( 5*d))/(14*d))/d)/(6*d^2)))/Sqrt[d^2 - e^2*x^4]
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] :> Sim p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart [p]) Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* c + a*d, 0] && !IntegerQ[p]
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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])) Int[Sqrt[1 + b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b *c + a*d, 0] && !(LtQ[a*c, 0] && GtQ[a*b, 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[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], 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] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Time = 1.77 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.44
| method | result | size |
| default | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{56 d^{4} e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{35 d^{5} e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {439 x \sqrt {-e^{2} x^{4}+d^{2}}}{3360 d^{6} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {59 \left (-e^{2} x^{2}+d e \right ) x}{160 e \,d^{7} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{96 d^{6} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {3 \left (-e^{2} x^{2}-d e \right ) x}{32 e \,d^{7} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {9 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{28 d^{6} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {11 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{40 d^{6} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(373\) |
| elliptic | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{56 d^{4} e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{35 d^{5} e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {439 x \sqrt {-e^{2} x^{4}+d^{2}}}{3360 d^{6} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {59 \left (-e^{2} x^{2}+d e \right ) x}{160 e \,d^{7} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{96 d^{6} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {3 \left (-e^{2} x^{2}-d e \right ) x}{32 e \,d^{7} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {9 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{28 d^{6} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {11 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{40 d^{6} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(373\) |
Input:
int(1/(e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/56/d^4*x/e^4*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^4+2/35/d^5/e^3*x*(-e^2*x^4+d ^2)^(1/2)/(x^2+d/e)^3+439/3360/d^6/e^2*x*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+ 59/160*(-e^2*x^2+d*e)/e/d^7*x/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)+1/96/d^6/e^ 2*x*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^2-3/32*(-e^2*x^2-d*e)/e/d^7*x/((x^2-d/e )*(-e^2*x^2-d*e))^(1/2)+9/28/d^6/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d) ^(1/2)/(-e^2*x^4+d^2)^(1/2)*EllipticF(x*(e/d)^(1/2),I)-11/40/d^6/(e/d)^(1/ 2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*(EllipticF(x*( e/d)^(1/2),I)-EllipticE(x*(e/d)^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {231 \, {\left (e^{7} x^{12} + 2 \, d e^{6} x^{10} - d^{2} e^{5} x^{8} - 4 \, d^{3} e^{4} x^{6} - d^{4} e^{3} x^{4} + 2 \, d^{5} e^{2} x^{2} + d^{6} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + 3 \, {\left ({\left (90 \, d e^{6} - 77 \, e^{7}\right )} x^{12} + 2 \, {\left (90 \, d^{2} e^{5} - 77 \, d e^{6}\right )} x^{10} - {\left (90 \, d^{3} e^{4} - 77 \, d^{2} e^{5}\right )} x^{8} + 90 \, d^{7} - 77 \, d^{6} e - 4 \, {\left (90 \, d^{4} e^{3} - 77 \, d^{3} e^{4}\right )} x^{6} - {\left (90 \, d^{5} e^{2} - 77 \, d^{4} e^{3}\right )} x^{4} + 2 \, {\left (90 \, d^{6} e - 77 \, d^{5} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (231 \, e^{6} x^{11} + 192 \, d e^{5} x^{9} - 694 \, d^{2} e^{4} x^{7} - 662 \, d^{3} e^{3} x^{5} + 503 \, d^{4} e^{2} x^{3} + 570 \, d^{5} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{840 \, {\left (d^{7} e^{7} x^{12} + 2 \, d^{8} e^{6} x^{10} - d^{9} e^{5} x^{8} - 4 \, d^{10} e^{4} x^{6} - d^{11} e^{3} x^{4} + 2 \, d^{12} e^{2} x^{2} + d^{13} e\right )}} \] Input:
integrate(1/(e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
Output:
1/840*(231*(e^7*x^12 + 2*d*e^6*x^10 - d^2*e^5*x^8 - 4*d^3*e^4*x^6 - d^4*e^ 3*x^4 + 2*d^5*e^2*x^2 + d^6*e)*sqrt(e/d)*elliptic_e(arcsin(x*sqrt(e/d)), - 1) + 3*((90*d*e^6 - 77*e^7)*x^12 + 2*(90*d^2*e^5 - 77*d*e^6)*x^10 - (90*d^ 3*e^4 - 77*d^2*e^5)*x^8 + 90*d^7 - 77*d^6*e - 4*(90*d^4*e^3 - 77*d^3*e^4)* x^6 - (90*d^5*e^2 - 77*d^4*e^3)*x^4 + 2*(90*d^6*e - 77*d^5*e^2)*x^2)*sqrt( e/d)*elliptic_f(arcsin(x*sqrt(e/d)), -1) + (231*e^6*x^11 + 192*d*e^5*x^9 - 694*d^2*e^4*x^7 - 662*d^3*e^3*x^5 + 503*d^4*e^2*x^3 + 570*d^5*e*x)*sqrt(- e^2*x^4 + d^2))/(d^7*e^7*x^12 + 2*d^8*e^6*x^10 - d^9*e^5*x^8 - 4*d^10*e^4* x^6 - d^11*e^3*x^4 + 2*d^12*e^2*x^2 + d^13*e)
\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}} \left (d + e x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(e*x**2+d)**2/(-e**2*x**4+d**2)**(5/2),x)
Output:
Integral(1/((-(-d + e*x**2)*(d + e*x**2))**(5/2)*(d + e*x**2)**2), x)
\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(1/(e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
Output:
integrate(1/((-e^2*x^4 + d^2)^(5/2)*(e*x^2 + d)^2), x)
\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(1/(e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
Output:
integrate(1/((-e^2*x^4 + d^2)^(5/2)*(e*x^2 + d)^2), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (d^2-e^2\,x^4\right )}^{5/2}\,{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int(1/((d^2 - e^2*x^4)^(5/2)*(d + e*x^2)^2),x)
Output:
int(1/((d^2 - e^2*x^4)^(5/2)*(d + e*x^2)^2), x)
\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{8} x^{16}-2 d \,e^{7} x^{14}+2 d^{2} e^{6} x^{12}+6 d^{3} e^{5} x^{10}-6 d^{5} e^{3} x^{6}-2 d^{6} e^{2} x^{4}+2 d^{7} e \,x^{2}+d^{8}}d x \] Input:
int(1/(e*x^2+d)^2/(-e^2*x^4+d^2)^(5/2),x)
Output:
int(sqrt(d**2 - e**2*x**4)/(d**8 + 2*d**7*e*x**2 - 2*d**6*e**2*x**4 - 6*d* *5*e**3*x**6 + 6*d**3*e**5*x**10 + 2*d**2*e**6*x**12 - 2*d*e**7*x**14 - e* *8*x**16),x)