Integrand size = 27, antiderivative size = 295 \[ \int \frac {1}{\left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {x}{18 d^2 \left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}}+\frac {x}{9 d^3 \left (d-e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2}}+\frac {11 x}{60 d^4 \left (d-e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2}}+\frac {x \left (39 d+77 e x^2\right )}{360 d^6 \left (d^2-e^2 x^4\right )^{3/2}}+\frac {x \left (65 d+77 e x^2\right )}{240 d^8 \sqrt {d^2-e^2 x^4}}-\frac {77 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{240 d^{13/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {71 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{120 d^{13/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
1/18*x/d^2/(-e*x^2+d)^3/(-e^2*x^4+d^2)^(3/2)+1/9*x/d^3/(-e*x^2+d)^2/(-e^2* x^4+d^2)^(3/2)+11/60*x/d^4/(-e*x^2+d)/(-e^2*x^4+d^2)^(3/2)+1/360*x*(77*e*x ^2+39*d)/d^6/(-e^2*x^4+d^2)^(3/2)+1/240*x*(77*e*x^2+65*d)/d^8/(-e^2*x^4+d^ 2)^(1/2)-77/240*(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/d^(13 /2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+71/120*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^ (1/2)*x/d^(1/2),I)/d^(13/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.97 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {\frac {x \left (525 d^6-778 d^5 e x^2-399 d^4 e^2 x^4+1236 d^3 e^3 x^6-277 d^2 e^4 x^8-498 d e^5 x^{10}+231 e^6 x^{12}\right )}{\left (d-e x^2\right )^4 \left (d+e x^2\right )}+\frac {3 i d \sqrt {1-\frac {e^2 x^4}{d^2}} \left (77 E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-142 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\sqrt {-\frac {e}{d}}}}{720 d^8 \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[1/((d - e*x^2)^3*(d^2 - e^2*x^4)^(5/2)),x]
Output:
((x*(525*d^6 - 778*d^5*e*x^2 - 399*d^4*e^2*x^4 + 1236*d^3*e^3*x^6 - 277*d^ 2*e^4*x^8 - 498*d*e^5*x^10 + 231*e^6*x^12))/((d - e*x^2)^4*(d + e*x^2)) + ((3*I)*d*Sqrt[1 - (e^2*x^4)/d^2]*(77*EllipticE[I*ArcSinh[Sqrt[-(e/d)]*x], -1] - 142*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1]))/Sqrt[-(e/d)])/(720*d^ 8*Sqrt[d^2 - e^2*x^4])
Time = 1.10 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.49, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {1396, 316, 27, 402, 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.
| \(\displaystyle \int \frac {1}{\left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {1}{\left (d-e x^2\right )^{11/2} \left (e x^2+d\right )^{5/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (11 e x^2+17 d\right )}{\left (d-e x^2\right )^{9/2} \left (e x^2+d\right )^{5/2}}dx}{18 d^2 e}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {11 e x^2+17 d}{\left (d-e x^2\right )^{9/2} \left (e x^2+d\right )^{5/2}}dx}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {\int \frac {42 d e \left (6 e x^2+5 d\right )}{\left (d-e x^2\right )^{7/2} \left (e x^2+d\right )^{5/2}}dx}{14 d^2 e}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \int \frac {6 e x^2+5 d}{\left (d-e x^2\right )^{7/2} \left (e x^2+d\right )^{5/2}}dx}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\int \frac {d e \left (77 e x^2+39 d\right )}{\left (d-e x^2\right )^{5/2} \left (e x^2+d\right )^{5/2}}dx}{10 d^2 e}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\int \frac {77 e x^2+39 d}{\left (d-e x^2\right )^{5/2} \left (e x^2+d\right )^{5/2}}dx}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\int \frac {2 d e \left (290 e x^2+59 d\right )}{\left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{5/2}}dx}{6 d^2 e}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\int \frac {290 e x^2+59 d}{\left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{5/2}}dx}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {\int -\frac {3 d e \left (77 d-349 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{2 d^2 e}+\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \int \frac {77 d-349 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {6 d e \left (6 d-71 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {\int \frac {6 d-71 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{d}+\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {\frac {77 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\int \frac {d e \left (65 d-77 e x^2\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}}{d}+\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {\frac {77 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\int \frac {65 d-77 e x^2}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}}{d}+\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {\frac {77 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {142 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-77 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d}}{d}+\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {\frac {77 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\frac {142 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-77 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx}{2 d}}{d}+\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {\frac {77 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\frac {142 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {77 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}}{d}+\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {\frac {77 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\frac {142 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}}{d}+\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {\frac {77 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\frac {142 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}}{d}+\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \left (\frac {\frac {\frac {349 x}{2 d \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}}-\frac {3 \left (\frac {\frac {77 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\frac {142 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}}{d}+\frac {71 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{2 d}}{3 d}+\frac {58 x}{3 d \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}}}{10 d}+\frac {11 x}{10 d \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}}\right )}{d}+\frac {2 x}{d \left (d-e x^2\right )^{7/2} \left (d+e x^2\right )^{3/2}}}{18 d^2}+\frac {x}{18 d^2 \left (d-e x^2\right )^{9/2} \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[1/((d - e*x^2)^3*(d^2 - e^2*x^4)^(5/2)),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(18*d^2*(d - e*x^2)^(9/2)*(d + e*x^2)^ (3/2)) + ((2*x)/(d*(d - e*x^2)^(7/2)*(d + e*x^2)^(3/2)) + (3*((11*x)/(10*d *(d - e*x^2)^(5/2)*(d + e*x^2)^(3/2)) + ((58*x)/(3*d*(d - e*x^2)^(3/2)*(d + e*x^2)^(3/2)) + ((349*x)/(2*d*Sqrt[d - e*x^2]*(d + e*x^2)^(3/2)) - (3*(( 71*x*Sqrt[d - e*x^2])/(d*(d + e*x^2)^(3/2)) + ((77*x*Sqrt[d - e*x^2])/(2*d *Sqrt[d + e*x^2]) - ((-77*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSin [(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]) + (1 42*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]))/(2*d))/d))/(2*d))/(3*d))/( 10*d)))/d)/(18*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 = 2.36 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.39
| method | result | size |
| default | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{192 d^{7} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {7 \left (-e^{2} x^{2}+d e \right ) x}{128 e \,d^{8} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{72 d^{4} e^{5} \left (x^{2}-\frac {d}{e}\right )^{5}}+\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{24 d^{5} e^{4} \left (x^{2}-\frac {d}{e}\right )^{4}}-\frac {121 x \sqrt {-e^{2} x^{4}+d^{2}}}{1440 d^{6} e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {37 x \sqrt {-e^{2} x^{4}+d^{2}}}{240 d^{7} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {721 \left (-e^{2} x^{2}-d e \right ) x}{1920 e \,d^{8} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {13 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{48 d^{7} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {77 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{240 d^{7} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(410\) |
| elliptic | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{192 d^{7} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {7 \left (-e^{2} x^{2}+d e \right ) x}{128 e \,d^{8} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}-\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{72 d^{4} e^{5} \left (x^{2}-\frac {d}{e}\right )^{5}}+\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{24 d^{5} e^{4} \left (x^{2}-\frac {d}{e}\right )^{4}}-\frac {121 x \sqrt {-e^{2} x^{4}+d^{2}}}{1440 d^{6} e^{3} \left (x^{2}-\frac {d}{e}\right )^{3}}+\frac {37 x \sqrt {-e^{2} x^{4}+d^{2}}}{240 d^{7} e^{2} \left (x^{2}-\frac {d}{e}\right )^{2}}-\frac {721 \left (-e^{2} x^{2}-d e \right ) x}{1920 e \,d^{8} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {13 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{48 d^{7} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {77 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{240 d^{7} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(410\) |
Input:
int(1/(-e*x^2+d)^3/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/192/d^7/e^2*x*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+7/128*(-e^2*x^2+d*e)/e/d^ 8*x/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)-1/72/d^4*x/e^5*(-e^2*x^4+d^2)^(1/2)/( x^2-d/e)^5+1/24/d^5/e^4*x*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^4-121/1440/d^6/e^ 3*x*(-e^2*x^4+d^2)^(1/2)/(x^2-d/e)^3+37/240/d^7/e^2*x*(-e^2*x^4+d^2)^(1/2) /(x^2-d/e)^2-721/1920*(-e^2*x^2-d*e)/e/d^8*x/((x^2-d/e)*(-e^2*x^2-d*e))^(1 /2)+13/48/d^7/(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)+77/240/d^7/(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)-Elli pticE(x*(e/d)^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=-\frac {231 \, {\left (e^{8} x^{14} - 3 \, d e^{7} x^{12} + d^{2} e^{6} x^{10} + 5 \, d^{3} e^{5} x^{8} - 5 \, d^{4} e^{4} x^{6} - d^{5} e^{3} x^{4} + 3 \, d^{6} e^{2} x^{2} - d^{7} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - 3 \, {\left ({\left (65 \, d e^{7} + 77 \, e^{8}\right )} x^{14} - 3 \, {\left (65 \, d^{2} e^{6} + 77 \, d e^{7}\right )} x^{12} + {\left (65 \, d^{3} e^{5} + 77 \, d^{2} e^{6}\right )} x^{10} + 5 \, {\left (65 \, d^{4} e^{4} + 77 \, d^{3} e^{5}\right )} x^{8} - 65 \, d^{8} - 77 \, d^{7} e - 5 \, {\left (65 \, d^{5} e^{3} + 77 \, d^{4} e^{4}\right )} x^{6} - {\left (65 \, d^{6} e^{2} + 77 \, d^{5} e^{3}\right )} x^{4} + 3 \, {\left (65 \, d^{7} e + 77 \, d^{6} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (231 \, e^{7} x^{13} - 498 \, d e^{6} x^{11} - 277 \, d^{2} e^{5} x^{9} + 1236 \, d^{3} e^{4} x^{7} - 399 \, d^{4} e^{3} x^{5} - 778 \, d^{5} e^{2} x^{3} + 525 \, d^{6} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{720 \, {\left (d^{8} e^{8} x^{14} - 3 \, d^{9} e^{7} x^{12} + d^{10} e^{6} x^{10} + 5 \, d^{11} e^{5} x^{8} - 5 \, d^{12} e^{4} x^{6} - d^{13} e^{3} x^{4} + 3 \, d^{14} e^{2} x^{2} - d^{15} e\right )}} \] Input:
integrate(1/(-e*x^2+d)^3/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
Output:
-1/720*(231*(e^8*x^14 - 3*d*e^7*x^12 + d^2*e^6*x^10 + 5*d^3*e^5*x^8 - 5*d^ 4*e^4*x^6 - d^5*e^3*x^4 + 3*d^6*e^2*x^2 - d^7*e)*sqrt(e/d)*elliptic_e(arcs in(x*sqrt(e/d)), -1) - 3*((65*d*e^7 + 77*e^8)*x^14 - 3*(65*d^2*e^6 + 77*d* e^7)*x^12 + (65*d^3*e^5 + 77*d^2*e^6)*x^10 + 5*(65*d^4*e^4 + 77*d^3*e^5)*x ^8 - 65*d^8 - 77*d^7*e - 5*(65*d^5*e^3 + 77*d^4*e^4)*x^6 - (65*d^6*e^2 + 7 7*d^5*e^3)*x^4 + 3*(65*d^7*e + 77*d^6*e^2)*x^2)*sqrt(e/d)*elliptic_f(arcsi n(x*sqrt(e/d)), -1) + (231*e^7*x^13 - 498*d*e^6*x^11 - 277*d^2*e^5*x^9 + 1 236*d^3*e^4*x^7 - 399*d^4*e^3*x^5 - 778*d^5*e^2*x^3 + 525*d^6*e*x)*sqrt(-e ^2*x^4 + d^2))/(d^8*e^8*x^14 - 3*d^9*e^7*x^12 + d^10*e^6*x^10 + 5*d^11*e^5 *x^8 - 5*d^12*e^4*x^6 - d^13*e^3*x^4 + 3*d^14*e^2*x^2 - d^15*e)
\[ \int \frac {1}{\left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=- \int \frac {1}{- d^{7} \sqrt {d^{2} - e^{2} x^{4}} + 3 d^{6} e x^{2} \sqrt {d^{2} - e^{2} x^{4}} - d^{5} e^{2} x^{4} \sqrt {d^{2} - e^{2} x^{4}} - 5 d^{4} e^{3} x^{6} \sqrt {d^{2} - e^{2} x^{4}} + 5 d^{3} e^{4} x^{8} \sqrt {d^{2} - e^{2} x^{4}} + d^{2} e^{5} x^{10} \sqrt {d^{2} - e^{2} x^{4}} - 3 d e^{6} x^{12} \sqrt {d^{2} - e^{2} x^{4}} + e^{7} x^{14} \sqrt {d^{2} - e^{2} x^{4}}}\, dx \] Input:
integrate(1/(-e*x**2+d)**3/(-e**2*x**4+d**2)**(5/2),x)
Output:
-Integral(1/(-d**7*sqrt(d**2 - e**2*x**4) + 3*d**6*e*x**2*sqrt(d**2 - e**2 *x**4) - d**5*e**2*x**4*sqrt(d**2 - e**2*x**4) - 5*d**4*e**3*x**6*sqrt(d** 2 - e**2*x**4) + 5*d**3*e**4*x**8*sqrt(d**2 - e**2*x**4) + d**2*e**5*x**10 *sqrt(d**2 - e**2*x**4) - 3*d*e**6*x**12*sqrt(d**2 - e**2*x**4) + e**7*x** 14*sqrt(d**2 - e**2*x**4)), x)
\[ \int \frac {1}{\left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { -\frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}} {\left (e x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate(1/(-e*x^2+d)^3/(-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)^3), x)
\[ \int \frac {1}{\left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { -\frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}} {\left (e x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate(1/(-e*x^2+d)^3/(-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)^3), x)
Timed out. \[ \int \frac {1}{\left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (d^2-e^2\,x^4\right )}^{5/2}\,{\left (d-e\,x^2\right )}^3} \,d x \] Input:
int(1/((d^2 - e^2*x^4)^(5/2)*(d - e*x^2)^3),x)
Output:
int(1/((d^2 - e^2*x^4)^(5/2)*(d - e*x^2)^3), x)
\[ \int \frac {1}{\left (d-e x^2\right )^3 \left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e^{9} x^{18}-3 d \,e^{8} x^{16}+8 d^{3} e^{6} x^{12}-6 d^{4} e^{5} x^{10}-6 d^{5} e^{4} x^{8}+8 d^{6} e^{3} x^{6}-3 d^{8} e \,x^{2}+d^{9}}d x \] Input:
int(1/(-e*x^2+d)^3/(-e^2*x^4+d^2)^(5/2),x)
Output:
int(sqrt(d**2 - e**2*x**4)/(d**9 - 3*d**8*e*x**2 + 8*d**6*e**3*x**6 - 6*d* *5*e**4*x**8 - 6*d**4*e**5*x**10 + 8*d**3*e**6*x**12 - 3*d*e**8*x**16 + e* *9*x**18),x)