Integrand size = 26, antiderivative size = 239 \[ \int \left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {4}{715} d^4 x \left (65 d+77 e x^2\right ) \sqrt {d^2-e^2 x^4}+\frac {2}{429} d^2 x \left (39 d+77 e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2}-\frac {3}{11} d x \left (d^2-e^2 x^4\right )^{5/2}-\frac {1}{13} e x^3 \left (d^2-e^2 x^4\right )^{5/2}+\frac {56 d^{15/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{65 \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {96 d^{15/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{715 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
4/715*d^4*x*(77*e*x^2+65*d)*(-e^2*x^4+d^2)^(1/2)+2/429*d^2*x*(77*e*x^2+39* d)*(-e^2*x^4+d^2)^(3/2)-3/11*d*x*(-e^2*x^4+d^2)^(5/2)-1/13*e*x^3*(-e^2*x^4 +d^2)^(5/2)+56/65*d^(15/2)*(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/ 2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)-96/715*d^(15/2)*(1-e^2*x^4/d^2)^(1/2)*E llipticF(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.16 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.56 \[ \int \left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2} \, dx=-\frac {x \sqrt {d^2-e^2 x^4} \left (\left (39 d+11 e x^2\right ) \left (d^2-e^2 x^4\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}}-182 d^5 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )-154 d^4 e x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )\right )}{143 \sqrt {1-\frac {e^2 x^4}{d^2}}} \] Input:
Integrate[(d + e*x^2)^3*(d^2 - e^2*x^4)^(3/2),x]
Output:
-1/143*(x*Sqrt[d^2 - e^2*x^4]*((39*d + 11*e*x^2)*(d^2 - e^2*x^4)^2*Sqrt[1 - (e^2*x^4)/d^2] - 182*d^5*Hypergeometric2F1[-3/2, 1/4, 5/4, (e^2*x^4)/d^2 ] - 154*d^4*e*x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, (e^2*x^4)/d^2]))/Sqrt[ 1 - (e^2*x^4)/d^2]
Time = 0.99 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.56, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.808, Rules used = {1396, 318, 27, 403, 25, 27, 403, 27, 403, 25, 27, 403, 27, 403, 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.
| \(\displaystyle \int \left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{9/2}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (-\frac {\int -14 d e \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{5/2} \left (2 e x^2+d\right )dx}{13 e}-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \int \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{5/2} \left (2 e x^2+d\right )dx-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (-\frac {\int -d e \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{3/2} \left (33 e x^2+13 d\right )dx}{11 e}-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {\int d e \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{3/2} \left (33 e x^2+13 d\right )dx}{11 e}-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \int \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{3/2} \left (33 e x^2+13 d\right )dx-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (-\frac {\int -6 d e \left (d-e x^2\right )^{3/2} \sqrt {e x^2+d} \left (58 e x^2+25 d\right )dx}{9 e}-\frac {11}{3} x \left (d+e x^2\right )^{3/2} \left (d-e x^2\right )^{5/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \int \left (d-e x^2\right )^{3/2} \sqrt {e x^2+d} \left (58 e x^2+25 d\right )dx-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (-\frac {\int -\frac {d e \left (d-e x^2\right )^{3/2} \left (349 e x^2+233 d\right )}{\sqrt {e x^2+d}}dx}{7 e}-\frac {58}{7} x \sqrt {d+e x^2} \left (d-e x^2\right )^{5/2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {\int \frac {d e \left (d-e x^2\right )^{3/2} \left (349 e x^2+233 d\right )}{\sqrt {e x^2+d}}dx}{7 e}-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \int \frac {\left (d-e x^2\right )^{3/2} \left (349 e x^2+233 d\right )}{\sqrt {e x^2+d}}dx-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {\int \frac {6 d e \sqrt {d-e x^2} \left (213 e x^2+136 d\right )}{\sqrt {e x^2+d}}dx}{5 e}+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {6}{5} d \int \frac {\sqrt {d-e x^2} \left (213 e x^2+136 d\right )}{\sqrt {e x^2+d}}dx+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {6}{5} d \left (\frac {\int \frac {3 d e \left (77 e x^2+65 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 e}+71 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {6}{5} d \left (d \int \frac {77 e x^2+65 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx+71 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {6}{5} d \left (d \left (77 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-12 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx\right )+71 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {6}{5} d \left (d \left (77 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {12 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+71 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {6}{5} d \left (d \left (\frac {77 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+71 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {6}{5} d \left (d \left (\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+71 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {6}{5} d \left (d \left (\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+71 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {14}{13} d \left (\frac {1}{11} d \left (\frac {2}{3} d \left (\frac {1}{7} d \left (\frac {6}{5} d \left (d \left (\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {12 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+71 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {349}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {58}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{3} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {2}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )-\frac {1}{13} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{7/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[(d + e*x^2)^3*(d^2 - e^2*x^4)^(3/2),x]
Output:
(Sqrt[d^2 - e^2*x^4]*(-1/13*(x*(d - e*x^2)^(5/2)*(d + e*x^2)^(7/2)) + (14* d*((-2*x*(d - e*x^2)^(5/2)*(d + e*x^2)^(5/2))/11 + (d*((-11*x*(d - e*x^2)^ (5/2)*(d + e*x^2)^(3/2))/3 + (2*d*((-58*x*(d - e*x^2)^(5/2)*Sqrt[d + e*x^2 ])/7 + (d*((349*x*(d - e*x^2)^(3/2)*Sqrt[d + e*x^2])/5 + (6*d*(71*x*Sqrt[d - e*x^2]*Sqrt[d + e*x^2] + d*((77*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*Ellipti cE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x ^2]) - (12*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sq rt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]))))/5))/7))/3))/11)) /13))/(Sqrt[d - e*x^2]*Sqrt[d + e*x^2])
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[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
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 = 7.94 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.90
| method | result | size |
| risch | \(\frac {x \left (-165 e^{5} x^{10}-585 d \,e^{4} x^{8}-440 d^{2} e^{3} x^{6}+780 d^{3} e^{2} x^{4}+1529 d^{4} e \,x^{2}+585 d^{5}\right ) \sqrt {-e^{2} x^{4}+d^{2}}}{2145}+\frac {8 d^{6} \left (\frac {65 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {77 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )}{715}\) | \(216\) |
| elliptic | \(-\frac {e^{5} x^{11} \sqrt {-e^{2} x^{4}+d^{2}}}{13}-\frac {3 e^{4} d \,x^{9} \sqrt {-e^{2} x^{4}+d^{2}}}{11}-\frac {8 d^{2} e^{3} x^{7} \sqrt {-e^{2} x^{4}+d^{2}}}{39}+\frac {4 d^{3} e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{11}+\frac {139 d^{4} e \,x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{195}+\frac {3 d^{5} x \sqrt {-e^{2} x^{4}+d^{2}}}{11}+\frac {8 d^{7} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{11 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {56 d^{7} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{65 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(281\) |
| default | \(d^{3} \left (-\frac {e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {3 d^{2} x \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {4 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+e^{3} \left (-\frac {e^{2} x^{11} \sqrt {-e^{2} x^{4}+d^{2}}}{13}+\frac {5 d^{2} x^{7} \sqrt {-e^{2} x^{4}+d^{2}}}{39}-\frac {4 d^{4} x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{195 e^{2}}-\frac {4 d^{7} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{65 e^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+3 d \,e^{2} \left (-\frac {e^{2} x^{9} \sqrt {-e^{2} x^{4}+d^{2}}}{11}+\frac {13 d^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{77}-\frac {4 d^{4} x \sqrt {-e^{2} x^{4}+d^{2}}}{77 e^{2}}+\frac {4 d^{6} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{77 e^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+3 d^{2} e \left (-\frac {e^{2} x^{7} \sqrt {-e^{2} x^{4}+d^{2}}}{9}+\frac {11 d^{2} x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{45}-\frac {4 d^{5} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{15 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\) | \(539\) |
Input:
int((e*x^2+d)^3*(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2145*x*(-165*e^5*x^10-585*d*e^4*x^8-440*d^2*e^3*x^6+780*d^3*e^2*x^4+1529 *d^4*e*x^2+585*d^5)*(-e^2*x^4+d^2)^(1/2)+8/715*d^6*(65*d/(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*d/(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 = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.71 \[ \int \left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2} \, dx=-\frac {1848 \, \sqrt {-e^{2}} d^{7} x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 24 \, {\left (77 \, d^{7} + 65 \, d^{6} e\right )} \sqrt {-e^{2}} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) + {\left (165 \, e^{7} x^{12} + 585 \, d e^{6} x^{10} + 440 \, d^{2} e^{5} x^{8} - 780 \, d^{3} e^{4} x^{6} - 1529 \, d^{4} e^{3} x^{4} - 585 \, d^{5} e^{2} x^{2} + 1848 \, d^{6} e\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{2145 \, e^{2} x} \] Input:
integrate((e*x^2+d)^3*(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
Output:
-1/2145*(1848*sqrt(-e^2)*d^7*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), - 1) - 24*(77*d^7 + 65*d^6*e)*sqrt(-e^2)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt( d/e)/x), -1) + (165*e^7*x^12 + 585*d*e^6*x^10 + 440*d^2*e^5*x^8 - 780*d^3* e^4*x^6 - 1529*d^4*e^3*x^4 - 585*d^5*e^2*x^2 + 1848*d^6*e)*sqrt(-e^2*x^4 + d^2))/(e^2*x)
Time = 2.59 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.17 \[ \int \left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {d^{6} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {3 d^{5} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {d^{4} e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} - \frac {d^{3} e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} - \frac {3 d^{2} e^{4} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} - \frac {d e^{5} x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} \] Input:
integrate((e*x**2+d)**3*(-e**2*x**4+d**2)**(3/2),x)
Output:
d**6*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), e**2*x**4*exp_polar(2*I*pi)/d **2)/(4*gamma(5/4)) + 3*d**5*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/(4*gamma(7/4)) + d**4*e**2*x**5*gamma(5/ 4)*hyper((-1/2, 5/4), (9/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/(2*gamma(9 /4)) - d**3*e**3*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), e**2*x**4*exp _polar(2*I*pi)/d**2)/(2*gamma(11/4)) - 3*d**2*e**4*x**9*gamma(9/4)*hyper(( -1/2, 9/4), (13/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/(4*gamma(13/4)) - d *e**5*x**11*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), e**2*x**4*exp_polar(2 *I*pi)/d**2)/(4*gamma(15/4))
\[ \int \left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int { {\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3} \,d x } \] Input:
integrate((e*x^2+d)^3*(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)*(e*x^2 + d)^3, x)
\[ \int \left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int { {\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3} \,d x } \] Input:
integrate((e*x^2+d)^3*(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)*(e*x^2 + d)^3, x)
Timed out. \[ \int \left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int {\left (d^2-e^2\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^3 \,d x \] Input:
int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^3,x)
Output:
int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^3, x)
\[ \int \left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {3 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{5} x}{11}+\frac {139 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{4} e \,x^{3}}{195}+\frac {4 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{3} e^{2} x^{5}}{11}-\frac {8 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{2} e^{3} x^{7}}{39}-\frac {3 \sqrt {-e^{2} x^{4}+d^{2}}\, d \,e^{4} x^{9}}{11}-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{5} x^{11}}{13}+\frac {8 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{7}}{11}+\frac {56 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{6} e}{65} \] Input:
int((e*x^2+d)^3*(-e^2*x^4+d^2)^(3/2),x)
Output:
(585*sqrt(d**2 - e**2*x**4)*d**5*x + 1529*sqrt(d**2 - e**2*x**4)*d**4*e*x* *3 + 780*sqrt(d**2 - e**2*x**4)*d**3*e**2*x**5 - 440*sqrt(d**2 - e**2*x**4 )*d**2*e**3*x**7 - 585*sqrt(d**2 - e**2*x**4)*d*e**4*x**9 - 165*sqrt(d**2 - e**2*x**4)*e**5*x**11 + 1560*int(sqrt(d**2 - e**2*x**4)/(d**2 - e**2*x** 4),x)*d**7 + 1848*int((sqrt(d**2 - e**2*x**4)*x**2)/(d**2 - e**2*x**4),x)* d**6*e)/2145