Integrand size = 30, antiderivative size = 606 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt {c-d x^2}}-\frac {\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac {d^{3/4} \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 c^{11/4} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}}+\frac {b^3 \sqrt [4]{c} (7 b c-17 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}}+\frac {b^3 \sqrt [4]{c} (7 b c-17 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}} \] Output:
1/6*d*(2*a*d+3*b*c)/a/c/(-a*d+b*c)^2/e/(e*x)^(3/2)/(-d*x^2+c)^(3/2)+1/2*b/ a/(-a*d+b*c)/e/(e*x)^(3/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2)+1/2*d*(-3*a^2*d^2+7 *a*b*c*d+b^2*c^2)/a/c^2/(-a*d+b*c)^3/e/(e*x)^(3/2)/(-d*x^2+c)^(1/2)-1/6*(- 15*a^3*d^3+35*a^2*b*c*d^2-12*a*b^2*c^2*d+7*b^3*c^3)*(-d*x^2+c)^(1/2)/a^2/c ^3/(-a*d+b*c)^3/e/(e*x)^(3/2)+1/6*d^(3/4)*(-15*a^3*d^3+35*a^2*b*c*d^2-12*a *b^2*c^2*d+7*b^3*c^3)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1 /4)/e^(1/2),I)/a^2/c^(11/4)/(-a*d+b*c)^3/e^(5/2)/(-d*x^2+c)^(1/2)+1/4*b^3* c^(1/4)*(-17*a*d+7*b*c)*(1-d*x^2/c)^(1/2)*EllipticPi(d^(1/4)*(e*x)^(1/2)/c ^(1/4)/e^(1/2),-b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)/a^3/d^(1/4)/(-a*d+b*c)^ 3/e^(5/2)/(-d*x^2+c)^(1/2)+1/4*b^3*c^(1/4)*(-17*a*d+7*b*c)*(1-d*x^2/c)^(1/ 2)*EllipticPi(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),b^(1/2)*c^(1/2)/a^(1/2)/ d^(1/2),I)/a^3/d^(1/4)/(-a*d+b*c)^3/e^(5/2)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.93 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {x \left (-\frac {5 a \left (7 b^4 c^3 x^2 \left (c-d x^2\right )^2-4 a b^3 c^2 \left (c-d x^2\right )^2 \left (c+3 d x^2\right )+a^4 d^3 \left (4 c^2-21 c d x^2+15 d^2 x^4\right )-a^3 b d^2 \left (12 c^3-45 c^2 d x^2+14 c d^2 x^4+15 d^3 x^6\right )+a^2 b^2 c d \left (12 c^3-12 c^2 d x^2-37 c d^2 x^4+35 d^3 x^6\right )\right )}{(-b c+a d)^3 \left (a-b x^2\right ) \left (c-d x^2\right )}+\frac {5 \left (21 b^4 c^4-44 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+35 a^3 b c d^3-15 a^4 d^4\right ) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{(b c-a d)^3}-\frac {b d \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{(b c-a d)^3}\right )}{30 a^3 c^3 (e x)^{5/2} \sqrt {c-d x^2}} \] Input:
Integrate[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
Output:
(x*((-5*a*(7*b^4*c^3*x^2*(c - d*x^2)^2 - 4*a*b^3*c^2*(c - d*x^2)^2*(c + 3* d*x^2) + a^4*d^3*(4*c^2 - 21*c*d*x^2 + 15*d^2*x^4) - a^3*b*d^2*(12*c^3 - 4 5*c^2*d*x^2 + 14*c*d^2*x^4 + 15*d^3*x^6) + a^2*b^2*c*d*(12*c^3 - 12*c^2*d* x^2 - 37*c*d^2*x^4 + 35*d^3*x^6)))/((-(b*c) + a*d)^3*(a - b*x^2)*(c - d*x^ 2)) + (5*(21*b^4*c^4 - 44*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 35*a^3*b*c*d^ 3 - 15*a^4*d^4)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2) /c, (b*x^2)/a])/(b*c - a*d)^3 - (b*d*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2* b*c*d^2 - 15*a^3*d^3)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, ( d*x^2)/c, (b*x^2)/a])/(b*c - a*d)^3))/(30*a^3*c^3*(e*x)^(5/2)*Sqrt[c - d*x ^2])
Time = 1.35 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.567, Rules used = {368, 27, 972, 27, 1049, 27, 1049, 27, 1053, 25, 1021, 765, 762, 925, 27, 1543, 1542}
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.
| \(\displaystyle \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^2}{x^2 \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {1}{e^2 x^2 \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(7 b c-4 a d) e^2-13 b d e^2 x^2}{e^4 x^2 \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(7 b c-4 a d) e^2-13 b d e^2 x^2}{e^2 x^2 \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {6 \left (\left (7 b^2 c^2-8 a b d c+6 a^2 d^2\right ) e^2-3 b d (3 b c+2 a d) e^2 x^2\right )}{e^2 x^2 \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{6 c (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {\left (7 b^2 c^2-8 a b d c+6 a^2 d^2\right ) e^2-3 b d (3 b c+2 a d) e^2 x^2}{e^2 x^2 \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {2 \left (\left (7 b^3 c^3-12 a b^2 d c^2+35 a^2 b d^2 c-15 a^3 d^3\right ) e^2-5 b d \left (b^2 c^2+7 a b d c-3 a^2 d^2\right ) e^2 x^2\right )}{e^2 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\int \frac {\left (7 b^3 c^3-12 a b^2 d c^2+35 a^2 b d^2 c-15 a^3 d^3\right ) e^2-5 b d \left (b^2 c^2+7 a b d c-3 a^2 d^2\right ) e^2 x^2}{e^2 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {-\frac {\int -\frac {\left (21 b^4 c^4-44 a b^3 d c^3-12 a^2 b^2 d^2 c^2+35 a^3 b d^3 c-15 a^4 d^4\right ) e^2-b d \left (7 b^3 c^3-12 a b^2 d c^2+35 a^2 b d^2 c-15 a^3 d^3\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{3 a c (e x)^{3/2}}}{c (b c-a d)}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {\int \frac {\left (21 b^4 c^4-44 a b^3 d c^3-12 a^2 b^2 d^2 c^2+35 a^3 b d^3 c-15 a^4 d^4\right ) e^2-b d \left (7 b^3 c^3-12 a b^2 d c^2+35 a^2 b d^2 c-15 a^3 d^3\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{3 a c (e x)^{3/2}}}{c (b c-a d)}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {d \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}+3 b^3 c^3 e^2 (7 b c-17 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{3 a c (e x)^{3/2}}}{c (b c-a d)}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {\frac {d \sqrt {1-\frac {d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{\sqrt {c-d x^2}}+3 b^3 c^3 e^2 (7 b c-17 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{3 a c (e x)^{3/2}}}{c (b c-a d)}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {3 b^3 c^3 e^2 (7 b c-17 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{3 a c (e x)^{3/2}}}{c (b c-a d)}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {3 b^3 c^3 e^2 (7 b c-17 a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{3 a c (e x)^{3/2}}}{c (b c-a d)}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {3 b^3 c^3 e^2 (7 b c-17 a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{3 a c (e x)^{3/2}}}{c (b c-a d)}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {3 b^3 c^3 e^2 (7 b c-17 a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{3 a c (e x)^{3/2}}}{c (b c-a d)}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}+\frac {\frac {\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}+3 b^3 c^3 e^2 (7 b c-17 a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{3 a c (e x)^{3/2}}}{c (b c-a d)}}{c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
Input:
Int[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
Output:
2*e^3*(b/(4*a*(b*c - a*d)*e^2*(e*x)^(3/2)*(c - d*x^2)^(3/2)*(a*e^2 - b*e^2 *x^2)) + ((d*(3*b*c + 2*a*d))/(3*c*(b*c - a*d)*(e*x)^(3/2)*(c - d*x^2)^(3/ 2)) + ((d*(b^2*c^2 + 7*a*b*c*d - 3*a^2*d^2))/(c*(b*c - a*d)*(e*x)^(3/2)*Sq rt[c - d*x^2]) + (-1/3*((7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15* a^3*d^3)*Sqrt[c - d*x^2])/(a*c*(e*x)^(3/2)) + ((c^(1/4)*d^(3/4)*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[e]*Sqrt[1 - (d*x^2)/ c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x^2] + 3*b^3*c^3*(7*b*c - 17*a*d)*e^2*((c^(1/4)*Sqrt[1 - (d*x^2)/c]*Elli pticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/ (c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2]) + (c^(1/4)* Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin [(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2])))/(3*a*c*e^2))/(c*(b*c - a*d)))/(c*(b*c - a*d)))/(4*a*(b*c - a*d )*e^4))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
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[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 1/(2*c) Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 *c) Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x _)^(n_)]), x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* e - a*f)/b Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1463\) vs. \(2(506)=1012\).
Time = 3.90 (sec) , antiderivative size = 1464, normalized size of antiderivative = 2.42
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1464\) |
| default | \(\text {Expression too large to display}\) | \(5236\) |
Input:
int(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(1/2*b^4*d/e^3/(a^2*d^ 2-2*a*b*c*d+b^2*c^2)/a^2/(a*d-b*c)*(-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d)+1/ 3*d/e^3/c^2/(a*d-b*c)^2*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2+1/6*d^3/e^2*x/c ^3*(11*a*d-23*b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(-(x^2-c/d)*d*e*x )^(1/2)-2/3/c^3/e^3/a^2*(-d*e*x^3+c*e*x)^(1/2)/x^2-1/4*(c*d)^(1/2)*(1+x*d/ (c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(- d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1 /2*2^(1/2))*b^3/e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^2/(a*d-b*c)+11/12*d^3*(c *d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d )^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c* d)^(1/2))^(1/2),1/2*2^(1/2))/c^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/e^2 *a-23/12*d^2*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^( 1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c* d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2 )/(a*d-b*c)/e^2*b+1/3*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d) ^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF((( x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))/c^3/e^2/a^2-17/8*b^3/ e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a/(a*d-b*c)/(a*b)^(1/2)*(c*d)^(1/2)*(1+x*d /(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/( -d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x...
Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)**(5/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="maxima ")
Output:
integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)*(e*x)^(5/2)), x)
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="giac")
Output:
integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)*(e*x)^(5/2)), x)
Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \] Input:
int(1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x)
Output:
int(1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)), x)
\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)
Output:
(sqrt(e)*( - 2*sqrt(c - d*x**2) + 9*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)* a**2*c**3 - 3*sqrt(x)*a**2*c**2*d*x**2 + 3*sqrt(x)*a**2*c*d**2*x**4 - sqrt (x)*a**2*d**3*x**6 - 2*sqrt(x)*a*b*c**3*x**2 + 6*sqrt(x)*a*b*c**2*d*x**4 - 6*sqrt(x)*a*b*c*d**2*x**6 + 2*sqrt(x)*a*b*d**3*x**8 + sqrt(x)*b**2*c**3*x **4 - 3*sqrt(x)*b**2*c**2*d*x**6 + 3*sqrt(x)*b**2*c*d**2*x**8 - sqrt(x)*b* *2*d**3*x**10),x)*a**2*c**2*d*x - 18*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x) *a**2*c**3 - 3*sqrt(x)*a**2*c**2*d*x**2 + 3*sqrt(x)*a**2*c*d**2*x**4 - sqr t(x)*a**2*d**3*x**6 - 2*sqrt(x)*a*b*c**3*x**2 + 6*sqrt(x)*a*b*c**2*d*x**4 - 6*sqrt(x)*a*b*c*d**2*x**6 + 2*sqrt(x)*a*b*d**3*x**8 + sqrt(x)*b**2*c**3* x**4 - 3*sqrt(x)*b**2*c**2*d*x**6 + 3*sqrt(x)*b**2*c*d**2*x**8 - sqrt(x)*b **2*d**3*x**10),x)*a**2*c*d**2*x**3 + 9*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt (x)*a**2*c**3 - 3*sqrt(x)*a**2*c**2*d*x**2 + 3*sqrt(x)*a**2*c*d**2*x**4 - sqrt(x)*a**2*d**3*x**6 - 2*sqrt(x)*a*b*c**3*x**2 + 6*sqrt(x)*a*b*c**2*d*x* *4 - 6*sqrt(x)*a*b*c*d**2*x**6 + 2*sqrt(x)*a*b*d**3*x**8 + sqrt(x)*b**2*c* *3*x**4 - 3*sqrt(x)*b**2*c**2*d*x**6 + 3*sqrt(x)*b**2*c*d**2*x**8 - sqrt(x )*b**2*d**3*x**10),x)*a**2*d**3*x**5 + 7*sqrt(x)*int(sqrt(c - d*x**2)/(sqr t(x)*a**2*c**3 - 3*sqrt(x)*a**2*c**2*d*x**2 + 3*sqrt(x)*a**2*c*d**2*x**4 - sqrt(x)*a**2*d**3*x**6 - 2*sqrt(x)*a*b*c**3*x**2 + 6*sqrt(x)*a*b*c**2*d*x **4 - 6*sqrt(x)*a*b*c*d**2*x**6 + 2*sqrt(x)*a*b*d**3*x**8 + sqrt(x)*b**2*c **3*x**4 - 3*sqrt(x)*b**2*c**2*d*x**6 + 3*sqrt(x)*b**2*c*d**2*x**8 - sq...