Integrand size = 30, antiderivative size = 735 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e \sqrt {e x} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+19 a b c d-7 a^2 d^2\right )}{6 a c^2 (b c-a d)^3 e \sqrt {e x} \sqrt {c-d x^2}}-\frac {\left (5 b^3 c^3-12 a b^2 c^2 d+19 a^2 b c d^2-7 a^3 d^3\right ) \sqrt {c-d x^2}}{2 a^2 c^3 (b c-a d)^3 e \sqrt {e x}}-\frac {\sqrt [4]{d} \left (5 b^3 c^3-12 a b^2 c^2 d+19 a^2 b c d^2-7 a^3 d^3\right ) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{9/4} (b c-a d)^3 e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (5 b^3 c^3-12 a b^2 c^2 d+19 a^2 b c d^2-7 a^3 d^3\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 c^{9/4} (b c-a d)^3 e^{3/2} \sqrt {c-d x^2}}-\frac {5 b^{5/2} \sqrt [4]{c} (b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^3 e^{3/2} \sqrt {c-d x^2}}+\frac {5 b^{5/2} \sqrt [4]{c} (b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^3 e^{3/2} \sqrt {c-d x^2}} \] Output:
1/6*d*(2*a*d+3*b*c)/a/c/(-a*d+b*c)^2/e/(e*x)^(1/2)/(-d*x^2+c)^(3/2)+1/2*b/ a/(-a*d+b*c)/e/(e*x)^(1/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2)+1/6*d*(-7*a^2*d^2+1 9*a*b*c*d+3*b^2*c^2)/a/c^2/(-a*d+b*c)^3/e/(e*x)^(1/2)/(-d*x^2+c)^(1/2)-1/2 *(-7*a^3*d^3+19*a^2*b*c*d^2-12*a*b^2*c^2*d+5*b^3*c^3)*(-d*x^2+c)^(1/2)/a^2 /c^3/(-a*d+b*c)^3/e/(e*x)^(1/2)-1/2*d^(1/4)*(-7*a^3*d^3+19*a^2*b*c*d^2-12* a*b^2*c^2*d+5*b^3*c^3)*(1-d*x^2/c)^(1/2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^( 1/4)/e^(1/2),I)/a^2/c^(9/4)/(-a*d+b*c)^3/e^(3/2)/(-d*x^2+c)^(1/2)+1/2*d^(1 /4)*(-7*a^3*d^3+19*a^2*b*c*d^2-12*a*b^2*c^2*d+5*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^(9/4)/(-a*d+b*c)^3 /e^(3/2)/(-d*x^2+c)^(1/2)-5/4*b^(5/2)*c^(1/4)*(-3*a*d+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^(5/2)/d^(1/4)/(-a*d+b*c)^3/e^(3/2)/(-d*x^2+c)^(1/2)+5/4*b^(5 /2)*c^(1/4)*(-3*a*d+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^(5/2)/d^(1/4)/(-a*d+b *c)^3/e^(3/2)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.86 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {x \left (-\frac {7 a \left (15 b^4 c^3 x^2 \left (c-d x^2\right )^2-12 a b^3 c^2 \left (c-d x^2\right )^2 \left (c+3 d x^2\right )+a^4 d^3 \left (12 c^2-35 c d x^2+21 d^2 x^4\right )-a^3 b d^2 \left (36 c^3-83 c^2 d x^2+22 c d^2 x^4+21 d^3 x^6\right )+a^2 b^2 c d \left (36 c^3-36 c^2 d x^2-59 c d^2 x^4+57 d^3 x^6\right )\right )}{\left (a-b x^2\right ) \left (c-d x^2\right )}-7 \left (5 b^4 c^4-20 a b^3 c^3 d+12 a^2 b^2 c^2 d^2-19 a^3 b c d^3+7 a^4 d^4\right ) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d \left (-5 b^3 c^3+12 a b^2 c^2 d-19 a^2 b c d^2+7 a^3 d^3\right ) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^3 c^3 (-b c+a d)^3 (e x)^{3/2} \sqrt {c-d x^2}} \] Input:
Integrate[1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
Output:
(x*((-7*a*(15*b^4*c^3*x^2*(c - d*x^2)^2 - 12*a*b^3*c^2*(c - d*x^2)^2*(c + 3*d*x^2) + a^4*d^3*(12*c^2 - 35*c*d*x^2 + 21*d^2*x^4) - a^3*b*d^2*(36*c^3 - 83*c^2*d*x^2 + 22*c*d^2*x^4 + 21*d^3*x^6) + a^2*b^2*c*d*(36*c^3 - 36*c^2 *d*x^2 - 59*c*d^2*x^4 + 57*d^3*x^6)))/((a - b*x^2)*(c - d*x^2)) - 7*(5*b^4 *c^4 - 20*a*b^3*c^3*d + 12*a^2*b^2*c^2*d^2 - 19*a^3*b*c*d^3 + 7*a^4*d^4)*x ^2*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(-5*b^3*c^3 + 12*a*b^2*c^2*d - 19*a^2*b*c*d^2 + 7*a^3*d^3)*x^4*Sqrt[ 1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^3 *c^3*(-(b*c) + a*d)^3*(e*x)^(3/2)*Sqrt[c - d*x^2])
Time = 1.36 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {368, 27, 972, 27, 1049, 27, 1049, 27, 1053, 25, 1054, 2009}
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.
| \(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^3}{x \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {1}{e x \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(5 b c-4 a d) e^2-11 b d e^2 x^2}{e^3 x \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(5 b c-4 a d) e^2-11 b d e^2 x^2}{e x \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {d (2 a d+3 b c)}{3 c \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {2 \left (\left (15 b^2 c^2-24 a b d c+14 a^2 d^2\right ) e^2-7 b d (3 b c+2 a d) e^2 x^2\right )}{e x \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{6 c (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {\left (15 b^2 c^2-24 a b d c+14 a^2 d^2\right ) e^2-7 b d (3 b c+2 a d) e^2 x^2}{e x \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {d \left (-7 a^2 d^2+19 a b c d+3 b^2 c^2\right )}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {6 \left (\left (5 b^3 c^3-12 a b^2 d c^2+19 a^2 b d^2 c-7 a^3 d^3\right ) e^2-b d \left (3 b^2 c^2+19 a b d c-7 a^2 d^2\right ) e^2 x^2\right )}{e x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}}{3 c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 \int \frac {\left (5 b^3 c^3-12 a b^2 d c^2+19 a^2 b d^2 c-7 a^3 d^3\right ) e^2-b d \left (3 b^2 c^2+19 a b d c-7 a^2 d^2\right ) e^2 x^2}{e x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d \left (-7 a^2 d^2+19 a b c d+3 b^2 c^2\right )}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 \left (-\frac {\int -\frac {e x \left (b d \left (5 b^3 c^3-12 a b^2 d c^2+19 a^2 b d^2 c-7 a^3 d^3\right ) x^2 e^2+\left (5 b^4 c^4-20 a b^3 d c^3+12 a^2 b^2 d^2 c^2-19 a^3 b d^3 c+7 a^4 d^4\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right )}{a c \sqrt {e x}}\right )}{c (b c-a d)}+\frac {d \left (-7 a^2 d^2+19 a b c d+3 b^2 c^2\right )}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 \left (\frac {\int \frac {e x \left (b d \left (5 b^3 c^3-12 a b^2 d c^2+19 a^2 b d^2 c-7 a^3 d^3\right ) x^2 e^2+\left (5 b^4 c^4-20 a b^3 d c^3+12 a^2 b^2 d^2 c^2-19 a^3 b d^3 c+7 a^4 d^4\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right )}{a c \sqrt {e x}}\right )}{c (b c-a d)}+\frac {d \left (-7 a^2 d^2+19 a b c d+3 b^2 c^2\right )}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 \left (\frac {\int \left (\frac {5 e \left (b^4 c^4 e^2-3 a b^3 c^3 d e^2\right ) x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d \left (5 b^3 c^3-12 a b^2 d c^2+19 a^2 b d^2 c-7 a^3 d^3\right ) e x}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right )}{a c \sqrt {e x}}\right )}{c (b c-a d)}+\frac {d \left (-7 a^2 d^2+19 a b c d+3 b^2 c^2\right )}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {d \left (-7 a^2 d^2+19 a b c d+3 b^2 c^2\right )}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}+\frac {3 \left (\frac {\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}-\frac {5 b^{5/2} c^{13/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-3 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {5 b^{5/2} c^{13/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-3 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}}{a c e^2}-\frac {\sqrt {c-d x^2} \left (-7 a^3 d^3+19 a^2 b c d^2-12 a b^2 c^2 d+5 b^3 c^3\right )}{a c \sqrt {e x}}\right )}{c (b c-a d)}}{3 c (b c-a d)}+\frac {d (2 a d+3 b c)}{3 c \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
Input:
Int[1/((e*x)^(3/2)*(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*Sqrt[e*x]*(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)*Sqrt[e*x]*(c - d*x^2)^(3/2)) + ((d*(3*b^2*c^2 + 19*a*b*c*d - 7*a^2*d^2))/(c*(b*c - a*d)*Sqrt[e*x]*Sqrt[ c - d*x^2]) + (3*(-(((5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^3* d^3)*Sqrt[c - d*x^2])/(a*c*Sqrt[e*x])) + (-((c^(3/4)*d^(1/4)*(5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^3*d^3)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*E llipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x^ 2]) + (c^(3/4)*d^(1/4)*(5*b^3*c^3 - 12*a*b^2*c^2*d + 19*a^2*b*c*d^2 - 7*a^ 3*d^3)*e^(3/2)*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] - (5*b^(5/2)*c^(13/4)*(b*c - 3*a*d) *e^(3/2)*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*Sqrt[a]*d^(1/ 4)*Sqrt[c - d*x^2]) + (5*b^(5/2)*c^(13/4)*(b*c - 3*a*d)*e^(3/2)*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*Sqrt[a]*d^(1/4)*Sqrt[c - d*x^2]))/( a*c*e^2)))/(c*(b*c - a*d)))/(3*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[((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[((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[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1887\) vs. \(2(607)=1214\).
Time = 3.73 (sec) , antiderivative size = 1888, normalized size of antiderivative = 2.57
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1888\) |
| default | \(\text {Expression too large to display}\) | \(6322\) |
Input:
int(1/(e*x)^(3/2)/(-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^2/(a^2*d^ 2-2*a*b*c*d+b^2*c^2)/a^2/(a*d-b*c)*x*(-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d)+ 1/3*d/e^2/(a*d-b*c)^2/c^2*x*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2+1/2*d^3/e*x ^2/c^3*(3*a*d-7*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*(-d*e*x^2+c*e)/c^3/e^2/a^2/(x*(-d*e*x^2+c*e))^(1/2)-1/2*c*(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)*b^3/e/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^2/(a*d-b*c)*El lipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/4*c*(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)*b^3/e/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^2/(a*d-b*c)*Ell ipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+3/2*d^3/c^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)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/e*a*Elli pticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-3/4*d^3/c^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)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/e*a*Ellip ticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-7/2*d^2/c*(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)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/e*b*Elliptic E(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+7/4*d^2/c*(1+x...
Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {3}{2}} \left (- a + b x^{2}\right )^{2} \left (c - d x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(e*x)**(3/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)
Output:
Integral(1/((e*x)**(3/2)*(-a + b*x**2)**2*(c - d*x**2)**(5/2)), x)
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(3/2)/(-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)^(3/2)), x)
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(3/2)/(-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)^(3/2)), x)
Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \] Input:
int(1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x)
Output:
int(1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)), x)
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int(1/(e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)
Output:
(sqrt(e)*( - 2*sqrt(c - d*x**2) - 11*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2) *x**2)/(a**2*c**3 - 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 - a**2*d**3*x* *6 - 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 - 6*a*b*c*d**2*x**6 + 2*a*b*d**3* x**8 + b**2*c**3*x**4 - 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 - b**2*d** 3*x**10),x)*a*b*c**2*d + 22*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a **2*c**3 - 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 - a**2*d**3*x**6 - 2*a* b*c**3*x**2 + 6*a*b*c**2*d*x**4 - 6*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b* *2*c**3*x**4 - 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 - b**2*d**3*x**10), x)*a*b*c*d**2*x**2 - 11*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a**2* c**3 - 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 - a**2*d**3*x**6 - 2*a*b*c* *3*x**2 + 6*a*b*c**2*d*x**4 - 6*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c **3*x**4 - 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 - b**2*d**3*x**10),x)*a *b*d**3*x**4 + 11*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a**2*c**3 - 3*a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 - a**2*d**3*x**6 - 2*a*b*c**3*x** 2 + 6*a*b*c**2*d*x**4 - 6*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x* *4 - 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 - b**2*d**3*x**10),x)*b**2*c* *2*d*x**2 - 22*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a**2*c**3 - 3* a**2*c**2*d*x**2 + 3*a**2*c*d**2*x**4 - a**2*d**3*x**6 - 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 - 6*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x**4 - 3*b**2*c**2*d*x**6 + 3*b**2*c*d**2*x**8 - b**2*d**3*x**10),x)*b**2*c*...