Integrand size = 30, antiderivative size = 514 \[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {d (3 b c+2 a d) \sqrt {e x}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b \sqrt {e x}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) \sqrt {e x}}{6 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}+\frac {d^{3/4} \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a c^{7/4} (b c-a d)^3 \sqrt {e} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (3 b c-13 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^2 \sqrt [4]{d} (b c-a d)^3 \sqrt {e} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (3 b c-13 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^2 \sqrt [4]{d} (b c-a d)^3 \sqrt {e} \sqrt {c-d x^2}} \] Output:
1/6*d*(2*a*d+3*b*c)*(e*x)^(1/2)/a/c/(-a*d+b*c)^2/e/(-d*x^2+c)^(3/2)+1/2*b* (e*x)^(1/2)/a/(-a*d+b*c)/e/(-b*x^2+a)/(-d*x^2+c)^(3/2)+1/6*d*(-5*a^2*d^2+1 7*a*b*c*d+3*b^2*c^2)*(e*x)^(1/2)/a/c^2/(-a*d+b*c)^3/e/(-d*x^2+c)^(1/2)+1/6 *d^(3/4)*(-5*a^2*d^2+17*a*b*c*d+3*b^2*c^2)*(1-d*x^2/c)^(1/2)*EllipticF(d^( 1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/c^(7/4)/(-a*d+b*c)^3/e^(1/2)/(-d*x^2 +c)^(1/2)+1/4*b^2*c^(1/4)*(-13*a*d+3*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^2/d ^(1/4)/(-a*d+b*c)^3/e^(1/2)/(-d*x^2+c)^(1/2)+1/4*b^2*c^(1/4)*(-13*a*d+3*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^2/d^(1/4)/(-a*d+b*c)^3/e^(1/2)/(-d*x^2+c)^( 1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.37 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=-\frac {5 a x \left (3 b^3 c^2 \left (c-d x^2\right )^2+a^3 d^3 \left (-7 c+5 d x^2\right )+a b^2 c d^2 x^2 \left (-19 c+17 d x^2\right )+a^2 b d^2 \left (19 c^2-10 c d x^2-5 d^2 x^4\right )\right )-5 \left (-9 b^3 c^3+36 a b^2 c^2 d-17 a^2 b c d^2+5 a^3 d^3\right ) x \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b d \left (3 b^2 c^2+17 a b c d-5 a^2 d^2\right ) x^3 \left (-a+b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{30 a^2 c^2 (b c-a d)^3 \sqrt {e x} \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \] Input:
Integrate[1/(Sqrt[e*x]*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
Output:
-1/30*(5*a*x*(3*b^3*c^2*(c - d*x^2)^2 + a^3*d^3*(-7*c + 5*d*x^2) + a*b^2*c *d^2*x^2*(-19*c + 17*d*x^2) + a^2*b*d^2*(19*c^2 - 10*c*d*x^2 - 5*d^2*x^4)) - 5*(-9*b^3*c^3 + 36*a*b^2*c^2*d - 17*a^2*b*c*d^2 + 5*a^3*d^3)*x*(a - b*x ^2)*(c - d*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*d*(3*b^2*c^2 + 17*a*b*c*d - 5*a^2*d^2)*x^3*(-a + b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2) /a])/(a^2*c^2*(b*c - a*d)^3*Sqrt[e*x]*(-a + b*x^2)*(c - d*x^2)^(3/2))
Time = 1.04 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {368, 27, 931, 27, 1024, 27, 1024, 27, 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.
| \(\displaystyle \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^4}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {1}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(3 b c-4 a d) e^2-9 b d e^2 x^2}{e^2 \left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(3 b c-4 a d) e^2-9 b d e^2 x^2}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {2 \left (\left (9 b^2 c^2-24 a b d c+10 a^2 d^2\right ) e^2-5 b d (3 b c+2 a d) e^2 x^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{6 c (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {\left (9 b^2 c^2-24 a b d c+10 a^2 d^2\right ) e^2-5 b d (3 b c+2 a d) e^2 x^2}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {2 \left (\left (9 b^3 c^3-36 a b^2 d c^2+17 a^2 b d^2 c-5 a^3 d^3\right ) e^2-b d \left (3 b^2 c^2+17 a b d c-5 a^2 d^2\right ) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\int \frac {\left (9 b^3 c^3-36 a b^2 d c^2+17 a^2 b d^2 c-5 a^3 d^3\right ) e^2-b d \left (3 b^2 c^2+17 a b d c-5 a^2 d^2\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {d \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right ) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}+3 b^2 c^2 e^2 (3 b c-13 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {d \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{\sqrt {c-d x^2}}+3 b^2 c^2 e^2 (3 b c-13 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 b^2 c^2 e^2 (3 b c-13 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{c (b c-a d)}+\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 b^2 c^2 e^2 (3 b c-13 a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{c (b c-a d)}+\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 b^2 c^2 e^2 (3 b c-13 a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{c (b c-a d)}+\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 b^2 c^2 e^2 (3 b c-13 a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{c (b c-a d)}+\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}+3 b^2 c^2 e^2 (3 b c-13 a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{c (b c-a d)}+\frac {d \sqrt {e x} \left (-5 a^2 d^2+17 a b c d+3 b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{3 c (b c-a d)}+\frac {d \sqrt {e x} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {e x}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
Input:
Int[1/(Sqrt[e*x]*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
Output:
2*e^3*((b*Sqrt[e*x])/(4*a*(b*c - a*d)*e^2*(c - d*x^2)^(3/2)*(a*e^2 - b*e^2 *x^2)) + ((d*(3*b*c + 2*a*d)*Sqrt[e*x])/(3*c*(b*c - a*d)*(c - d*x^2)^(3/2) ) + ((d*(3*b^2*c^2 + 17*a*b*c*d - 5*a^2*d^2)*Sqrt[e*x])/(c*(b*c - a*d)*Sqr t[c - d*x^2]) + ((c^(1/4)*d^(3/4)*(3*b^2*c^2 + 17*a*b*c*d - 5*a^2*d^2)*Sqr t[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqr t[e])], -1])/Sqrt[c - d*x^2] + 3*b^2*c^2*(3*b*c - 13*a*d)*e^2*((c^(1/4)*Sq rt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSi n[(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])/(Sq rt[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*(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[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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f _.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -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. \(1310\) vs. \(2(420)=840\).
Time = 3.78 (sec) , antiderivative size = 1311, normalized size of antiderivative = 2.55
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1311\) |
| default | \(\text {Expression too large to display}\) | \(4764\) |
Input:
int(1/(e*x)^(1/2)/(-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^3*d/e/(a*d-b*c) /a/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d)+1/3/e/ c/(a*d-b*c)^2*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2+1/6*d^2*x/c^2*(5*a*d-17*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)-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^2/(a*d-b*c)/a/(a^2*d^2-2*a*b*c*d+b^2*c^2)+5/ 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)*a-17/12*d*(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*b/(a^2*d^2-2*a*b*c*d+b^2* c^2)/(a*d-b*c)-13/8*b^2/(a*d-b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(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+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1 /d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))+3/8*b^3/(a*d-b*c)/a/(a^2*d^2- 2*a*b*c*d+b^2*c^2)/(a*b)^(1/2)/d*(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...
Timed out. \[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {e x} \left (- a + b x^{2}\right )^{2} \left (c - d x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(e*x)**(1/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)
Output:
Integral(1/(sqrt(e*x)*(-a + b*x**2)**2*(c - d*x**2)**(5/2)), x)
\[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {e x}} \,d x } \] Input:
integrate(1/(e*x)^(1/2)/(-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)*sqrt(e*x)), x)
\[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {e x}} \,d x } \] Input:
integrate(1/(e*x)^(1/2)/(-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)*sqrt(e*x)), x)
Timed out. \[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {e\,x}\,{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \] Input:
int(1/((e*x)^(1/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x)
Output:
int(1/((e*x)^(1/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)), x)
\[ \int \frac {1}{\sqrt {e x} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a^{2} c^{3}-3 \sqrt {x}\, a^{2} c^{2} d \,x^{2}+3 \sqrt {x}\, a^{2} c \,d^{2} x^{4}-\sqrt {x}\, a^{2} d^{3} x^{6}-2 \sqrt {x}\, a b \,c^{3} x^{2}+6 \sqrt {x}\, a b \,c^{2} d \,x^{4}-6 \sqrt {x}\, a b c \,d^{2} x^{6}+2 \sqrt {x}\, a b \,d^{3} x^{8}+\sqrt {x}\, b^{2} c^{3} x^{4}-3 \sqrt {x}\, b^{2} c^{2} d \,x^{6}+3 \sqrt {x}\, b^{2} c \,d^{2} x^{8}-\sqrt {x}\, b^{2} d^{3} x^{10}}d x \right )}{e} \] Input:
int(1/(e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)
Output:
(sqrt(e)*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))/e