Integrand size = 30, antiderivative size = 505 \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {c (2 b c+3 a d) e^5 \sqrt {e x}}{6 b d (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{5/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac {\left (b^2 c^2-13 a b c d-3 a^2 d^2\right ) e^5 \sqrt {e x}}{6 b d (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} \left (b^2 c^2-13 a b c d-3 a^2 d^2\right ) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 b d^{5/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {a \sqrt [4]{c} (9 b c+a d) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {a \sqrt [4]{c} (9 b c+a d) e^{11/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \] Output:
1/6*c*(3*a*d+2*b*c)*e^5*(e*x)^(1/2)/b/d/(-a*d+b*c)^2/(-d*x^2+c)^(3/2)+1/2* a*e^3*(e*x)^(5/2)/b/(-a*d+b*c)/(-b*x^2+a)/(-d*x^2+c)^(3/2)-1/6*(-3*a^2*d^2 -13*a*b*c*d+b^2*c^2)*e^5*(e*x)^(1/2)/b/d/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-1/6 *c^(1/4)*(-3*a^2*d^2-13*a*b*c*d+b^2*c^2)*e^(11/2)*(1-d*x^2/c)^(1/2)*Ellipt icF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/b/d^(5/4)/(-a*d+b*c)^3/(-d*x^2+ c)^(1/2)-1/4*a*c^(1/4)*(a*d+9*b*c)*e^(11/2)*(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)/b/d ^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-1/4*a*c^(1/4)*(a*d+9*b*c)*e^(11/2)*(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)/b/d^(1/4)/(-a*d+b*c)^3/(-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 = 280, normalized size of antiderivative = 0.55 \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {e^5 \sqrt {e x} \left (5 a \left (b^2 c^2 x^2 \left (c+d x^2\right )+a^2 d \left (-14 c^2+19 c d x^2-3 d^2 x^4\right )-a b c \left (c^2-10 c d x^2+13 d^2 x^4\right )\right )+5 a c (b c+14 a d) \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+\left (-b^2 c^2+13 a b c d+3 a^2 d^2\right ) x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a d (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \] Input:
Integrate[(e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
Output:
(e^5*Sqrt[e*x]*(5*a*(b^2*c^2*x^2*(c + d*x^2) + a^2*d*(-14*c^2 + 19*c*d*x^2 - 3*d^2*x^4) - a*b*c*(c^2 - 10*c*d*x^2 + 13*d^2*x^4)) + 5*a*c*(b*c + 14*a *d)*(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^2*c^2) + 13*a*b*c*d + 3*a^2*d^2)*x^2*(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]))/(30*a*d*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))
Time = 1.13 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {368, 27, 970, 1048, 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.
| \(\displaystyle \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^{10} x^6}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {e^6 x^6}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 970 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\int \frac {e^2 x^2 \left ((4 b c+a d) x^2 e^2+5 a c e^2\right )}{\left (c-d x^2\right )^{5/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1048 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {-\frac {e^2 \int -\frac {2 \left (a c (2 b c+3 a d) e^2-\left (2 b^2 c^2-24 a b d c-3 a^2 d^2\right ) e^2 x^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{6 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \int \frac {a c (2 b c+3 a d) e^2-\left (2 b^2 c^2-24 a b d c-3 a^2 d^2\right ) e^2 x^2}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {\sqrt {e x} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right )}{\sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {2 b c \left (a c (b c+14 a d) e^2-\left (b^2 c^2-13 a b d c-3 a^2 d^2\right ) e^2 x^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}\right )}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {b \int \frac {a c (b c+14 a d) e^2-\left (b^2 c^2-13 a b d c-3 a^2 d^2\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b c-a d}+\frac {\sqrt {e x} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right )}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {b \left (\frac {\left (-3 a^2 d^2-13 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}+\frac {3 a^2 d e^2 (a d+9 b c) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{b c-a d}+\frac {\sqrt {e x} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right )}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {b \left (\frac {\sqrt {1-\frac {d x^2}{c}} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}+\frac {3 a^2 d e^2 (a d+9 b c) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}\right )}{b c-a d}+\frac {\sqrt {e x} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right )}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {b \left (\frac {3 a^2 d e^2 (a d+9 b c) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{b c-a d}+\frac {\sqrt {e x} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right )}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {b \left (\frac {3 a^2 d e^2 (a d+9 b c) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{b c-a d}+\frac {\sqrt {e x} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right )}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {b \left (\frac {3 a^2 d e^2 (a d+9 b c) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{b c-a d}+\frac {\sqrt {e x} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right )}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {b \left (\frac {3 a^2 d e^2 (a d+9 b c) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}\right )}{b c-a d}+\frac {\sqrt {e x} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right )}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 e^3 \left (\frac {a e^2 (e x)^{5/2}}{4 b \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {\frac {e^2 \left (\frac {b \left (\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {3 a^2 d e^2 (a d+9 b c) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}\right )}{b c-a d}+\frac {\sqrt {e x} \left (-3 a^2 d^2-13 a b c d+b^2 c^2\right )}{\sqrt {c-d x^2} (b c-a d)}\right )}{3 d (b c-a d)}-\frac {c e^2 \sqrt {e x} (3 a d+2 b c)}{3 d \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 b (b c-a d)}\right )\) |
Input:
Int[(e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
Output:
2*e^3*((a*e^2*(e*x)^(5/2))/(4*b*(b*c - a*d)*(c - d*x^2)^(3/2)*(a*e^2 - b*e ^2*x^2)) - (-1/3*(c*(2*b*c + 3*a*d)*e^2*Sqrt[e*x])/(d*(b*c - a*d)*(c - d*x ^2)^(3/2)) + (e^2*(((b^2*c^2 - 13*a*b*c*d - 3*a^2*d^2)*Sqrt[e*x])/((b*c - a*d)*Sqrt[c - d*x^2]) + (b*((c^(1/4)*(b^2*c^2 - 13*a*b*c*d - 3*a^2*d^2)*Sq rt[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sq rt[e])], -1])/(b*d^(1/4)*Sqrt[c - d*x^2]) + (3*a^2*d*(9*b*c + a*d)*e^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]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqr t[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])))/b))/(b*c - a*d)))/(3*d*(b*c - a *d)))/(4*b*(b*c - a*d)))
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[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) ^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) /(b*n*(b*c - a*d)*(p + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d *x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && 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[((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[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[g^(n - 1)*(b*e - a*f)*( g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)* (p + 1))), x] - Simp[g^n/(b*n*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a* f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]
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. \(1338\) vs. \(2(411)=822\).
Time = 3.25 (sec) , antiderivative size = 1339, normalized size of antiderivative = 2.65
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1339\) |
| default | \(\text {Expression too large to display}\) | \(4752\) |
Input:
int((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*d*a^2*e^5/( a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*(-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d)+ 1/3*c^2*e^5/(a*d-b*c)^2/d^3*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2-1/6/d*e^6*x *c*(13*a*d-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))*a^2*e^6/(a^2*d^2-2*a*b*c*d+b^2*c^2 )/(a*d-b*c)/b-13/12/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*e^6/(a^2*d^2-2*a*b* c*d+b^2*c^2)/(a*d-b*c)*a+1/12/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*e^6/( a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*b-1/8*a^3*e^6/b/(a^2*d^2-2*a*b*c*d+b^ 2*c^2)/(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+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))- 9/8*a^2*e^6/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(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/...
Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="fricas" )
Output:
Timed out
Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(11/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)
Output:
Timed out
\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {11}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="maxima" )
Output:
integrate((e*x)^(11/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)
Exception generated. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Exception raised: AttributeError} \] Input:
integrate((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="giac")
Output:
Exception raised: AttributeError >> type
Timed out. \[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{11/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \] Input:
int((e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x)
Output:
int((e*x)^(11/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x)
\[ \int \frac {(e x)^{11/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((e*x)^(11/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)
Output:
(sqrt(e)*e**5*(10*sqrt(x)*sqrt(c - d*x**2)*a*c - 10*sqrt(x)*sqrt(c - d*x** 2)*a*d*x**2 - 6*sqrt(x)*sqrt(c - d*x**2)*b*c*x**2 - 25*int(sqrt(c - d*x**2 )/(5*sqrt(x)*a**3*c**3*d - 15*sqrt(x)*a**3*c**2*d**2*x**2 + 15*sqrt(x)*a** 3*c*d**3*x**4 - 5*sqrt(x)*a**3*d**4*x**6 + 3*sqrt(x)*a**2*b*c**4 - 19*sqrt (x)*a**2*b*c**3*d*x**2 + 39*sqrt(x)*a**2*b*c**2*d**2*x**4 - 33*sqrt(x)*a** 2*b*c*d**3*x**6 + 10*sqrt(x)*a**2*b*d**4*x**8 - 6*sqrt(x)*a*b**2*c**4*x**2 + 23*sqrt(x)*a*b**2*c**3*d*x**4 - 33*sqrt(x)*a*b**2*c**2*d**2*x**6 + 21*s qrt(x)*a*b**2*c*d**3*x**8 - 5*sqrt(x)*a*b**2*d**4*x**10 + 3*sqrt(x)*b**3*c **4*x**4 - 9*sqrt(x)*b**3*c**3*d*x**6 + 9*sqrt(x)*b**3*c**2*d**2*x**8 - 3* sqrt(x)*b**3*c*d**3*x**10),x)*a**4*c**4*d + 50*int(sqrt(c - d*x**2)/(5*sqr t(x)*a**3*c**3*d - 15*sqrt(x)*a**3*c**2*d**2*x**2 + 15*sqrt(x)*a**3*c*d**3 *x**4 - 5*sqrt(x)*a**3*d**4*x**6 + 3*sqrt(x)*a**2*b*c**4 - 19*sqrt(x)*a**2 *b*c**3*d*x**2 + 39*sqrt(x)*a**2*b*c**2*d**2*x**4 - 33*sqrt(x)*a**2*b*c*d* *3*x**6 + 10*sqrt(x)*a**2*b*d**4*x**8 - 6*sqrt(x)*a*b**2*c**4*x**2 + 23*sq rt(x)*a*b**2*c**3*d*x**4 - 33*sqrt(x)*a*b**2*c**2*d**2*x**6 + 21*sqrt(x)*a *b**2*c*d**3*x**8 - 5*sqrt(x)*a*b**2*d**4*x**10 + 3*sqrt(x)*b**3*c**4*x**4 - 9*sqrt(x)*b**3*c**3*d*x**6 + 9*sqrt(x)*b**3*c**2*d**2*x**8 - 3*sqrt(x)* b**3*c*d**3*x**10),x)*a**4*c**3*d**2*x**2 - 25*int(sqrt(c - d*x**2)/(5*sqr t(x)*a**3*c**3*d - 15*sqrt(x)*a**3*c**2*d**2*x**2 + 15*sqrt(x)*a**3*c*d**3 *x**4 - 5*sqrt(x)*a**3*d**4*x**6 + 3*sqrt(x)*a**2*b*c**4 - 19*sqrt(x)*a...