\(\int \frac {1}{(e x)^{5/2} (a-b x^2)^2 (c-d x^2)^{3/2}} \, dx\) [1151]

Optimal result

Integrand size = 30, antiderivative size = 512 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e (e x)^{3/2} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) \sqrt {c-d x^2}}{6 a^2 c^2 (b c-a d)^2 e (e x)^{3/2}}+\frac {d^{3/4} \left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 c^{7/4} (b c-a d)^2 e^{5/2} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (7 b c-13 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^2 e^{5/2} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (7 b c-13 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^2 e^{5/2} \sqrt {c-d x^2}} \] Output:

1/2*d*(2*a*d+b*c)/a/c/(-a*d+b*c)^2/e/(e*x)^(3/2)/(-d*x^2+c)^(1/2)+1/2*b/a/ 
(-a*d+b*c)/e/(e*x)^(3/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2)-1/6*(10*a^2*d^2-8*a*b 
*c*d+7*b^2*c^2)*(-d*x^2+c)^(1/2)/a^2/c^2/(-a*d+b*c)^2/e/(e*x)^(3/2)+1/6*d^ 
(3/4)*(10*a^2*d^2-8*a*b*c*d+7*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^2/c^(7/4)/(-a*d+b*c)^2/e^(5/2)/(-d*x^2+c 
)^(1/2)+1/4*b^2*c^(1/4)*(-13*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)^2/e^(5/2)/(-d*x^2+c)^(1/2)+1/4*b^2*c^(1/4)*(-13*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)^2/e^(5/2)/(-d*x^2+c)^(1/ 
2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.34 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {x \left (5 a \left (2 a^3 d^2 \left (2 c-5 d x^2\right )-7 b^3 c^2 x^2 \left (c-d x^2\right )+4 a b^2 c \left (c^2+c d x^2-2 d^2 x^4\right )+2 a^2 b d \left (-4 c^2+2 c d x^2+5 d^2 x^4\right )\right )+5 \left (21 b^3 c^3-32 a b^2 c^2 d-8 a^2 b c d^2+10 a^3 d^3\right ) x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b d \left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a^3 c^2 (b c-a d)^2 (e x)^{5/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

Integrate[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]
 

Output:

(x*(5*a*(2*a^3*d^2*(2*c - 5*d*x^2) - 7*b^3*c^2*x^2*(c - d*x^2) + 4*a*b^2*c 
*(c^2 + c*d*x^2 - 2*d^2*x^4) + 2*a^2*b*d*(-4*c^2 + 2*c*d*x^2 + 5*d^2*x^4)) 
 + 5*(21*b^3*c^3 - 32*a*b^2*c^2*d - 8*a^2*b*c*d^2 + 10*a^3*d^3)*x^2*(-a + 
b*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*(7*b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*x^4*(a - b*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^3*c^2*(b*c 
 - a*d)^2*(e*x)^(5/2)*(-a + b*x^2)*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 507, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {368, 27, 972, 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.

\(\displaystyle \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle \frac {2 \int \frac {e^2}{x^2 \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \int \frac {1}{e^2 x^2 \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\)

\(\Big \downarrow \) 972

\(\displaystyle 2 e^3 \left (\frac {\int \frac {(7 b c-4 a d) e^2-9 b d e^2 x^2}{e^4 x^2 \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\int \frac {(7 b c-4 a d) e^2-9 b d e^2 x^2}{e^2 x^2 \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1049

\(\displaystyle 2 e^3 \left (\frac {\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {2 \left (\left (7 b^2 c^2-8 a b d c+10 a^2 d^2\right ) e^2-5 b d (b c+2 a d) e^2 x^2\right )}{e^2 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {\left (7 b^2 c^2-8 a b d c+10 a^2 d^2\right ) e^2-5 b d (b c+2 a d) e^2 x^2}{e^2 x^2 \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1053

\(\displaystyle 2 e^3 \left (\frac {\frac {-\frac {\int -\frac {\left (21 b^3 c^3-32 a b^2 d c^2-8 a^2 b d^2 c+10 a^3 d^3\right ) e^2-b d \left (7 b^2 c^2-8 a b d c+10 a^2 d^2\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {7 b^2 c}{a}+\frac {10 a d^2}{c}-8 b d\right )}{3 (e x)^{3/2}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\int \frac {\left (21 b^3 c^3-32 a b^2 d c^2-8 a^2 b d^2 c+10 a^3 d^3\right ) e^2-b d \left (7 b^2 c^2-8 a b d c+10 a^2 d^2\right ) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {7 b^2 c}{a}+\frac {10 a d^2}{c}-8 b d\right )}{3 (e x)^{3/2}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1021

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {d \left (10 a^2 d^2-8 a b c d+7 b^2 c^2\right ) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}+3 b^2 c^2 e^2 (7 b c-13 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {7 b^2 c}{a}+\frac {10 a d^2}{c}-8 b d\right )}{3 (e x)^{3/2}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 765

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {d \sqrt {1-\frac {d x^2}{c}} \left (10 a^2 d^2-8 a b c d+7 b^2 c^2\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{\sqrt {c-d x^2}}+3 b^2 c^2 e^2 (7 b c-13 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {7 b^2 c}{a}+\frac {10 a d^2}{c}-8 b d\right )}{3 (e x)^{3/2}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 762

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 b^2 c^2 e^2 (7 b c-13 a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (10 a^2 d^2-8 a b c d+7 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {7 b^2 c}{a}+\frac {10 a d^2}{c}-8 b d\right )}{3 (e x)^{3/2}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 925

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 b^2 c^2 e^2 (7 b c-13 a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (10 a^2 d^2-8 a b c d+7 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {7 b^2 c}{a}+\frac {10 a d^2}{c}-8 b d\right )}{3 (e x)^{3/2}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 b^2 c^2 e^2 (7 b c-13 a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (10 a^2 d^2-8 a b c d+7 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {7 b^2 c}{a}+\frac {10 a d^2}{c}-8 b d\right )}{3 (e x)^{3/2}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1543

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {3 b^2 c^2 e^2 (7 b c-13 a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (10 a^2 d^2-8 a b c d+7 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {7 b^2 c}{a}+\frac {10 a d^2}{c}-8 b d\right )}{3 (e x)^{3/2}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1542

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (10 a^2 d^2-8 a b c d+7 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}+3 b^2 c^2 e^2 (7 b c-13 a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{3 a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {7 b^2 c}{a}+\frac {10 a d^2}{c}-8 b d\right )}{3 (e x)^{3/2}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

Input:

Int[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]
 

Output:

2*e^3*(b/(4*a*(b*c - a*d)*e^2*(e*x)^(3/2)*Sqrt[c - d*x^2]*(a*e^2 - b*e^2*x 
^2)) + ((d*(b*c + 2*a*d))/(c*(b*c - a*d)*(e*x)^(3/2)*Sqrt[c - d*x^2]) + (- 
1/3*(((7*b^2*c)/a - 8*b*d + (10*a*d^2)/c)*Sqrt[c - d*x^2])/(e*x)^(3/2) + ( 
(c^(1/4)*d^(3/4)*(7*b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*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^2*c^2*(7*b*c - 13*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]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), A 
rcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sq 
rt[c - d*x^2])))/(3*a*c*e^2))/(c*(b*c - a*d)))/(4*a*(b*c - a*d)*e^4))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 368

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]
 

rule 762

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]
 

rule 765

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]
 

rule 925

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]
 

rule 972

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]
 

rule 1021

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]
 

rule 1049

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]
 

rule 1053

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]
 

rule 1542

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]
 

rule 1543

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1093\) vs. \(2(418)=836\).

Time = 3.36 (sec) , antiderivative size = 1094, normalized size of antiderivative = 2.14

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1094\)
default	\(\text {Expression too large to display}\)	\(2859\)

Input:

int(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(d^3/e^2*x/c^2/(a*d-b* 
c)^2/(-(x^2-c/d)*d*e*x)^(1/2)+1/2*b^3/a^2/(a*d-b*c)^2/e^3*(-d*e*x^3+c*e*x) 
^(1/2)/(-b*x^2+a)-2/3/c^2/e^3/a^2*(-d*e*x^3+c*e*x)^(1/2)/x^2+1/2*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^2/(a*d-b*c)^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/e^2/a^2/(a*d-b*c)^2+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^2/e^2/ 
a^2+13/8*b^2/e^2/a/(a*d-b*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))-7/8*b^3/e^2/a^2/(a*d-b*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)/(-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))*c-13/8*b^2/e^2/a/(a*d-b*c)^2/(a*b)^(1/2)*(c*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)**(5/2)/(-b*x**2+a)**2/(-d*x**2+c)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="maxima 
")
 

Output:

integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*(e*x)^(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*(e*x)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \] Input:

int(1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x)
 

Output:

int(1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x)
 

Output:

(sqrt(e)*( - 2*sqrt(c - d*x**2) + 5*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)* 
a**2*c**2 - 2*sqrt(x)*a**2*c*d*x**2 + sqrt(x)*a**2*d**2*x**4 - 2*sqrt(x)*a 
*b*c**2*x**2 + 4*sqrt(x)*a*b*c*d*x**4 - 2*sqrt(x)*a*b*d**2*x**6 + sqrt(x)* 
b**2*c**2*x**4 - 2*sqrt(x)*b**2*c*d*x**6 + sqrt(x)*b**2*d**2*x**8),x)*a**2 
*c*d*x - 5*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c**2 - 2*sqrt(x)*a** 
2*c*d*x**2 + sqrt(x)*a**2*d**2*x**4 - 2*sqrt(x)*a*b*c**2*x**2 + 4*sqrt(x)* 
a*b*c*d*x**4 - 2*sqrt(x)*a*b*d**2*x**6 + sqrt(x)*b**2*c**2*x**4 - 2*sqrt(x 
)*b**2*c*d*x**6 + sqrt(x)*b**2*d**2*x**8),x)*a**2*d**2*x**3 + 7*sqrt(x)*in 
t(sqrt(c - d*x**2)/(sqrt(x)*a**2*c**2 - 2*sqrt(x)*a**2*c*d*x**2 + sqrt(x)* 
a**2*d**2*x**4 - 2*sqrt(x)*a*b*c**2*x**2 + 4*sqrt(x)*a*b*c*d*x**4 - 2*sqrt 
(x)*a*b*d**2*x**6 + sqrt(x)*b**2*c**2*x**4 - 2*sqrt(x)*b**2*c*d*x**6 + sqr 
t(x)*b**2*d**2*x**8),x)*a*b*c**2*x - 12*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt 
(x)*a**2*c**2 - 2*sqrt(x)*a**2*c*d*x**2 + sqrt(x)*a**2*d**2*x**4 - 2*sqrt( 
x)*a*b*c**2*x**2 + 4*sqrt(x)*a*b*c*d*x**4 - 2*sqrt(x)*a*b*d**2*x**6 + sqrt 
(x)*b**2*c**2*x**4 - 2*sqrt(x)*b**2*c*d*x**6 + sqrt(x)*b**2*d**2*x**8),x)* 
a*b*c*d*x**3 + 5*sqrt(x)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c**2 - 2*sqrt( 
x)*a**2*c*d*x**2 + sqrt(x)*a**2*d**2*x**4 - 2*sqrt(x)*a*b*c**2*x**2 + 4*sq 
rt(x)*a*b*c*d*x**4 - 2*sqrt(x)*a*b*d**2*x**6 + sqrt(x)*b**2*c**2*x**4 - 2* 
sqrt(x)*b**2*c*d*x**6 + sqrt(x)*b**2*d**2*x**8),x)*a*b*d**2*x**5 - 7*sqrt( 
x)*int(sqrt(c - d*x**2)/(sqrt(x)*a**2*c**2 - 2*sqrt(x)*a**2*c*d*x**2 + ...