Integrand size = 30, antiderivative size = 628 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}-\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}} \]
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Time = 0.99 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {477, 483, 593, 597, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=-\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (5 b c-11 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2} (b c-a d)^2}+\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (5 b c-11 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2} (b c-a d)^2}+\frac {\sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \left (6 a^2 d^2-8 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 c^{5/4} e^{3/2} \sqrt {c-d x^2} (b c-a d)^2}-\frac {\sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \left (6 a^2 d^2-8 a b c d+5 b^2 c^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} e^{3/2} \sqrt {c-d x^2} (b c-a d)^2}-\frac {\sqrt {c-d x^2} \left (6 a^2 d^2-8 a b c d+5 b^2 c^2\right )}{2 a^2 c^2 e \sqrt {e x} (b c-a d)^2}+\frac {b}{2 a e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2} (b c-a d)}+\frac {d (2 a d+b c)}{2 a c e \sqrt {e x} \sqrt {c-d x^2} (b c-a d)^2} \]
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 483
Rule 504
Rule 593
Rule 597
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{x^2 \left (a-\frac {b x^4}{e^2}\right )^2 \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e \text {Subst}\left (\int \frac {\frac {5 b c-4 a d}{e^2}-\frac {7 b d x^4}{e^4}}{x^2 \left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{2 a (b c-a d)} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {e^3 \text {Subst}\left (\int \frac {-\frac {2 \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right )}{e^4}+\frac {6 b d (b c+2 a d) x^4}{e^6}}{x^2 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a c (b c-a d)^2} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {e^3 \text {Subst}\left (\int \frac {x^2 \left (\frac {2 \left (5 b^3 c^3-16 a b^2 c^2 d+8 a^2 b c d^2-6 a^3 d^3\right )}{e^6}+\frac {2 b d \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) x^4}{e^8}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^2 c^2 (b c-a d)^2} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {e^3 \text {Subst}\left (\int \left (-\frac {2 d \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) x^2}{e^6 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {2 \left (5 b^3 c^3-11 a b^2 c^2 d\right ) x^2}{e^6 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{4 a^2 c^2 (b c-a d)^2} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {\left (b^2 (5 b c-11 a d)\right ) \text {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 (b c-a d)^2 e^3}-\frac {\left (d \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 c^2 (b c-a d)^2 e^3} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 c^{3/2} (b c-a d)^2 e^2}-\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 c^{3/2} (b c-a d)^2 e^2}+\frac {\left (b^{3/2} (5 b c-11 a d)\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^2 (b c-a d)^2 e}-\frac {\left (b^{3/2} (5 b c-11 a d)\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^2 (b c-a d)^2 e} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 c^{3/2} (b c-a d)^2 e^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 c^{3/2} (b c-a d)^2 e^2 \sqrt {c-d x^2}}+\frac {\left (b^{3/2} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^2 (b c-a d)^2 e \sqrt {c-d x^2}}-\frac {\left (b^{3/2} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^2 (b c-a d)^2 e \sqrt {c-d x^2}} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}}{\sqrt {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 c^{3/2} (b c-a d)^2 e^2 \sqrt {c-d x^2}} \\ & = \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}-\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.31 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {x \left (21 a \left (2 a^3 d^2 \left (2 c-3 d x^2\right )-5 b^3 c^2 x^2 \left (c-d x^2\right )+4 a b^2 c \left (c^2+c d x^2-2 d^2 x^4\right )+2 a^2 b d \left (-4 c^2+2 c d x^2+3 d^2 x^4\right )\right )+7 \left (-5 b^3 c^3+16 a b^2 c^2 d-8 a^2 b c d^2+6 a^3 d^3\right ) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) x^4 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^3 c^2 (b c-a d)^2 (e x)^{3/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1348\) vs. \(2(506)=1012\).
Time = 3.09 (sec) , antiderivative size = 1349, normalized size of antiderivative = 2.15
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1349\) |
default | \(\text {Expression too large to display}\) | \(3373\) |
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Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]
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