Integrand size = 30, antiderivative size = 535 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=-\frac {(5 b c-4 a d) \sqrt {c-d x^2}}{2 a^2 c (b c-a d) e \sqrt {e x}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}-\frac {\sqrt [4]{d} (5 b c-4 a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (5 b c-4 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (5 b c-7 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (5 b c-7 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}} \]
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Time = 0.78 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {477, 483, 597, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=-\frac {\sqrt {b} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2} (b c-a d)}+\frac {\sqrt {b} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2} (b c-a d)}+\frac {\sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-4 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-4 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 \sqrt [4]{c} e^{3/2} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt {c-d x^2} (5 b c-4 a d)}{2 a^2 c e \sqrt {e x} (b c-a d)}+\frac {b \sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right ) (b c-a d)} \]
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 483
Rule 504
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 \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}+\frac {e \text {Subst}\left (\int \frac {\frac {5 b c-4 a d}{e^2}-\frac {3 b d x^4}{e^4}}{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 (b c-a d)} \\ & = -\frac {(5 b c-4 a d) \sqrt {c-d x^2}}{2 a^2 c (b c-a d) e \sqrt {e x}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {x^2 \left (-\frac {(b c-2 a d) (5 b c-2 a d)}{e^4}-\frac {b d (5 b c-4 a d) x^4}{e^6}\right )}{\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 c (b c-a d)} \\ & = -\frac {(5 b c-4 a d) \sqrt {c-d x^2}}{2 a^2 c (b c-a d) e \sqrt {e x}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \left (\frac {d (5 b c-4 a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {\left (5 b^2 c^2-7 a b c d\right ) x^2}{e^4 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{2 a^2 c (b c-a d)} \\ & = -\frac {(5 b c-4 a d) \sqrt {c-d x^2}}{2 a^2 c (b c-a d) e \sqrt {e x}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}+\frac {(b (5 b c-7 a d)) \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) e^3}-\frac {(d (5 b c-4 a d)) \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 (b c-a d) e^3} \\ & = -\frac {(5 b c-4 a d) \sqrt {c-d x^2}}{2 a^2 c (b c-a d) e \sqrt {e x}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}+\frac {\left (\sqrt {d} (5 b c-4 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 \sqrt {c} (b c-a d) e^2}-\frac {\left (\sqrt {d} (5 b c-4 a d)\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 \sqrt {c} (b c-a d) e^2}+\frac {\left (\sqrt {b} (5 b c-7 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) e}-\frac {\left (\sqrt {b} (5 b c-7 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) e} \\ & = -\frac {(5 b c-4 a d) \sqrt {c-d x^2}}{2 a^2 c (b c-a d) e \sqrt {e x}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}+\frac {\left (\sqrt {d} (5 b c-4 a d) \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 \sqrt {c} (b c-a d) e^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} (5 b c-4 a d) \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 \sqrt {c} (b c-a d) e^2 \sqrt {c-d x^2}}+\frac {\left (\sqrt {b} (5 b c-7 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) e \sqrt {c-d x^2}}-\frac {\left (\sqrt {b} (5 b c-7 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) e \sqrt {c-d x^2}} \\ & = -\frac {(5 b c-4 a d) \sqrt {c-d x^2}}{2 a^2 c (b c-a d) e \sqrt {e x}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}+\frac {\sqrt [4]{d} (5 b c-4 a d) \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 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (5 b c-7 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) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (5 b c-7 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) e^{3/2} \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} (5 b c-4 a d) \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 \sqrt {c} (b c-a d) e^2 \sqrt {c-d x^2}} \\ & = -\frac {(5 b c-4 a d) \sqrt {c-d x^2}}{2 a^2 c (b c-a d) e \sqrt {e x}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}-\frac {\sqrt [4]{d} (5 b c-4 a d) \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 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (5 b c-4 a d) \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 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (5 b c-7 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) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (5 b c-7 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) 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 = 11.20 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {x \left (-21 a \left (c-d x^2\right ) \left (4 a^2 d+5 b^2 c x^2-4 a b \left (c+d x^2\right )\right )+7 \left (5 b^2 c^2-12 a b c d+4 a^2 d^2\right ) x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d (-5 b c+4 a d) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^3 c (b c-a d) (e x)^{3/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1099\) vs. \(2(419)=838\).
Time = 3.08 (sec) , antiderivative size = 1100, normalized size of antiderivative = 2.06
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1100\) |
default | \(\text {Expression too large to display}\) | \(2970\) |
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Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {3}{2}} \left (- a + b x^{2}\right )^{2} \sqrt {c - d x^{2}}}\, dx \]
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\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^2\,\sqrt {c-d\,x^2}} \,d x \]
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