Integrand size = 30, antiderivative size = 412 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=-\frac {(7 b c-3 a d) \sqrt {c-d x^2}}{6 a^2 b e (e x)^{3/2}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e (e x)^{3/2} \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} (7 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 b e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (7 b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (7 b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}} \]
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Time = 0.56 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {477, 479, 597, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d) (7 b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d) (7 b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {1-\frac {d x^2}{c}} (7 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 b e^{5/2} \sqrt {c-d x^2}}-\frac {\sqrt {c-d x^2} (7 b c-3 a d)}{6 a^2 b e (e x)^{3/2}}+\frac {\sqrt {c-d x^2} (b c-a d)}{2 a b e (e x)^{3/2} \left (a-b x^2\right )} \]
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Rule 227
Rule 230
Rule 418
Rule 477
Rule 479
Rule 537
Rule 597
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\left (c-\frac {d x^4}{e^2}\right )^{3/2}}{x^4 \left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e (e x)^{3/2} \left (a-b x^2\right )}+\frac {e \text {Subst}\left (\int \frac {\frac {c (7 b c-3 a d)}{e^2}-\frac {d (5 b c-a d) x^4}{e^4}}{x^4 \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} \\ & = -\frac {(7 b c-3 a d) \sqrt {c-d x^2}}{6 a^2 b e (e x)^{3/2}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e (e x)^{3/2} \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {-\frac {b c^2 (21 b c-17 a d)}{e^4}+\frac {b c d (7 b c-3 a d) x^4}{e^6}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 a^2 b c} \\ & = -\frac {(7 b c-3 a d) \sqrt {c-d x^2}}{6 a^2 b e (e x)^{3/2}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e (e x)^{3/2} \left (a-b x^2\right )}+\frac {(d (7 b c-3 a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 a^2 b e^3}+\frac {((b c-a d) (7 b c-a d)) \text {Subst}\left (\int \frac {1}{\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 e^3} \\ & = -\frac {(7 b c-3 a d) \sqrt {c-d x^2}}{6 a^2 b e (e x)^{3/2}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e (e x)^{3/2} \left (a-b x^2\right )}+\frac {((b c-a d) (7 b c-a d)) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^3 b e^3}+\frac {((b c-a d) (7 b c-a d)) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^3 b e^3}+\frac {\left (d (7 b c-3 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 )}{6 a^2 b e^3 \sqrt {c-d x^2}} \\ & = -\frac {(7 b c-3 a d) \sqrt {c-d x^2}}{6 a^2 b e (e x)^{3/2}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e (e x)^{3/2} \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} (7 b c-3 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 )}{6 a^2 b e^{5/2} \sqrt {c-d x^2}}+\frac {\left ((b c-a d) (7 b c-a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^3 b e^3 \sqrt {c-d x^2}}+\frac {\left ((b c-a d) (7 b c-a d) \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a^3 b e^3 \sqrt {c-d x^2}} \\ & = -\frac {(7 b c-3 a d) \sqrt {c-d x^2}}{6 a^2 b e (e x)^{3/2}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e (e x)^{3/2} \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} (7 b c-3 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 )}{6 a^2 b e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (7 b c-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^3 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (7 b c-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^3 b \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.18 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.48 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\frac {x \left (5 a \left (c-d x^2\right ) \left (4 a c-7 b c x^2+3 a d x^2\right )+5 c (-21 b c+17 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-d (-7 b c+3 a d) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a^3 (e x)^{5/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1190\) vs. \(2(324)=648\).
Time = 3.07 (sec) , antiderivative size = 1191, normalized size of antiderivative = 2.89
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1191\) |
default | \(\text {Expression too large to display}\) | \(3472\) |
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Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\int \frac {\left (c - d x^{2}\right )^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}} \left (- a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx=\int \frac {{\left (c-d\,x^2\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}\,{\left (a-b\,x^2\right )}^2} \,d x \]
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