Integrand size = 30, antiderivative size = 474 \[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} (b c-5 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} (b c-5 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} \left (b^2 c^2+4 a b c d-5 a^2 d^2\right ) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \left (b^2 c^2+4 a b c d-5 a^2 d^2\right ) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]
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Time = 0.60 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {477, 479, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+4 a b c d+b^2 c^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+4 a b c d+b^2 c^2\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a b^2 \sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-5 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {c-d x^2}}+\frac {(e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}{2 a b e \left (a-b x^2\right )} \]
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 479
Rule 504
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^2 \left (c-\frac {d x^4}{e^2}\right )^{3/2}}{\left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {e \text {Subst}\left (\int \frac {x^2 \left (\frac {c (b c+3 a d)}{e^2}+\frac {d (b c-5 a d) x^4}{e^4}\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 b} \\ & = \frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {e \text {Subst}\left (\int \left (-\frac {d (b c-5 a d) x^2}{b e^2 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {\left (b^2 c^2+4 a b c d-5 a^2 d^2\right ) x^2}{b e^2 \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 b} \\ & = \frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}-\frac {(d (b c-5 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 b^2 e}+\frac {((b c-a d) (b c+5 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 b^2 e} \\ & = \frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {d} (b c-5 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 b^2}-\frac {\left (\sqrt {c} \sqrt {d} (b c-5 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 b^2}+\frac {((b c-a d) (b c+5 a d) e) \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 b^{5/2}}-\frac {((b c-a d) (b c+5 a d) e) \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 b^{5/2}} \\ & = \frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {d} (b c-5 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 b^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {d} (b c-5 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 b^2 \sqrt {c-d x^2}}+\frac {\left ((b c-a d) (b c+5 a d) e \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 b^{5/2} \sqrt {c-d x^2}}-\frac {\left ((b c-a d) (b c+5 a d) e \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 b^{5/2} \sqrt {c-d x^2}} \\ & = \frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {c^{3/4} \sqrt [4]{d} (b c-5 a d) \sqrt {e} \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 b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) (b c+5 a d) \sqrt {e} \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^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (b c+5 a d) \sqrt {e} \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^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {d} (b c-5 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 b^2 \sqrt {c-d x^2}} \\ & = \frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} (b c-5 a d) \sqrt {e} \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 b^2 \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} (b c-5 a d) \sqrt {e} \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 b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) (b c+5 a d) \sqrt {e} \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^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (b c+5 a d) \sqrt {e} \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^{3/2} b^{5/2} \sqrt [4]{d} \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 = 189, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {\sqrt {e x} \left (21 a (-b c+a d) x \left (c-d x^2\right )+7 c (b c+3 a d) x \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 d (b c-5 a d) x^3 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^2 b \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1297\) vs. \(2(364)=728\).
Time = 3.22 (sec) , antiderivative size = 1298, normalized size of antiderivative = 2.74
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1298\) |
default | \(\text {Expression too large to display}\) | \(3846\) |
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Timed out. \[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {\sqrt {e x} \left (c - d x^{2}\right )^{\frac {3}{2}}}{\left (- a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {e x}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {e x}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {\sqrt {e\,x}\,{\left (c-d\,x^2\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2} \,d x \]
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