Integrand size = 30, antiderivative size = 444 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=-\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}-\frac {5 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 e^{3/2} \sqrt {c-d x^2}}+\frac {5 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (5 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \]
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Time = 0.62 (sec) , antiderivative size = 444, 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, 480, 597, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=-\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (5 b c-3 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (5 b c-3 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {5 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 e^{3/2} \sqrt {c-d x^2}}-\frac {5 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 e^{3/2} \sqrt {c-d x^2}}-\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )} \]
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 480
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 {\sqrt {c-\frac {d x^4}{e^2}}}{x^2 \left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}-\frac {\text {Subst}\left (\int \frac {-5 c+\frac {3 d x^4}{e^2}}{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 e} \\ & = -\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (\frac {c (5 b c-8 a d)}{e^2}+\frac {5 b c 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^2 c e} \\ & = -\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}+\frac {\text {Subst}\left (\int \left (-\frac {5 c d x^2}{e^2 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {\left (5 b c^2-3 a c d\right ) x^2}{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^2 c e} \\ & = -\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}-\frac {(5 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 e^3}+\frac {(5 b c-3 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 e^3} \\ & = -\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}+\frac {\left (5 \sqrt {c} \sqrt {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 e^2}-\frac {\left (5 \sqrt {c} \sqrt {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 e^2}+\frac {(5 b c-3 a d) \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 \sqrt {b} e}-\frac {(5 b c-3 a d) \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 \sqrt {b} e} \\ & = -\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}+\frac {\left (5 \sqrt {c} \sqrt {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 e^2 \sqrt {c-d x^2}}-\frac {\left (5 \sqrt {c} \sqrt {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 e^2 \sqrt {c-d x^2}}+\frac {\left ((5 b c-3 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 \sqrt {b} e \sqrt {c-d x^2}}-\frac {\left ((5 b c-3 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 \sqrt {b} e \sqrt {c-d x^2}} \\ & = -\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}+\frac {5 c^{3/4} \sqrt [4]{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 e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 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 {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (5 b c-3 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 {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}-\frac {\left (5 \sqrt {c} \sqrt {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 e^2 \sqrt {c-d x^2}} \\ & = -\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}-\frac {5 c^{3/4} \sqrt [4]{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 e^{3/2} \sqrt {c-d x^2}}+\frac {5 c^{3/4} \sqrt [4]{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 e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 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 {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (5 b c-3 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 {b} \sqrt [4]{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 = 11.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\frac {x \left (21 a \left (4 a-5 b x^2\right ) \left (c-d x^2\right )+7 (-5 b c+8 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+15 b d x^4 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^3 (e x)^{3/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(858\) vs. \(2(328)=656\).
Time = 3.09 (sec) , antiderivative size = 859, normalized size of antiderivative = 1.93
method | result | size |
elliptic | \(\frac {\sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {2 \left (-d e \,x^{2}+c e \right )}{e^{2} a^{2} \sqrt {x \left (-d e \,x^{2}+c e \right )}}+\frac {b x \sqrt {-d e \,x^{3}+c e x}}{2 a^{2} e^{2} \left (-b \,x^{2}+a \right )}+\frac {5 c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 e \,a^{2} \sqrt {-d e \,x^{3}+c e x}}-\frac {5 c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{4 e \,a^{2} \sqrt {-d e \,x^{3}+c e x}}+\frac {3 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{8 a e b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {5 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 a^{2} e d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{8 a e b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {5 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 a^{2} e d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\) | \(859\) |
default | \(\text {Expression too large to display}\) | \(2556\) |
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Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int \frac {\sqrt {c - d x^{2}}}{\left (e x\right )^{\frac {3}{2}} \left (- a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int \frac {\sqrt {c-d\,x^2}}{{\left (e\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^2} \,d x \]
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