Integrand size = 30, antiderivative size = 457 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=-\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-2 a d) \sqrt {c-d x^2}}{5 a^2 c e^3 \sqrt {e x}}-\frac {2 \sqrt [4]{d} (5 b c-2 a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} (5 b c-2 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}} \]
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Time = 0.71 (sec) , antiderivative size = 457, 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, 486, 597, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=-\frac {\sqrt {b} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt {c-d x^2}}-\frac {2 \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-2 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt {c-d x^2}}-\frac {2 \sqrt {c-d x^2} (5 b c-2 a d)}{5 a^2 c e^3 \sqrt {e x}}-\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}} \]
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 486
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^6 \left (a-\frac {b x^4}{e^2}\right )} \, dx,x,\sqrt {e x}\right )}{e} \\ & = -\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}+\frac {2 \text {Subst}\left (\int \frac {\frac {5 b c-2 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 )}{5 a e} \\ & = -\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-2 a d) \sqrt {c-d x^2}}{5 a^2 c e^3 \sqrt {e x}}-\frac {2 \text {Subst}\left (\int \frac {x^2 \left (-\frac {5 b^2 c^2-10 a b c d+2 a^2 d^2}{e^4}-\frac {b d (5 b c-2 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 )}{5 a^2 c e} \\ & = -\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-2 a d) \sqrt {c-d x^2}}{5 a^2 c e^3 \sqrt {e x}}-\frac {2 \text {Subst}\left (\int \left (\frac {d (5 b c-2 a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {5 \left (b^2 c^2-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 )}{5 a^2 c e} \\ & = -\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-2 a d) \sqrt {c-d x^2}}{5 a^2 c e^3 \sqrt {e x}}-\frac {(2 d (5 b c-2 a d)) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^2 c e^5}+\frac {(2 b (b c-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 )}{a^2 e^5} \\ & = -\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-2 a d) \sqrt {c-d x^2}}{5 a^2 c e^3 \sqrt {e x}}+\frac {\left (2 \sqrt {d} (5 b c-2 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^2 \sqrt {c} e^4}-\frac {\left (2 \sqrt {d} (5 b c-2 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 )}{5 a^2 \sqrt {c} e^4}+\frac {\left (\sqrt {b} (b c-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 )}{a^2 e^3}-\frac {\left (\sqrt {b} (b c-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 )}{a^2 e^3} \\ & = -\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-2 a d) \sqrt {c-d x^2}}{5 a^2 c e^3 \sqrt {e x}}+\frac {\left (2 \sqrt {d} (5 b c-2 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 )}{5 a^2 \sqrt {c} e^4 \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {d} (5 b c-2 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 )}{5 a^2 \sqrt {c} e^4 \sqrt {c-d x^2}}+\frac {\left (\sqrt {b} (b c-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 )}{a^2 e^3 \sqrt {c-d x^2}}-\frac {\left (\sqrt {b} (b c-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 )}{a^2 e^3 \sqrt {c-d x^2}} \\ & = -\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-2 a d) \sqrt {c-d x^2}}{5 a^2 c e^3 \sqrt {e x}}+\frac {2 \sqrt [4]{d} (5 b c-2 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 )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {d} (5 b c-2 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 )}{5 a^2 \sqrt {c} e^4 \sqrt {c-d x^2}} \\ & = -\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-2 a d) \sqrt {c-d x^2}}{5 a^2 c e^3 \sqrt {e x}}-\frac {2 \sqrt [4]{d} (5 b c-2 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 )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} (5 b c-2 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 )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \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 = 190, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\frac {x \left (14 \left (5 b^2 c^2-10 a b c d+2 a^2 d^2\right ) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-6 \left (7 a \left (c-d x^2\right ) \left (a c+5 b c x^2-2 a d x^2\right )+b d (-5 b c+2 a d) x^6 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )\right )}{105 a^3 c (e x)^{7/2} \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1029\) vs. \(2(345)=690\).
Time = 3.06 (sec) , antiderivative size = 1030, normalized size of antiderivative = 2.25
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1030\) |
default | \(\text {Expression too large to display}\) | \(1542\) |
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Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=- \int \frac {\sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {7}{2}} + b x^{2} \left (e x\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\int { -\frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\int { -\frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\int \frac {\sqrt {c-d\,x^2}}{{\left (e\,x\right )}^{7/2}\,\left (a-b\,x^2\right )} \,d x \]
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