Integrand size = 30, antiderivative size = 429 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=-\frac {(7 b c-4 a d) \sqrt {c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac {d^{3/4} (7 b c-4 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 c^{3/4} (b c-a d) e^{5/2} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} (7 b c-9 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} (7 b c-9 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt {c-d x^2}} \]
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Time = 0.57 (sec) , antiderivative size = 429, 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, 483, 597, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {b \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-9 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2} (b c-a d)}+\frac {b \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-9 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2} (b c-a d)}+\frac {d^{3/4} \sqrt {1-\frac {d x^2}{c}} (7 b c-4 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 c^{3/4} e^{5/2} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt {c-d x^2} (7 b c-4 a d)}{6 a^2 c e (e x)^{3/2} (b c-a d)}+\frac {b \sqrt {c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right ) (b c-a d)} \]
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Rule 227
Rule 230
Rule 418
Rule 477
Rule 483
Rule 537
Rule 597
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {1}{x^4 \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 (e x)^{3/2} \left (a-b x^2\right )}+\frac {e \text {Subst}\left (\int \frac {\frac {7 b c-4 a d}{e^2}-\frac {5 b 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 c-a d)} \\ & = -\frac {(7 b c-4 a d) \sqrt {c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {-\frac {21 b^2 c^2-20 a b c d-4 a^2 d^2}{e^4}+\frac {b d (7 b c-4 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 c (b c-a d)} \\ & = -\frac {(7 b c-4 a d) \sqrt {c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac {(b (7 b c-9 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 c-a d) e^3}+\frac {(d (7 b c-4 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 c (b c-a d) e^3} \\ & = -\frac {(7 b c-4 a d) \sqrt {c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac {(b (7 b c-9 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 c-a d) e^3}+\frac {(b (7 b c-9 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 c-a d) e^3}+\frac {\left (d (7 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 )}{6 a^2 c (b c-a d) e^3 \sqrt {c-d x^2}} \\ & = -\frac {(7 b c-4 a d) \sqrt {c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac {d^{3/4} (7 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 )}{6 a^2 c^{3/4} (b c-a d) e^{5/2} \sqrt {c-d x^2}}+\frac {\left (b (7 b c-9 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 c-a d) e^3 \sqrt {c-d x^2}}+\frac {\left (b (7 b c-9 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 c-a d) e^3 \sqrt {c-d x^2}} \\ & = -\frac {(7 b c-4 a d) \sqrt {c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac {d^{3/4} (7 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 )}{6 a^2 c^{3/4} (b c-a d) e^{5/2} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} (7 b c-9 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 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} (7 b c-9 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 \sqrt [4]{d} (b c-a 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 = 11.22 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {x \left (-5 a \left (c-d x^2\right ) \left (4 a^2 d+7 b^2 c x^2-4 a b \left (c+d x^2\right )\right )+5 \left (-21 b^2 c^2+20 a b c d+4 a^2 d^2\right ) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-b d (-7 b c+4 a d) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a^3 c (b c-a d) (e x)^{5/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(947\) vs. \(2(341)=682\).
Time = 3.17 (sec) , antiderivative size = 948, normalized size of antiderivative = 2.21
method | result | size |
elliptic | \(\frac {\sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {b^{2} \sqrt {-d e \,x^{3}+c e x}}{2 \left (a d -b c \right ) a^{2} e^{3} \left (-b \,x^{2}+a \right )}-\frac {2 \sqrt {-d e \,x^{3}+c e x}}{3 e^{3} c \,a^{2} x^{2}}-\frac {\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}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b}{4 \sqrt {-d e \,x^{3}+c e x}\, \left (a d -b c \right ) a^{2} e^{2}}+\frac {\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}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {-d e \,x^{3}+c e x}\, c \,e^{2} a^{2}}-\frac {9 b \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 \left (a d -b c \right ) a \,e^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {7 b^{2} \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 \left (a d -b c \right ) a^{2} e^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {9 b \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 \left (a d -b c \right ) a \,e^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {7 b^{2} \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 \left (a d -b c \right ) a^{2} e^{2} \sqrt {a b}\, 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}}\) | \(948\) |
default | \(\text {Expression too large to display}\) | \(2610\) |
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Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {5}{2}} \left (- a + b x^{2}\right )^{2} \sqrt {c - d x^{2}}}\, dx \]
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\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,{\left (a-b\,x^2\right )}^2\,\sqrt {c-d\,x^2}} \,d x \]
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