Integrand size = 30, antiderivative size = 551 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt [4]{d} (4 b c+a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 \sqrt [4]{c} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (4 b c+a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt [4]{c} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \]
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Time = 0.97 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {477, 482, 593, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {\sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (a d+4 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt [4]{c} \sqrt {c-d x^2} (b c-a d)^3}-\frac {\sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (a d+4 b c) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 \sqrt [4]{c} \sqrt {c-d x^2} (b c-a d)^3}+\frac {\sqrt {b} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (7 a d+3 b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}-\frac {\sqrt {b} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (7 a d+3 b c) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}+\frac {d e (e x)^{3/2} (a d+4 b c)}{2 c \sqrt {c-d x^2} (b c-a d)^3}+\frac {5 d e (e x)^{3/2}}{6 \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac {e (e x)^{3/2}}{2 \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 482
Rule 504
Rule 593
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^6}{\left (a-\frac {b x^4}{e^2}\right )^2 \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac {e \text {Subst}\left (\int \frac {x^2 \left (3 c+\frac {7 d x^4}{e^2}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)} \\ & = \frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {e^3 \text {Subst}\left (\int \frac {x^2 \left (-\frac {6 c (3 b c+2 a d)}{e^2}-\frac {30 b c d x^4}{e^4}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{12 c (b c-a d)^2} \\ & = \frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {e^5 \text {Subst}\left (\int \frac {x^2 \left (\frac {12 c \left (3 b^2 c^2+11 a b c d+a^2 d^2\right )}{e^4}-\frac {12 b c d (4 b c+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 )}{24 c^2 (b c-a d)^3} \\ & = \frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {e^5 \text {Subst}\left (\int \left (\frac {12 c d (4 b c+a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {12 \left (3 b^2 c^3+7 a b c^2 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 )}{24 c^2 (b c-a d)^3} \\ & = \frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {(d (4 b c+a d) e) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 c (b c-a d)^3}-\frac {(b (3 b c+7 a d) e) \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 (b c-a d)^3} \\ & = \frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\left (\sqrt {d} (4 b c+a d) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 \sqrt {c} (b c-a d)^3}-\frac {\left (\sqrt {d} (4 b c+a d) e^2\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 \sqrt {c} (b c-a d)^3}-\frac {\left (\sqrt {b} (3 b c+7 a d) e^3\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 )}{4 (b c-a d)^3}+\frac {\left (\sqrt {b} (3 b c+7 a d) e^3\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 )}{4 (b c-a d)^3} \\ & = \frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\left (\sqrt {d} (4 b c+a d) e^2 \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 \sqrt {c} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} (4 b c+a d) e^2 \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 \sqrt {c} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (\sqrt {b} (3 b c+7 a d) e^3 \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 (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\left (\sqrt {b} (3 b c+7 a d) e^3 \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 (b c-a d)^3 \sqrt {c-d x^2}} \\ & = \frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (4 b c+a d) e^{5/2} \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 \sqrt [4]{c} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \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 \sqrt {a} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \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 \sqrt {a} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} (4 b c+a d) e^2 \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 \sqrt {c} (b c-a d)^3 \sqrt {c-d x^2}} \\ & = \frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt [4]{d} (4 b c+a d) e^{5/2} \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 \sqrt [4]{c} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (4 b c+a d) e^{5/2} \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 \sqrt [4]{c} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \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 \sqrt {a} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \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 \sqrt {a} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.33 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.50 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=-\frac {e (e x)^{3/2} \left (7 a \left (a^2 d^2 \left (c-3 d x^2\right )+a b d \left (11 c^2-10 c d x^2+3 d^2 x^4\right )+b^2 c \left (3 c^2-17 c d x^2+12 d^2 x^4\right )\right )+7 \left (3 b^2 c^2+11 a b c d+a^2 d^2\right ) \left (-a+b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d (4 b c+a d) x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a c (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1392\) vs. \(2(429)=858\).
Time = 3.12 (sec) , antiderivative size = 1393, normalized size of antiderivative = 2.53
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1393\) |
default | \(\text {Expression too large to display}\) | \(5066\) |
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Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \]
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