Integrand size = 30, antiderivative size = 485 \[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=-\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {3 c^{3/4} \sqrt [4]{d} (11 b c-15 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{10 b^3 \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} (11 b c-15 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{10 b^3 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]
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Time = 0.74 (sec) , antiderivative size = 485, 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, 478, 595, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {3 \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (11 b c-15 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{10 b^3 \sqrt {c-d x^2}}-\frac {3 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (11 b c-15 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{10 b^3 \sqrt {c-d x^2}}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2} \]
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 478
Rule 504
Rule 595
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 (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 {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {x^2 \left (3 c-\frac {9 d x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{2 b} \\ & = -\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {e^3 \text {Subst}\left (\int \frac {x^2 \left (-\frac {3 c (5 b c-9 a d)}{e^2}+\frac {3 d (11 b c-15 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 )}{10 b^2} \\ & = -\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {e^3 \text {Subst}\left (\int \left (-\frac {3 d (11 b c-15 a d) x^2}{b e^2 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {15 \left (b^2 c^2-4 a b c d+3 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 )}{10 b^2} \\ & = -\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {(3 d (11 b c-15 a d) e) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{10 b^3}-\frac {(3 (b c-3 a d) (b c-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^3} \\ & = -\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {\left (3 \sqrt {c} \sqrt {d} (11 b c-15 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 )}{10 b^3}-\frac {\left (3 \sqrt {c} \sqrt {d} (11 b c-15 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 )}{10 b^3}-\frac {\left (3 (b c-3 a d) (b c-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^{7/2}}+\frac {\left (3 (b c-3 a d) (b c-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^{7/2}} \\ & = -\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {\left (3 \sqrt {c} \sqrt {d} (11 b c-15 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 )}{10 b^3 \sqrt {c-d x^2}}-\frac {\left (3 \sqrt {c} \sqrt {d} (11 b c-15 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 )}{10 b^3 \sqrt {c-d x^2}}-\frac {\left (3 (b c-3 a d) (b c-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^{7/2} \sqrt {c-d x^2}}+\frac {\left (3 (b c-3 a d) (b c-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^{7/2} \sqrt {c-d x^2}} \\ & = -\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {3 c^{3/4} \sqrt [4]{d} (11 b c-15 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 )}{10 b^3 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c-3 a d) (b c-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} b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} (b c-3 a d) (b c-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} b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\left (3 \sqrt {c} \sqrt {d} (11 b c-15 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 )}{10 b^3 \sqrt {c-d x^2}} \\ & = -\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {3 c^{3/4} \sqrt [4]{d} (11 b c-15 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 )}{10 b^3 \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} (11 b c-15 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 )}{10 b^3 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} (b c-3 a d) (b c-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} b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} (b c-3 a d) (b c-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} b^{7/2} \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 = 196, normalized size of antiderivative = 0.40 \[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {e (e x)^{3/2} \left (7 a \left (c-d x^2\right ) \left (-5 b c+9 a d-4 b d x^2\right )-7 c (-5 b c+9 a d) \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 d (-11 b c+15 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{70 a b^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1337\) vs. \(2(369)=738\).
Time = 5.17 (sec) , antiderivative size = 1338, normalized size of antiderivative = 2.76
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1338\) |
risch | \(\text {Expression too large to display}\) | \(1408\) |
default | \(\text {Expression too large to display}\) | \(3874\) |
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Timed out. \[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
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\[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,{\left (c-d\,x^2\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2} \,d x \]
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