Optimal. Leaf size=485 \[ -\frac {9 d e (e x)^{3/2} \sqrt {c-d x^2}}{10 b^2}+\frac {e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {3 c^{3/4} \sqrt [4]{d} (11 b c-15 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{10 b^3 \sqrt {c-d x^2}}+\frac {3 c^{3/4} \sqrt [4]{d} (11 b c-15 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{10 b^3 \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 \sqrt {a} b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {3 \sqrt [4]{c} \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 \sqrt {a} b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.69, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {477, 478,
595, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} \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 ) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\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} 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 ) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\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 c^{3/4} \sqrt [4]{d} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (11 b c-15 a d) F\left (\left .\text {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} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (11 b c-15 a d) E\left (\left .\text {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} \end {gather*}
Antiderivative was successfully verified.
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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*} \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx &=\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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.21, size = 196, normalized size = 0.40 \begin {gather*} \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}} F_1\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}} F_1\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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3873\) vs.
\(2(369)=738\).
time = 0.20, size = 3874, normalized size = 7.99
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1338\) |
risch | \(\text {Expression too large to display}\) | \(1408\) |
default | \(\text {Expression too large to display}\) | \(3874\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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