Optimal. Leaf size=519 \[ -\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} (5 b c-a d) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} \left (5 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \left (5 b^2 c^2-4 a b c d-a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \]
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Rubi [A]
time = 0.76, antiderivative size = 519, 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, 479,
597, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} \frac {c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-a d) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-a d) E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {c-d x^2} (5 b c-a d)}{2 a^2 b e \sqrt {e x}}-\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \left (-a^2 d^2-4 a b c d+5 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 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \left (-a^2 d^2-4 a b c d+5 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 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {c-d x^2} (b c-a d)}{2 a b e \sqrt {e x} \left (a-b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 479
Rule 504
Rule 597
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps
\begin {align*} \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx &=\frac {2 \text {Subst}\left (\int \frac {\left (c-\frac {d x^4}{e^2}\right )^{3/2}}{x^2 \left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}+\frac {e \text {Subst}\left (\int \frac {\frac {c (5 b c-a d)}{e^2}-\frac {d (3 b c+a 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 )}{2 a b}\\ &=-\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {x^2 \left (-\frac {b c^2 (5 b c-9 a d)}{e^4}-\frac {b c d (5 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 )}{2 a^2 b c}\\ &=-\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \left (\frac {c d (5 b c-a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {\left (5 b^2 c^3-4 a b c^2 d-a^2 c d^2\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 )}{2 a^2 b c}\\ &=-\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {(d (5 b c-a d)) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b e^3}+\frac {((b c-a d) (5 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 )}{2 a^2 b e^3}\\ &=-\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {d} (5 b c-a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 b e^2}-\frac {\left (\sqrt {c} \sqrt {d} (5 b c-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 )}{2 a^2 b e^2}+\frac {((b c-a d) (5 b c+a d)) \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 a^2 b^{3/2} e}-\frac {((b c-a d) (5 b c+a d)) \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 a^2 b^{3/2} e}\\ &=-\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {d} (5 b c-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 )}{2 a^2 b e^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {d} (5 b c-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 )}{2 a^2 b e^2 \sqrt {c-d x^2}}+\frac {\left ((b c-a d) (5 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 )}{4 a^2 b^{3/2} e \sqrt {c-d x^2}}-\frac {\left ((b c-a d) (5 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 )}{4 a^2 b^{3/2} e \sqrt {c-d x^2}}\\ &=-\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}+\frac {c^{3/4} \sqrt [4]{d} (5 b c-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 )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) (5 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 )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (5 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 )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {d} (5 b c-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 )}{2 a^2 b e^2 \sqrt {c-d x^2}}\\ &=-\frac {(5 b c-a d) \sqrt {c-d x^2}}{2 a^2 b e \sqrt {e x}}+\frac {(b c-a d) \sqrt {c-d x^2}}{2 a b e \sqrt {e x} \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} (5 b c-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 )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} (5 b c-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 )}{2 a^2 b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) (5 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 )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (5 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 )}{4 a^{5/2} b^{3/2} \sqrt [4]{d} e^{3/2} \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 = 10.18, size = 197, normalized size = 0.38 \begin {gather*} \frac {x \left (21 a \left (c-d x^2\right ) \left (4 a c-5 b c x^2+a d x^2\right )+7 c (-5 b c+9 a d) x^2 \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 (-5 b c+a d) x^4 \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 )}{42 a^3 (e x)^{3/2} \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. \(3866\) vs.
\(2(403)=806\).
time = 0.13, size = 3867, normalized size = 7.45
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1345\) |
default | \(\text {Expression too large to display}\) | \(3867\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c - d x^{2}\right )^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}} \left (- a + b x^{2}\right )^{2}}\, dx \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 (c-d\,x^2\right )}^{3/2}}{{\left (e\,x\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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