Optimal. Leaf size=29 \[ -\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{3/2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {270}
\begin {gather*} -\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{5/2} \sqrt [4]{a-b x^2}} \, dx &=-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 27, normalized size = 0.93 \begin {gather*} -\frac {2 x \left (a-b x^2\right )^{3/4}}{3 a (c x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 22, normalized size = 0.76
method | result | size |
gosper | \(-\frac {2 x \left (-b \,x^{2}+a \right )^{\frac {3}{4}}}{3 a \left (c x \right )^{\frac {5}{2}}}\) | \(22\) |
risch | \(-\frac {2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}}}{3 c^{2} \sqrt {c x}\, a x}\) | \(27\) |
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 [A]
time = 1.29, size = 26, normalized size = 0.90 \begin {gather*} -\frac {2 \, {\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{3 \, a c^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.19, size = 88, normalized size = 3.03 \begin {gather*} \begin {cases} \frac {b^{\frac {3}{4}} \left (\frac {a}{b x^{2}} - 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{2 a c^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\- \frac {b^{\frac {3}{4}} \left (- \frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {3}{4}\right )}{2 a c^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \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 [B]
time = 5.12, size = 26, normalized size = 0.90 \begin {gather*} -\frac {2\,{\left (a-b\,x^2\right )}^{3/4}}{3\,a\,c^2\,x\,\sqrt {c\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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