Optimal. Leaf size=29 \[ -\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{5/2}} \]
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Rubi [A] time = 0.01, 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 = {264} \[ -\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{7/2}} \, dx &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.93 \[ -\frac {2 x \left (a-b x^2\right )^{5/4}}{5 a (c x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 35, normalized size = 1.21 \[ \frac {2 \, {\left (b x^{2} - a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{5 \, a c^{4} x^{3}} \]
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 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.76 \[ -\frac {2 \left (-b \,x^{2}+a \right )^{\frac {5}{4}} x}{5 \left (c x \right )^{\frac {7}{2}} a} \]
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 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.85, size = 38, normalized size = 1.31 \[ -\frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2}{5\,c^3}-\frac {2\,b\,x^2}{5\,a\,c^3}\right )}{x^2\,\sqrt {c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.76, size = 178, normalized size = 6.14 \[ \begin {cases} \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {5}{4}\right )}{2 c^{\frac {7}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right )} - \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {5}{4}\right )}{2 a c^{\frac {7}{2}} \Gamma \left (- \frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\\frac {\sqrt [4]{b} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {5}{4}\right )}{2 c^{\frac {7}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right )} - \frac {b^{\frac {5}{4}} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {5}{4}\right )}{2 a c^{\frac {7}{2}} \Gamma \left (- \frac {1}{4}\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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