Optimal. Leaf size=97 \[ -\frac {2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}+\frac {2 b^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 \sqrt {a} c^4 \left (a-b x^2\right )^{3/4}} \]
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Rubi [A]
time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {283, 335, 243,
342, 281, 238} \begin {gather*} \frac {2 b^{3/2} (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 \sqrt {a} c^4 \left (a-b x^2\right )^{3/4}}-\frac {2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 238
Rule 243
Rule 281
Rule 283
Rule 335
Rule 342
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{5/2}} \, dx &=-\frac {2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}-\frac {b \int \frac {1}{\sqrt {c x} \left (a-b x^2\right )^{3/4}} \, dx}{3 c^2}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{\left (a-\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{3 c^3}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}-\frac {\left (2 b \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {a c^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {c x}\right )}{3 c^3 \left (a-b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}+\frac {\left (2 b \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1-\frac {a c^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {c x}}\right )}{3 c^3 \left (a-b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}+\frac {\left (b \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {a c^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{c x}\right )}{3 c^3 \left (a-b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}+\frac {2 b^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 \sqrt {a} c^4 \left (a-b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 57, normalized size = 0.59 \begin {gather*} -\frac {2 x \sqrt [4]{a-b x^2} \, _2F_1\left (-\frac {3}{4},-\frac {1}{4};\frac {1}{4};\frac {b x^2}{a}\right )}{3 (c x)^{5/2} \sqrt [4]{1-\frac {b x^2}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}{\left (c x \right )^{\frac {5}{2}}}\, dx\]
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]
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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Sympy [C] Result contains complex when optimal does not.
time = 2.18, size = 36, normalized size = 0.37 \begin {gather*} - \frac {i \sqrt [4]{b} e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a}{b x^{2}}} \right )}}{c^{\frac {5}{2}} x} \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 = -0.01 \begin {gather*} \int \frac {{\left (a-b\,x^2\right )}^{1/4}}{{\left (c\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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