Optimal. Leaf size=76 \[ -\frac {\left (a-b x^2\right )^{3/4}}{x}-\frac {3 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {283, 235, 234}
\begin {gather*} -\frac {3 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}}-\frac {\left (a-b x^2\right )^{3/4}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 235
Rule 283
Rubi steps
\begin {align*} \int \frac {\left (a-b x^2\right )^{3/4}}{x^2} \, dx &=-\frac {\left (a-b x^2\right )^{3/4}}{x}-\frac {1}{2} (3 b) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{x}-\frac {\left (3 b \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{2 \sqrt [4]{a-b x^2}}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{x}-\frac {3 \sqrt {a} \sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt [4]{a-b x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 8.26, size = 50, normalized size = 0.66 \begin {gather*} -\frac {\left (a-b x^2\right )^{3/4} \, _2F_1\left (-\frac {3}{4},-\frac {1}{2};\frac {1}{2};\frac {b x^2}{a}\right )}{x \left (1-\frac {b x^2}{a}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}{x^{2}}\, dx\]
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 = 0.59, size = 31, normalized size = 0.41 \begin {gather*} - \frac {a^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{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 [B]
time = 5.17, size = 41, normalized size = 0.54 \begin {gather*} \frac {2\,{\left (a-b\,x^2\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},-\frac {1}{4};\ \frac {3}{4};\ \frac {a}{b\,x^2}\right )}{x\,{\left (1-\frac {a}{b\,x^2}\right )}^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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