Optimal. Leaf size=79 \[ -\frac {\left (a-b x^2\right )^{3/4}}{a x}-\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 79, 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 = {331, 235, 234}
\begin {gather*} -\frac {\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 {a} \sqrt [4]{a-b x^2}}-\frac {\left (a-b x^2\right )^{3/4}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 235
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx &=-\frac {\left (a-b x^2\right )^{3/4}}{a x}-\frac {b \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{2 a}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{a x}-\frac {\left (b \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{2 a \sqrt [4]{a-b x^2}}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{a x}-\frac {\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 {a} \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 = 6.69, size = 50, normalized size = 0.63 \begin {gather*} -\frac {\sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\frac {b x^2}{a}\right )}{x \sqrt [4]{a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (-b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx\]
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 = 0.44, size = 29, normalized size = 0.37 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{\sqrt [4]{a} 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 = 5.08, size = 41, normalized size = 0.52 \begin {gather*} -\frac {2\,{\left (1-\frac {a}{b\,x^2}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ \frac {a}{b\,x^2}\right )}{3\,x\,{\left (a-b\,x^2\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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