Optimal. Leaf size=83 \[ \frac {1}{2} x \sqrt [4]{a-b x^4}-\frac {\sqrt {a} \sqrt {b} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \left (a-b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {201, 243, 342,
281, 238} \begin {gather*} \frac {1}{2} x \sqrt [4]{a-b x^4}-\frac {\sqrt {a} \sqrt {b} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \left (a-b x^4\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 238
Rule 243
Rule 281
Rule 342
Rubi steps
\begin {align*} \int \sqrt [4]{a-b x^4} \, dx &=\frac {1}{2} x \sqrt [4]{a-b x^4}+\frac {1}{2} a \int \frac {1}{\left (a-b x^4\right )^{3/4}} \, dx\\ &=\frac {1}{2} x \sqrt [4]{a-b x^4}+\frac {\left (a \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{2 \left (a-b x^4\right )^{3/4}}\\ &=\frac {1}{2} x \sqrt [4]{a-b x^4}-\frac {\left (a \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1-\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{2 \left (a-b x^4\right )^{3/4}}\\ &=\frac {1}{2} x \sqrt [4]{a-b x^4}-\frac {\left (a \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{4 \left (a-b x^4\right )^{3/4}}\\ &=\frac {1}{2} x \sqrt [4]{a-b x^4}-\frac {\sqrt {a} \sqrt {b} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \left (a-b x^4\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 = 6.67, size = 47, normalized size = 0.57 \begin {gather*} \frac {x \sqrt [4]{a-b x^4} \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {b x^4}{a}\right )}{\sqrt [4]{1-\frac {b x^4}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (-b \,x^{4}+a \right )^{\frac {1}{4}}\, dx\]
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 = 0.08, size = 12, normalized size = 0.14 \begin {gather*} {\rm integral}\left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}}, x\right ) \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 = 0.40, size = 39, normalized size = 0.47 \begin {gather*} \frac {\sqrt [4]{a} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \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 = 1.07, size = 38, normalized size = 0.46 \begin {gather*} \frac {x\,{\left (a-b\,x^4\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ \frac {b\,x^4}{a}\right )}{{\left (1-\frac {b\,x^4}{a}\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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