Optimal. Leaf size=75 \[ -\frac {1}{x \sqrt [4]{a+b x^4}}+\frac {\sqrt {b} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {318, 287, 342,
281, 202} \begin {gather*} \frac {\sqrt {b} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a+b x^4}}-\frac {1}{x \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 281
Rule 287
Rule 318
Rule 342
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [4]{a+b x^4}} \, dx &=-\frac {1}{x \sqrt [4]{a+b x^4}}-b \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx\\ &=-\frac {1}{x \sqrt [4]{a+b x^4}}-\frac {\left (\sqrt [4]{1+\frac {a}{b x^4}} x\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{5/4} x^3} \, dx}{\sqrt [4]{a+b x^4}}\\ &=-\frac {1}{x \sqrt [4]{a+b x^4}}+\frac {\left (\sqrt [4]{1+\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{\sqrt [4]{a+b x^4}}\\ &=-\frac {1}{x \sqrt [4]{a+b x^4}}+\frac {\left (\sqrt [4]{1+\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x^2}\right )}{2 \sqrt [4]{a+b x^4}}\\ &=-\frac {1}{x \sqrt [4]{a+b x^4}}+\frac {\sqrt {b} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a+b x^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 = 49, normalized size = 0.65 \begin {gather*} -\frac {\sqrt [4]{1+\frac {b x^4}{a}} \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {3}{4};-\frac {b x^4}{a}\right )}{x \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx\]
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 = 0.07, size = 25, normalized size = 0.33 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{6} + a x^{2}}, 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 = 0.41, size = 39, normalized size = 0.52 \begin {gather*} \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} x \Gamma \left (\frac {3}{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 = 1.46, size = 40, normalized size = 0.53 \begin {gather*} -\frac {{\left (\frac {a}{b\,x^4}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ -\frac {a}{b\,x^4}\right )}{2\,x\,{\left (b\,x^4+a\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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