Optimal. Leaf size=85 \[ -\frac {\sqrt [4]{a-b x^4}}{3 x^3}+\frac {b^{3/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 \sqrt {a} \left (a-b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {283, 243, 342,
281, 238} \begin {gather*} \frac {b^{3/2} 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 )}{3 \sqrt {a} \left (a-b x^4\right )^{3/4}}-\frac {\sqrt [4]{a-b x^4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 238
Rule 243
Rule 281
Rule 283
Rule 342
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^4}}{x^4} \, dx &=-\frac {\sqrt [4]{a-b x^4}}{3 x^3}-\frac {1}{3} b \int \frac {1}{\left (a-b x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{a-b x^4}}{3 x^3}-\frac {\left (b \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}{3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{3 x^3}+\frac {\left (b \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 )}{3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{3 x^3}+\frac {\left (b \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 )}{6 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{3 x^3}+\frac {b^{3/2} \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 )}{3 \sqrt {a} \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 = 10.01, size = 52, normalized size = 0.61 \begin {gather*} -\frac {\sqrt [4]{a-b x^4} \, _2F_1\left (-\frac {3}{4},-\frac {1}{4};\frac {1}{4};\frac {b x^4}{a}\right )}{3 x^3 \sqrt [4]{1-\frac {b x^4}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{x^{4}}\, dx\]
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 = 16, normalized size = 0.19 \begin {gather*} {\rm integral}\left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{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 = 0.48, size = 34, normalized size = 0.40 \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^{4}}} \right )}}{2 x^{2}} \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^4\right )}^{1/4}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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