Optimal. Leaf size=82 \[ -\frac {\sqrt [4]{a-b x^4}}{2 x^2}-\frac {\sqrt {a} \sqrt {b} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \left (a-b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {281, 283, 239,
238} \begin {gather*} -\frac {\sqrt {a} \sqrt {b} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \left (a-b x^4\right )^{3/4}}-\frac {\sqrt [4]{a-b x^4}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 238
Rule 239
Rule 281
Rule 283
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a-b x^4}}{x^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt [4]{a-b x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [4]{a-b x^4}}{2 x^2}-\frac {1}{4} b \text {Subst}\left (\int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt [4]{a-b x^4}}{2 x^2}-\frac {\left (b \left (1-\frac {b x^4}{a}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{4 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{2 x^2}-\frac {\sqrt {a} \sqrt {b} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-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 = 10.02, size = 52, normalized size = 0.63 \begin {gather*} -\frac {\sqrt [4]{a-b x^4} \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {1}{2};\frac {b x^4}{a}\right )}{2 x^2 \sqrt [4]{1-\frac {b x^4}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{x^{3}}\, dx\]
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 = 0.20 \begin {gather*} {\rm integral}\left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{3}}, 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 = 0.44, size = 34, normalized size = 0.41 \begin {gather*} - \frac {\sqrt [4]{a} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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