Optimal. Leaf size=20 \[ -\frac {\sqrt [4]{a-b x^4}}{a x} \]
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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {270}
\begin {gather*} -\frac {\sqrt [4]{a-b x^4}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a-b x^4\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{a-b x^4}}{a x}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 20, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{a-b x^4}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 19, normalized size = 0.95
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{a x}\) | \(19\) |
trager | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{a x}\) | \(19\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}}}{a x \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 18, normalized size = 0.90 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 18, normalized size = 0.90 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a x} \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 = 0.39, size = 80, normalized size = 4.00 \begin {gather*} \begin {cases} \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{4}} - 1} \Gamma \left (- \frac {1}{4}\right )}{4 a \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {\sqrt [4]{b} \sqrt [4]{- \frac {a}{b x^{4}} + 1} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {1}{4}\right )}{4 a \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases} \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 = 1.11, size = 18, normalized size = 0.90 \begin {gather*} -\frac {{\left (a-b\,x^4\right )}^{1/4}}{a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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