Optimal. Leaf size=28 \[ -\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{2/3}} \]
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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {270}
\begin {gather*} -\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{5/3} \left (a+b x^2\right )^{2/3}} \, dx &=-\frac {3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.81, size = 26, normalized size = 0.93 \begin {gather*} -\frac {3 x \sqrt [3]{a+b x^2}}{2 a (c x)^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 21, normalized size = 0.75
method | result | size |
gosper | \(-\frac {3 x \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{2 a \left (c x \right )^{\frac {5}{3}}}\) | \(21\) |
risch | \(-\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}}}{2 a c \left (c x \right )^{\frac {2}{3}}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 1.25 \begin {gather*} -\frac {3 \, {\left (b c^{\frac {1}{3}} x^{3} + a c^{\frac {1}{3}} x\right )}}{2 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a c^{2} x^{\frac {5}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.22, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {1}{3}}}{2 \, a c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.13, size = 36, normalized size = 1.29 \begin {gather*} \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{3}\right )}{2 a c^{\frac {5}{3}} \Gamma \left (\frac {2}{3}\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 [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{5/3}\,{\left (b\,x^2+a\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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