Optimal. Leaf size=44 \[ -\frac {\left (a+b x^3\right )^{2/3}}{5 a x^5}+\frac {3 b \left (a+b x^3\right )^{2/3}}{10 a^2 x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270}
\begin {gather*} \frac {3 b \left (a+b x^3\right )^{2/3}}{10 a^2 x^2}-\frac {\left (a+b x^3\right )^{2/3}}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^6 \sqrt [3]{a+b x^3}} \, dx &=-\frac {\left (a+b x^3\right )^{2/3}}{5 a x^5}-\frac {(3 b) \int \frac {1}{x^3 \sqrt [3]{a+b x^3}} \, dx}{5 a}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{5 a x^5}+\frac {3 b \left (a+b x^3\right )^{2/3}}{10 a^2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 31, normalized size = 0.70 \begin {gather*} \frac {\left (a+b x^3\right )^{2/3} \left (-2 a+3 b x^3\right )}{10 a^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 28, normalized size = 0.64
method | result | size |
gosper | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (-3 b \,x^{3}+2 a \right )}{10 a^{2} x^{5}}\) | \(28\) |
trager | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (-3 b \,x^{3}+2 a \right )}{10 a^{2} x^{5}}\) | \(28\) |
risch | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (-3 b \,x^{3}+2 a \right )}{10 a^{2} x^{5}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 0.80 \begin {gather*} \frac {\frac {5 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}} - \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}}}{x^{5}}}{10 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 27, normalized size = 0.61 \begin {gather*} \frac {{\left (3 \, b x^{3} - 2 \, a\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{10 \, a^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 70, normalized size = 1.59 \begin {gather*} - \frac {2 b^{\frac {2}{3}} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{9 a x^{3} \Gamma \left (\frac {1}{3}\right )} + \frac {b^{\frac {5}{3}} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 a^{2} \Gamma \left (\frac {1}{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 [B]
time = 1.08, size = 27, normalized size = 0.61 \begin {gather*} -\frac {{\left (b\,x^3+a\right )}^{2/3}\,\left (2\,a-3\,b\,x^3\right )}{10\,a^2\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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