Optimal. Leaf size=68 \[ -\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8}+\frac {3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac {9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}+\frac {3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^9 \sqrt [3]{a+b x^3}} \, dx &=-\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8}-\frac {(3 b) \int \frac {1}{x^6 \sqrt [3]{a+b x^3}} \, dx}{4 a}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8}+\frac {3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}+\frac {\left (9 b^2\right ) \int \frac {1}{x^3 \sqrt [3]{a+b x^3}} \, dx}{20 a^2}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{8 a x^8}+\frac {3 b \left (a+b x^3\right )^{2/3}}{20 a^2 x^5}-\frac {9 b^2 \left (a+b x^3\right )^{2/3}}{40 a^3 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 42, normalized size = 0.62 \begin {gather*} \frac {\left (a+b x^3\right )^{2/3} \left (-5 a^2+6 a b x^3-9 b^2 x^6\right )}{40 a^3 x^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 39, normalized size = 0.57
method | result | size |
gosper | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (9 b^{2} x^{6}-6 a b \,x^{3}+5 a^{2}\right )}{40 a^{3} x^{8}}\) | \(39\) |
trager | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (9 b^{2} x^{6}-6 a b \,x^{3}+5 a^{2}\right )}{40 a^{3} x^{8}}\) | \(39\) |
risch | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (9 b^{2} x^{6}-6 a b \,x^{3}+5 a^{2}\right )}{40 a^{3} x^{8}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 52, normalized size = 0.76 \begin {gather*} -\frac {\frac {20 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{2}}{x^{2}} - \frac {16 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} b}{x^{5}} + \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}}}{x^{8}}}{40 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 38, normalized size = 0.56 \begin {gather*} -\frac {{\left (9 \, b^{2} x^{6} - 6 \, a b x^{3} + 5 \, a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{40 \, a^{3} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 406 vs.
\(2 (61) = 122\).
time = 0.66, size = 406, normalized size = 5.97 \begin {gather*} \frac {10 a^{4} b^{\frac {14}{3}} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {1}{3}\right )} + \frac {8 a^{3} b^{\frac {17}{3}} x^{3} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {1}{3}\right )} + \frac {4 a^{2} b^{\frac {20}{3}} x^{6} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {1}{3}\right )} + \frac {24 a b^{\frac {23}{3}} x^{9} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \Gamma \left (\frac {1}{3}\right )} + \frac {18 b^{\frac {26}{3}} x^{12} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 a^{5} b^{4} x^{6} \Gamma \left (\frac {1}{3}\right ) + 54 a^{4} b^{5} x^{9} \Gamma \left (\frac {1}{3}\right ) + 27 a^{3} b^{6} x^{12} \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 = 1.14, size = 38, normalized size = 0.56 \begin {gather*} -\frac {{\left (b\,x^3+a\right )}^{2/3}\,\left (5\,a^2-6\,a\,b\,x^3+9\,b^2\,x^6\right )}{40\,a^3\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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