Optimal. Leaf size=80 \[ -\frac {a^3 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac {3 a^2 \left (a+b x^3\right )^{8/3}}{8 b^4}-\frac {3 a \left (a+b x^3\right )^{11/3}}{11 b^4}+\frac {\left (a+b x^3\right )^{14/3}}{14 b^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 80, 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 = {272, 45}
\begin {gather*} -\frac {a^3 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac {3 a^2 \left (a+b x^3\right )^{8/3}}{8 b^4}+\frac {\left (a+b x^3\right )^{14/3}}{14 b^4}-\frac {3 a \left (a+b x^3\right )^{11/3}}{11 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{11} \left (a+b x^3\right )^{2/3} \, dx &=\frac {1}{3} \text {Subst}\left (\int x^3 (a+b x)^{2/3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{2/3}}{b^3}+\frac {3 a^2 (a+b x)^{5/3}}{b^3}-\frac {3 a (a+b x)^{8/3}}{b^3}+\frac {(a+b x)^{11/3}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^3 \left (a+b x^3\right )^{5/3}}{5 b^4}+\frac {3 a^2 \left (a+b x^3\right )^{8/3}}{8 b^4}-\frac {3 a \left (a+b x^3\right )^{11/3}}{11 b^4}+\frac {\left (a+b x^3\right )^{14/3}}{14 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 50, normalized size = 0.62 \begin {gather*} \frac {\left (a+b x^3\right )^{5/3} \left (-81 a^3+135 a^2 b x^3-180 a b^2 x^6+220 b^3 x^9\right )}{3080 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 47, normalized size = 0.59
method | result | size |
gosper | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {5}{3}} \left (-220 b^{3} x^{9}+180 a \,b^{2} x^{6}-135 a^{2} b \,x^{3}+81 a^{3}\right )}{3080 b^{4}}\) | \(47\) |
trager | \(-\frac {\left (-220 b^{4} x^{12}-40 a \,b^{3} x^{9}+45 a^{2} b^{2} x^{6}-54 a^{3} b \,x^{3}+81 a^{4}\right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{3080 b^{4}}\) | \(58\) |
risch | \(-\frac {\left (-220 b^{4} x^{12}-40 a \,b^{3} x^{9}+45 a^{2} b^{2} x^{6}-54 a^{3} b \,x^{3}+81 a^{4}\right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{3080 b^{4}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 64, normalized size = 0.80 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{\frac {14}{3}}}{14 \, b^{4}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {11}{3}} a}{11 \, b^{4}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} a^{2}}{8 \, b^{4}} - \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{3}}{5 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 57, normalized size = 0.71 \begin {gather*} \frac {{\left (220 \, b^{4} x^{12} + 40 \, a b^{3} x^{9} - 45 \, a^{2} b^{2} x^{6} + 54 \, a^{3} b x^{3} - 81 \, a^{4}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{3080 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.49, size = 110, normalized size = 1.38 \begin {gather*} \begin {cases} - \frac {81 a^{4} \left (a + b x^{3}\right )^{\frac {2}{3}}}{3080 b^{4}} + \frac {27 a^{3} x^{3} \left (a + b x^{3}\right )^{\frac {2}{3}}}{1540 b^{3}} - \frac {9 a^{2} x^{6} \left (a + b x^{3}\right )^{\frac {2}{3}}}{616 b^{2}} + \frac {a x^{9} \left (a + b x^{3}\right )^{\frac {2}{3}}}{77 b} + \frac {x^{12} \left (a + b x^{3}\right )^{\frac {2}{3}}}{14} & \text {for}\: b \neq 0 \\\frac {a^{\frac {2}{3}} x^{12}}{12} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.06, size = 57, normalized size = 0.71 \begin {gather*} \frac {220 \, {\left (b x^{3} + a\right )}^{\frac {14}{3}} - 840 \, {\left (b x^{3} + a\right )}^{\frac {11}{3}} a + 1155 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} a^{2} - 616 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{3}}{3080 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.11, size = 55, normalized size = 0.69 \begin {gather*} {\left (b\,x^3+a\right )}^{2/3}\,\left (\frac {x^{12}}{14}-\frac {81\,a^4}{3080\,b^4}+\frac {a\,x^9}{77\,b}+\frac {27\,a^3\,x^3}{1540\,b^3}-\frac {9\,a^2\,x^6}{616\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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