Optimal. Leaf size=47 \[ \frac {b^3 x^2}{2}+\frac {9}{7} a b^2 x^{7/3}+\frac {9}{8} a^2 b x^{8/3}+\frac {a^3 x^3}{3} \]
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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {269, 272, 45}
\begin {gather*} \frac {a^3 x^3}{3}+\frac {9}{8} a^2 b x^{8/3}+\frac {9}{7} a b^2 x^{7/3}+\frac {b^3 x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \left (a+\frac {b}{\sqrt [3]{x}}\right )^3 x^2 \, dx &=\int \left (b+a \sqrt [3]{x}\right )^3 x \, dx\\ &=3 \text {Subst}\left (\int x^5 (b+a x)^3 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (b^3 x^5+3 a b^2 x^6+3 a^2 b x^7+a^3 x^8\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {b^3 x^2}{2}+\frac {9}{7} a b^2 x^{7/3}+\frac {9}{8} a^2 b x^{8/3}+\frac {a^3 x^3}{3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 43, normalized size = 0.91 \begin {gather*} \frac {1}{168} \left (84 b^3 x^2+216 a b^2 x^{7/3}+189 a^2 b x^{8/3}+56 a^3 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 36, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {b^{3} x^{2}}{2}+\frac {9 a \,b^{2} x^{\frac {7}{3}}}{7}+\frac {9 a^{2} b \,x^{\frac {8}{3}}}{8}+\frac {a^{3} x^{3}}{3}\) | \(36\) |
default | \(\frac {b^{3} x^{2}}{2}+\frac {9 a \,b^{2} x^{\frac {7}{3}}}{7}+\frac {9 a^{2} b \,x^{\frac {8}{3}}}{8}+\frac {a^{3} x^{3}}{3}\) | \(36\) |
trager | \(\frac {\left (2 a^{3} x^{2}+2 a^{3} x +3 b^{3} x +2 a^{3}+3 b^{3}\right ) \left (x -1\right )}{6}+\frac {9 a \,b^{2} x^{\frac {7}{3}}}{7}+\frac {9 a^{2} b \,x^{\frac {8}{3}}}{8}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 37, normalized size = 0.79 \begin {gather*} \frac {1}{168} \, {\left (56 \, a^{3} + \frac {189 \, a^{2} b}{x^{\frac {1}{3}}} + \frac {216 \, a b^{2}}{x^{\frac {2}{3}}} + \frac {84 \, b^{3}}{x}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 35, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, a^{3} x^{3} + \frac {9}{8} \, a^{2} b x^{\frac {8}{3}} + \frac {9}{7} \, a b^{2} x^{\frac {7}{3}} + \frac {1}{2} \, b^{3} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 42, normalized size = 0.89 \begin {gather*} \frac {a^{3} x^{3}}{3} + \frac {9 a^{2} b x^{\frac {8}{3}}}{8} + \frac {9 a b^{2} x^{\frac {7}{3}}}{7} + \frac {b^{3} x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 35, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, a^{3} x^{3} + \frac {9}{8} \, a^{2} b x^{\frac {8}{3}} + \frac {9}{7} \, a b^{2} x^{\frac {7}{3}} + \frac {1}{2} \, b^{3} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 35, normalized size = 0.74 \begin {gather*} \frac {a^3\,x^3}{3}+\frac {b^3\,x^2}{2}+\frac {9\,a\,b^2\,x^{7/3}}{7}+\frac {9\,a^2\,b\,x^{8/3}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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