Optimal. Leaf size=34 \[ \frac {3}{4} b^2 x^{4/3}+\frac {6}{5} a b x^{5/3}+\frac {a^2 x^2}{2} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 45}
\begin {gather*} \frac {a^2 x^2}{2}+\frac {6}{5} a b x^{5/3}+\frac {3}{4} b^2 x^{4/3} \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 )^2 x \, dx &=\int \left (b+a \sqrt [3]{x}\right )^2 \sqrt [3]{x} \, dx\\ &=3 \text {Subst}\left (\int x^3 (b+a x)^2 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (b^2 x^3+2 a b x^4+a^2 x^5\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{4} b^2 x^{4/3}+\frac {6}{5} a b x^{5/3}+\frac {a^2 x^2}{2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} \frac {1}{20} \left (15 b^2+24 a b \sqrt [3]{x}+10 a^2 x^{2/3}\right ) x^{4/3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 25, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {3 b^{2} x^{\frac {4}{3}}}{4}+\frac {6 a b \,x^{\frac {5}{3}}}{5}+\frac {a^{2} x^{2}}{2}\) | \(25\) |
default | \(\frac {3 b^{2} x^{\frac {4}{3}}}{4}+\frac {6 a b \,x^{\frac {5}{3}}}{5}+\frac {a^{2} x^{2}}{2}\) | \(25\) |
trager | \(\frac {\left (x -1\right ) a^{2} \left (x +1\right )}{2}+\frac {3 b^{2} x^{\frac {4}{3}}}{4}+\frac {6 a b \,x^{\frac {5}{3}}}{5}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 26, normalized size = 0.76 \begin {gather*} \frac {1}{20} \, {\left (10 \, a^{2} + \frac {24 \, a b}{x^{\frac {1}{3}}} + \frac {15 \, b^{2}}{x^{\frac {2}{3}}}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 24, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, a^{2} x^{2} + \frac {6}{5} \, a b x^{\frac {5}{3}} + \frac {3}{4} \, b^{2} x^{\frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 31, normalized size = 0.91 \begin {gather*} \frac {a^{2} x^{2}}{2} + \frac {6 a b x^{\frac {5}{3}}}{5} + \frac {3 b^{2} x^{\frac {4}{3}}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 24, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, a^{2} x^{2} + \frac {6}{5} \, a b x^{\frac {5}{3}} + \frac {3}{4} \, b^{2} x^{\frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 24, normalized size = 0.71 \begin {gather*} \frac {a^2\,x^2}{2}+\frac {3\,b^2\,x^{4/3}}{4}+\frac {6\,a\,b\,x^{5/3}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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