Optimal. Leaf size=38 \[ \frac {3 \left (a+b x^2\right )^{8/3}}{16 b^2}-\frac {3 a \left (a+b x^2\right )^{5/3}}{10 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, 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 = {266, 43} \[ \frac {3 \left (a+b x^2\right )^{8/3}}{16 b^2}-\frac {3 a \left (a+b x^2\right )^{5/3}}{10 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right )^{2/3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^{2/3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^{2/3}}{b}+\frac {(a+b x)^{5/3}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 a \left (a+b x^2\right )^{5/3}}{10 b^2}+\frac {3 \left (a+b x^2\right )^{8/3}}{16 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.74 \[ \frac {3 \left (a+b x^2\right )^{5/3} \left (5 b x^2-3 a\right )}{80 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 35, normalized size = 0.92 \[ \frac {3 \, {\left (5 \, b^{2} x^{4} + 2 \, a b x^{2} - 3 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{80 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 29, normalized size = 0.76 \[ \frac {3 \, {\left (5 \, {\left (b x^{2} + a\right )}^{\frac {8}{3}} - 8 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}} a\right )}}{80 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.66 \[ -\frac {3 \left (b \,x^{2}+a \right )^{\frac {5}{3}} \left (-5 b \,x^{2}+3 a \right )}{80 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 30, normalized size = 0.79 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {8}{3}}}{16 \, b^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}} a}{10 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.71, size = 33, normalized size = 0.87 \[ {\left (b\,x^2+a\right )}^{2/3}\,\left (\frac {3\,x^4}{16}-\frac {9\,a^2}{80\,b^2}+\frac {3\,a\,x^2}{40\,b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 66, normalized size = 1.74 \[ \begin {cases} - \frac {9 a^{2} \left (a + b x^{2}\right )^{\frac {2}{3}}}{80 b^{2}} + \frac {3 a x^{2} \left (a + b x^{2}\right )^{\frac {2}{3}}}{40 b} + \frac {3 x^{4} \left (a + b x^{2}\right )^{\frac {2}{3}}}{16} & \text {for}\: b \neq 0 \\\frac {a^{\frac {2}{3}} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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