Optimal. Leaf size=38 \[ \frac {3 \left (a+b x^2\right )^{5/3}}{10 b^2}-\frac {3 a \left (a+b x^2\right )^{2/3}}{4 b^2} \]
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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} \[ \frac {3 \left (a+b x^2\right )^{5/3}}{10 b^2}-\frac {3 a \left (a+b x^2\right )^{2/3}}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt [3]{a+b x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{a+b x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a}{b \sqrt [3]{a+b x}}+\frac {(a+b x)^{2/3}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 a \left (a+b x^2\right )^{2/3}}{4 b^2}+\frac {3 \left (a+b x^2\right )^{5/3}}{10 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.74 \[ \frac {3 \left (a+b x^2\right )^{2/3} \left (2 b x^2-3 a\right )}{20 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 24, normalized size = 0.63 \[ \frac {3 \, {\left (2 \, b x^{2} - 3 \, a\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{20 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 30, normalized size = 0.79 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}}}{10 \, b^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.66 \[ -\frac {3 \left (b \,x^{2}+a \right )^{\frac {2}{3}} \left (-2 b \,x^{2}+3 a \right )}{20 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 30, normalized size = 0.79 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}}}{10 \, b^{2}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.01, size = 24, normalized size = 0.63 \[ -\frac {3\,{\left (b\,x^2+a\right )}^{2/3}\,\left (3\,a-2\,b\,x^2\right )}{20\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.14, size = 178, normalized size = 4.68 \[ - \frac {9 a^{\frac {11}{3}} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} + \frac {9 a^{\frac {11}{3}}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} - \frac {3 a^{\frac {8}{3}} b x^{2} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} + \frac {9 a^{\frac {8}{3}} b x^{2}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} + \frac {6 a^{\frac {5}{3}} b^{2} x^{4} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{20 a^{2} b^{2} + 20 a b^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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