Optimal. Leaf size=38 \[ \frac{3 \left (a+b x^2\right )^{7/3}}{14 b^2}-\frac{3 a \left (a+b x^2\right )^{4/3}}{8 b^2} \]
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Rubi [A] time = 0.0231805, antiderivative size = 38, normalized size of antiderivative = 1., 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} \[ \frac{3 \left (a+b x^2\right )^{7/3}}{14 b^2}-\frac{3 a \left (a+b x^2\right )^{4/3}}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \sqrt [3]{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \sqrt [3]{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a \sqrt [3]{a+b x}}{b}+\frac{(a+b x)^{4/3}}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a \left (a+b x^2\right )^{4/3}}{8 b^2}+\frac{3 \left (a+b x^2\right )^{7/3}}{14 b^2}\\ \end{align*}
Mathematica [A] time = 0.0132055, size = 28, normalized size = 0.74 \[ \frac{3 \left (a+b x^2\right )^{4/3} \left (4 b x^2-3 a\right )}{56 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-12\,b{x}^{2}+9\,a}{56\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62692, size = 41, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}}}{14 \, b^{2}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50011, size = 78, normalized size = 2.05 \begin{align*} \frac{3 \,{\left (4 \, b^{2} x^{4} + a b x^{2} - 3 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{56 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.08524, size = 223, normalized size = 5.87 \begin{align*} - \frac{9 a^{\frac{13}{3}} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} + \frac{9 a^{\frac{13}{3}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} - \frac{6 a^{\frac{10}{3}} b x^{2} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} + \frac{9 a^{\frac{10}{3}} b x^{2}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} + \frac{15 a^{\frac{7}{3}} b^{2} x^{4} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} + \frac{12 a^{\frac{4}{3}} b^{3} x^{6} \sqrt [3]{1 + \frac{b x^{2}}{a}}}{56 a^{2} b^{2} + 56 a b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.93942, size = 39, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (4 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a\right )}}{56 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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