Optimal. Leaf size=46 \[ \frac{2 \sqrt{a+b x^3} (A b-a B)}{3 b^2}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^2} \]
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Rubi [A] time = 0.0378856, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {444, 43} \[ \frac{2 \sqrt{a+b x^3} (A b-a B)}{3 b^2}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 \left (A+B x^3\right )}{\sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{\sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b \sqrt{a+b x}}+\frac{B \sqrt{a+b x}}{b}\right ) \, dx,x,x^3\right )\\ &=\frac{2 (A b-a B) \sqrt{a+b x^3}}{3 b^2}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.0221966, size = 33, normalized size = 0.72 \[ \frac{2 \sqrt{a+b x^3} \left (-2 a B+3 A b+b B x^3\right )}{9 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 30, normalized size = 0.7 \begin{align*}{\frac{2\,bB{x}^{3}+6\,Ab-4\,Ba}{9\,{b}^{2}}\sqrt{b{x}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938191, size = 65, normalized size = 1.41 \begin{align*} \frac{2}{9} \, B{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{b x^{3} + a} a}{b^{2}}\right )} + \frac{2 \, \sqrt{b x^{3} + a} A}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74681, size = 69, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (B b x^{3} - 2 \, B a + 3 \, A b\right )} \sqrt{b x^{3} + a}}{9 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.768402, size = 75, normalized size = 1.63 \begin{align*} \begin{cases} \frac{2 A \sqrt{a + b x^{3}}}{3 b} - \frac{4 B a \sqrt{a + b x^{3}}}{9 b^{2}} + \frac{2 B x^{3} \sqrt{a + b x^{3}}}{9 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{6}}{6}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11092, size = 58, normalized size = 1.26 \begin{align*} \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} B - 3 \, \sqrt{b x^{3} + a} B a + 3 \, \sqrt{b x^{3} + a} A b\right )}}{9 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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