Optimal. Leaf size=46 \[ \frac{2 \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^2}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b^2} \]
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Rubi [A] time = 0.0399213, 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} \[ \frac{2 \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^2}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int x^2 \sqrt{a+b x^3} \left (A+B x^3\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \sqrt{a+b x} (A+B x) \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{(A b-a B) \sqrt{a+b x}}{b}+\frac{B (a+b x)^{3/2}}{b}\right ) \, dx,x,x^3\right )\\ &=\frac{2 (A b-a B) \left (a+b x^3\right )^{3/2}}{9 b^2}+\frac{2 B \left (a+b x^3\right )^{5/2}}{15 b^2}\\ \end{align*}
Mathematica [A] time = 0.0229571, size = 34, normalized size = 0.74 \[ \frac{2 \left (a+b x^3\right )^{3/2} \left (-2 a B+5 A b+3 b B x^3\right )}{45 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 31, normalized size = 0.7 \begin{align*}{\frac{6\,bB{x}^{3}+10\,Ab-4\,Ba}{45\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.922227, size = 66, normalized size = 1.43 \begin{align*} \frac{2}{45} \, B{\left (\frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )} + \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A}{9 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63958, size = 113, normalized size = 2.46 \begin{align*} \frac{2 \,{\left (3 \, B b^{2} x^{6} +{\left (B a b + 5 \, A b^{2}\right )} x^{3} - 2 \, B a^{2} + 5 \, A a b\right )} \sqrt{b x^{3} + a}}{45 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.628639, size = 117, normalized size = 2.54 \begin{align*} \begin{cases} \frac{2 A a \sqrt{a + b x^{3}}}{9 b} + \frac{2 A x^{3} \sqrt{a + b x^{3}}}{9} - \frac{4 B a^{2} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 B a x^{3} \sqrt{a + b x^{3}}}{45 b} + \frac{2 B x^{6} \sqrt{a + b x^{3}}}{15} & \text{for}\: b \neq 0 \\\sqrt{a} \left (\frac{A x^{3}}{3} + \frac{B x^{6}}{6}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12564, size = 63, normalized size = 1.37 \begin{align*} \frac{2 \,{\left (5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A + \frac{{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a\right )} B}{b}\right )}}{45 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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