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 = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
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*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 0.72 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (-2 a B+3 A b+b B x^3\right )}{9 b^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 33, normalized size = 0.72 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (-2 a B+3 A b+b B x^3\right )}{9 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 29, normalized size = 0.63 \begin {gather*} \frac {2 \, {\left (B b x^{3} - 2 \, B a + 3 \, A b\right )} \sqrt {b x^{3} + a}}{9 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 38, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B}{9 \, b^{2}} - \frac {2 \, \sqrt {b x^{3} + a} {\left (B a - A b\right )}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 0.65 \begin {gather*} \frac {2 \sqrt {b \,x^{3}+a}\, \left (B b \,x^{3}+3 A b -2 B a \right )}{9 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 48, normalized size = 1.04 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 29, normalized size = 0.63 \begin {gather*} \frac {2\,\sqrt {b\,x^3+a}\,\left (B\,b\,x^3+3\,A\,b-2\,B\,a\right )}{9\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.98, size = 75, normalized size = 1.63 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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