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 = 0.04, 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 \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
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*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.74 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 34, normalized size = 0.74 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 50, normalized size = 1.09 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 44, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} B - 5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B a + 5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} A b\right )}}{45 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 0.67 \begin {gather*} \frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (3 B b \,x^{3}+5 A b -2 B a \right )}{45 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 49, normalized size = 1.07 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 44, normalized size = 0.96 \begin {gather*} \frac {6\,B\,{\left (b\,x^3+a\right )}^{5/2}+10\,A\,b\,{\left (b\,x^3+a\right )}^{3/2}-10\,B\,a\,{\left (b\,x^3+a\right )}^{3/2}}{45\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 117, normalized size = 2.54 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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