Optimal. Leaf size=18 \[ \frac {\left (a+b x^2\right )^{5/2}}{5 b} \]
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Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {261} \begin {gather*} \frac {\left (a+b x^2\right )^{5/2}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^{3/2} \, dx &=\frac {\left (a+b x^2\right )^{5/2}}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right )^{5/2}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right )^{5/2}}{5 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 32, normalized size = 1.78 \begin {gather*} \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {b x^{2} + a}}{5 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{5 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{5 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.57, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (b\,x^2+a\right )}^{5/2}}{5\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 61, normalized size = 3.39 \begin {gather*} \begin {cases} \frac {a^{2} \sqrt {a + b x^{2}}}{5 b} + \frac {2 a x^{2} \sqrt {a + b x^{2}}}{5} + \frac {b x^{4} \sqrt {a + b x^{2}}}{5} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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