Optimal. Leaf size=25 \[ \frac {2 \left (x^2 \left (a+b x^3\right )\right )^{3/2}}{9 b x^3} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1588} \begin {gather*} \frac {2 \left (x^2 \left (a+b x^3\right )\right )^{3/2}}{9 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin {align*} \int x \sqrt {x^2 \left (a+b x^3\right )} \, dx &=\frac {2 \left (x^2 \left (a+b x^3\right )\right )^{3/2}}{9 b x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2 \left (a+b x^3\right )\right )^{3/2}}{9 b x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \left (a x^2+b x^5\right )^{3/2}}{9 b x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 28, normalized size = 1.12 \begin {gather*} \frac {2 \, \sqrt {b x^{5} + a x^{2}} {\left (b x^{3} + a\right )}}{9 \, b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{9 \, b} - \frac {2 \, a^{\frac {3}{2}} \mathrm {sgn}\relax (x)}{9 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 1.16 \begin {gather*} \frac {2 \left (b \,x^{3}+a \right ) \sqrt {\left (b \,x^{3}+a \right ) x^{2}}}{9 b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 14, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{9 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.23, size = 22, normalized size = 0.88 \begin {gather*} \frac {2\,{\left (b\,x^3+a\right )}^{3/2}\,\sqrt {x^2}}{9\,b\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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