Optimal. Leaf size=36 \[ \frac {2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {368, 261} \begin {gather*} \frac {2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 261
Rule 368
Rubi steps
\begin {align*} \int x^2 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx &=\frac {x^3 \operatorname {Subst}\left (\int x^2 \sqrt {a+b x^3} \, dx,x,\sqrt {c x^2}\right )}{\left (c x^2\right )^{3/2}}\\ &=\frac {2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 1.00 \begin {gather*} \frac {2 x^3 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.83, size = 73, normalized size = 2.03 \begin {gather*} \frac {2 a \sqrt {c x^2} \sqrt {a+b c x^2 \sqrt {c x^2}}}{9 b c^2 x}+\frac {2}{9} x^3 \sqrt {a+b c x^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 46, normalized size = 1.28 \begin {gather*} \frac {2 \, {\left (b c^{2} x^{4} + \sqrt {c x^{2}} a\right )} \sqrt {\sqrt {c x^{2}} b c x^{2} + a}}{9 \, b c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 20, normalized size = 0.56 \begin {gather*} \frac {2 \, {\left (b c^{\frac {3}{2}} x^{3} + a\right )}^{\frac {3}{2}}}{9 \, b c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.81 \begin {gather*} \frac {2 \left (a +\left (c \,x^{2}\right )^{\frac {3}{2}} b \right )^{\frac {3}{2}} x^{3}}{9 \left (c \,x^{2}\right )^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\frac {2 \, {\left (b c^{\frac {3}{2}} x^{3} + a\right )}^{\frac {3}{2}} {\left (c - \sqrt {c}\right )}}{9 \, b c^{\frac {3}{2}}}}{c + 1} + \frac {{\left (b c^{\frac {3}{2}} x^{3} + a\right )}^{\frac {3}{2}}}{3 \, {\left (c^{2} + c\right )} b \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x^2\,\sqrt {a+b\,{\left (c\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt {a + b \left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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