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 = 0.01, 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 = {375, 267}
\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 267
Rule 375
Rubi steps
\begin {align*} \int x^2 \sqrt {a+b \left (c x^2\right )^{3/2}} \, dx &=\frac {x^3 \text {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 = 1.30, 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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Maple [A]
time = 0.24, size = 29, normalized size = 0.81
method | result | size |
default | \(\frac {2 x^{3} \left (a +b \left (c \,x^{2}\right )^{\frac {3}{2}}\right )^{\frac {3}{2}}}{9 b \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(29\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, 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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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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Giac [A]
time = 1.21, 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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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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