Optimal. Leaf size=21 \[ \frac {\left (a+\frac {b}{x^2}\right )^{3/2} x^3}{3 a} \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270}
\begin {gather*} \frac {x^3 \left (a+\frac {b}{x^2}\right )^{3/2}}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \sqrt {a+\frac {b}{x^2}} x^2 \, dx &=\frac {\left (a+\frac {b}{x^2}\right )^{3/2} x^3}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 26, normalized size = 1.24 \begin {gather*} \frac {\sqrt {a+\frac {b}{x^2}} x \left (b+a x^2\right )}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 27, normalized size = 1.29
method | result | size |
gosper | \(\frac {\left (a \,x^{2}+b \right ) x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{3 a}\) | \(27\) |
default | \(\frac {\left (a \,x^{2}+b \right ) x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{3 a}\) | \(27\) |
risch | \(\frac {\left (a \,x^{2}+b \right ) x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{3 a}\) | \(27\) |
trager | \(\frac {x \left (a \,x^{2}+b \right ) \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{3 a}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 17, normalized size = 0.81 \begin {gather*} \frac {{\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} x^{3}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 27, normalized size = 1.29 \begin {gather*} \frac {{\left (a x^{3} + b x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (15) = 30\).
time = 0.39, size = 41, normalized size = 1.95 \begin {gather*} \frac {\sqrt {b} x^{2} \sqrt {\frac {a x^{2}}{b} + 1}}{3} + \frac {b^{\frac {3}{2}} \sqrt {\frac {a x^{2}}{b} + 1}}{3 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 27, normalized size = 1.29 \begin {gather*} \frac {{\left (a x^{2} + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a} - \frac {b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.24, size = 23, normalized size = 1.10 \begin {gather*} \sqrt {a+\frac {b}{x^2}}\,\left (\frac {x^3}{3}+\frac {b\,x}{3\,a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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