Optimal. Leaf size=18 \[ \frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2}} \]
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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267}
\begin {gather*} \frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{5/2} x^3} \, dx &=\frac {1}{3 b \left (a+\frac {b}{x^2}\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 28, normalized size = 1.56 \begin {gather*} \frac {b+a x^2}{3 b \left (a+\frac {b}{x^2}\right )^{5/2} x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 29, normalized size = 1.61
method | result | size |
derivativedivides | \(\frac {1}{3 b \left (\frac {b}{x^{2}}+a \right )^{\frac {3}{2}}}\) | \(15\) |
gosper | \(\frac {a \,x^{2}+b}{3 x^{2} b \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}}}\) | \(29\) |
default | \(\frac {a \,x^{2}+b}{3 x^{2} b \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}}}\) | \(29\) |
trager | \(\frac {x^{4} \sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{3 \left (a \,x^{2}+b \right )^{2} b}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{3 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (14) = 28\).
time = 0.36, size = 41, normalized size = 2.28 \begin {gather*} \frac {x^{4} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{3 \, {\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )}} \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. 48 vs.
\(2 (14) = 28\).
time = 0.70, size = 48, normalized size = 2.67 \begin {gather*} \begin {cases} \frac {1}{3 a b \sqrt {a + \frac {b}{x^{2}}} + \frac {3 b^{2} \sqrt {a + \frac {b}{x^{2}}}}{x^{2}}} & \text {for}\: b \neq 0 \\- \frac {1}{2 a^{\frac {5}{2}} x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 21, normalized size = 1.17 \begin {gather*} \frac {x^{3}}{3 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} b \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{3\,b\,{\left (a+\frac {b}{x^2}\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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