Optimal. Leaf size=18 \[ -\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b} \]
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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 {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{x^3}}}{x^4} \, dx &=-\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 \left (a+\frac {b}{x^3}\right )^{3/2}}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs.
\(2(14)=28\).
time = 0.07, size = 51, normalized size = 2.83
method | result | size |
derivativedivides | \(-\frac {2 \left (a +\frac {b}{x^{3}}\right )^{\frac {3}{2}}}{9 b}\) | \(15\) |
gosper | \(-\frac {2 \left (a \,x^{3}+b \right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{9 x^{3} b}\) | \(29\) |
risch | \(-\frac {2 \left (a \,x^{3}+b \right ) \sqrt {\frac {a \,x^{3}+b}{x^{3}}}}{9 x^{3} b}\) | \(29\) |
trager | \(-\frac {2 \left (a \,x^{3}+b \right ) \sqrt {-\frac {-a \,x^{3}-b}{x^{3}}}}{9 x^{3} b}\) | \(33\) |
default | \(-\frac {2 \sqrt {\frac {a \,x^{3}+b}{x^{3}}}\, \sqrt {a \,x^{4}+b x}\, \left (a \,x^{3}+b \right )}{9 x^{3} \sqrt {x \left (a \,x^{3}+b \right )}\, b}\) | \(51\) |
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 {2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 28, normalized size = 1.56 \begin {gather*} -\frac {2 \, {\left (a x^{3} + b\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{9 \, b x^{3}} \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. 46 vs.
\(2 (15) = 30\).
time = 0.52, size = 46, normalized size = 2.56 \begin {gather*} - \frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}}{9 b} - \frac {2 \sqrt {a} \sqrt {1 + \frac {b}{a x^{3}}}}{9 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2 \, {\left (a + \frac {b}{x^{3}}\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 = 1.43, size = 24, normalized size = 1.33 \begin {gather*} -\frac {2\,\sqrt {a+\frac {b}{x^3}}\,\left (a\,x^3+b\right )}{9\,b\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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