Optimal. Leaf size=16 \[ -\frac {\left (a+\frac {b}{x^2}\right )^4}{8 b} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267}
\begin {gather*} -\frac {\left (a+\frac {b}{x^2}\right )^4}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x^3} \, dx &=-\frac {\left (a+\frac {b}{x^2}\right )^4}{8 b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(16)=32\).
time = 0.01, size = 43, normalized size = 2.69 \begin {gather*} -\frac {b^3}{8 x^8}-\frac {a b^2}{2 x^6}-\frac {3 a^2 b}{4 x^4}-\frac {a^3}{2 x^2} \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. \(35\) vs.
\(2(14)=28\).
time = 0.02, size = 36, normalized size = 2.25
method | result | size |
gosper | \(-\frac {4 a^{3} x^{6}+6 a^{2} b \,x^{4}+4 a \,b^{2} x^{2}+b^{3}}{8 x^{8}}\) | \(36\) |
default | \(-\frac {3 a^{2} b}{4 x^{4}}-\frac {a \,b^{2}}{2 x^{6}}-\frac {a^{3}}{2 x^{2}}-\frac {b^{3}}{8 x^{8}}\) | \(36\) |
norman | \(\frac {-\frac {1}{2} a^{3} x^{6}-\frac {3}{4} a^{2} b \,x^{4}-\frac {1}{2} a \,b^{2} x^{2}-\frac {1}{8} b^{3}}{x^{8}}\) | \(37\) |
risch | \(\frac {-\frac {1}{2} a^{3} x^{6}-\frac {3}{4} a^{2} b \,x^{4}-\frac {1}{2} a \,b^{2} x^{2}-\frac {1}{8} b^{3}}{x^{8}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 14, normalized size = 0.88 \begin {gather*} -\frac {{\left (a + \frac {b}{x^{2}}\right )}^{4}}{8 \, 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. 35 vs.
\(2 (14) = 28\).
time = 0.36, size = 35, normalized size = 2.19 \begin {gather*} -\frac {4 \, a^{3} x^{6} + 6 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} + b^{3}}{8 \, x^{8}} \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. 37 vs.
\(2 (12) = 24\).
time = 0.11, size = 37, normalized size = 2.31 \begin {gather*} \frac {- 4 a^{3} x^{6} - 6 a^{2} b x^{4} - 4 a b^{2} x^{2} - b^{3}}{8 x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (14) = 28\).
time = 0.53, size = 35, normalized size = 2.19 \begin {gather*} -\frac {4 \, a^{3} x^{6} + 6 \, a^{2} b x^{4} + 4 \, a b^{2} x^{2} + b^{3}}{8 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 37, normalized size = 2.31 \begin {gather*} -\frac {\frac {a^3\,x^6}{2}+\frac {3\,a^2\,b\,x^4}{4}+\frac {a\,b^2\,x^2}{2}+\frac {b^3}{8}}{x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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