Optimal. Leaf size=19 \[ \frac {x^4}{4 a \left (a+b x^2\right )^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, 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 = {270}
\begin {gather*} \frac {x^4}{4 a \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right )^3} \, dx &=\frac {x^4}{4 a \left (a+b x^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.26 \begin {gather*} -\frac {a+2 b x^2}{4 b^2 \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 31, normalized size = 1.63
method | result | size |
gosper | \(-\frac {2 b \,x^{2}+a}{4 b^{2} \left (b \,x^{2}+a \right )^{2}}\) | \(23\) |
norman | \(\frac {-\frac {x^{2}}{2 b}-\frac {a}{4 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}\) | \(26\) |
risch | \(\frac {-\frac {x^{2}}{2 b}-\frac {a}{4 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}\) | \(26\) |
default | \(-\frac {1}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {a}{4 b^{2} \left (b \,x^{2}+a \right )^{2}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (17) = 34\).
time = 0.27, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \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. 36 vs.
\(2 (17) = 34\).
time = 1.13, size = 36, normalized size = 1.89 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\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. 36 vs.
\(2 (14) = 28\).
time = 0.11, size = 36, normalized size = 1.89 \begin {gather*} \frac {- a - 2 b x^{2}}{4 a^{2} b^{2} + 8 a b^{3} x^{2} + 4 b^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.39, size = 22, normalized size = 1.16 \begin {gather*} -\frac {2 \, b x^{2} + a}{4 \, {\left (b x^{2} + a\right )}^{2} b^{2}} \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 = 1.95 \begin {gather*} -\frac {\frac {a}{4\,b^2}+\frac {x^2}{2\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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