Optimal. Leaf size=16 \[ -\frac {1}{10 b \left (a+b x^2\right )^5} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {28, 267}
\begin {gather*} -\frac {1}{10 b \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 267
Rubi steps
\begin {align*} \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {1}{10 b \left (a+b x^2\right )^5}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{10 b \left (a+b x^2\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 15, normalized size = 0.94
method | result | size |
default | \(-\frac {1}{10 b \left (b \,x^{2}+a \right )^{5}}\) | \(15\) |
norman | \(-\frac {1}{10 b \left (b \,x^{2}+a \right )^{5}}\) | \(15\) |
gosper | \(-\frac {1}{10 \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} b}\) | \(35\) |
risch | \(-\frac {1}{10 \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} b}\) | \(35\) |
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. 59 vs.
\(2 (14) = 28\).
time = 0.29, size = 59, normalized size = 3.69 \begin {gather*} -\frac {1}{10 \, {\left (b^{6} x^{10} + 5 \, a b^{5} x^{8} + 10 \, a^{2} b^{4} x^{6} + 10 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{2} + a^{5} b\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. 59 vs.
\(2 (14) = 28\).
time = 0.36, size = 59, normalized size = 3.69 \begin {gather*} -\frac {1}{10 \, {\left (b^{6} x^{10} + 5 \, a b^{5} x^{8} + 10 \, a^{2} b^{4} x^{6} + 10 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{2} + a^{5} b\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. 63 vs.
\(2 (14) = 28\).
time = 0.23, size = 63, normalized size = 3.94 \begin {gather*} - \frac {1}{10 a^{5} b + 50 a^{4} b^{2} x^{2} + 100 a^{3} b^{3} x^{4} + 100 a^{2} b^{4} x^{6} + 50 a b^{5} x^{8} + 10 b^{6} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.52, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{10 \, {\left (b x^{2} + a\right )}^{5} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 61, normalized size = 3.81 \begin {gather*} -\frac {1}{10\,a^5\,b+50\,a^4\,b^2\,x^2+100\,a^3\,b^3\,x^4+100\,a^2\,b^4\,x^6+50\,a\,b^5\,x^8+10\,b^6\,x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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