Optimal. Leaf size=16 \[ -\frac {1}{6 b \left (a+b x^2\right )^3} \]
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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, 261} \begin {gather*} -\frac {1}{6 b \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 261
Rubi steps
\begin {align*} \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {x}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {1}{6 b \left (a+b x^2\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{6 b \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.80, size = 37, normalized size = 2.31 \begin {gather*} -\frac {1}{6 \, {\left (b^{4} x^{6} + 3 \, a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{6 \, {\left (b x^{2} + a\right )}^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{6 \left (b \,x^{2}+a \right )^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.34, size = 37, normalized size = 2.31 \begin {gather*} -\frac {1}{6 \, {\left (b^{4} x^{6} + 3 \, a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.28, size = 39, normalized size = 2.44 \begin {gather*} -\frac {1}{6\,a^3\,b+18\,a^2\,b^2\,x^2+18\,a\,b^3\,x^4+6\,b^4\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.33, size = 39, normalized size = 2.44 \begin {gather*} - \frac {1}{6 a^{3} b + 18 a^{2} b^{2} x^{2} + 18 a b^{3} x^{4} + 6 b^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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