Optimal. Leaf size=19 \[ \frac{x^6}{6 a \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.006774, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 264} \[ \frac{x^6}{6 a \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 264
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{x^5}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{x^6}{6 a \left (a+b x^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0132677, size = 35, normalized size = 1.84 \[ -\frac{a^2+3 a b x^2+3 b^2 x^4}{6 b^3 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 48, normalized size = 2.5 \begin{align*} -{\frac{{a}^{2}}{6\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{a}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{1}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.979301, size = 78, normalized size = 4.11 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58219, size = 116, normalized size = 6.11 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.58234, size = 60, normalized size = 3.16 \begin{align*} - \frac{a^{2} + 3 a b x^{2} + 3 b^{2} x^{4}}{6 a^{3} b^{3} + 18 a^{2} b^{4} x^{2} + 18 a b^{5} x^{4} + 6 b^{6} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13551, size = 45, normalized size = 2.37 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \,{\left (b x^{2} + a\right )}^{3} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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