Optimal. Leaf size=34 \[ \frac{a}{6 b^2 \left (a+b x^2\right )^3}-\frac{1}{4 b^2 \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.0292993, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ \frac{a}{6 b^2 \left (a+b x^2\right )^3}-\frac{1}{4 b^2 \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{x^3}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{1}{2} b^4 \operatorname{Subst}\left (\int \frac{x}{\left (a b+b^2 x\right )^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^4 \operatorname{Subst}\left (\int \left (-\frac{a}{b^5 (a+b x)^4}+\frac{1}{b^5 (a+b x)^3}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{6 b^2 \left (a+b x^2\right )^3}-\frac{1}{4 b^2 \left (a+b x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0074374, size = 24, normalized size = 0.71 \[ -\frac{a+3 b x^2}{12 b^2 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 31, normalized size = 0.9 \begin{align*}{\frac{a}{6\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{1}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993504, size = 63, normalized size = 1.85 \begin{align*} -\frac{3 \, b x^{2} + a}{12 \,{\left (b^{5} x^{6} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64251, size = 96, normalized size = 2.82 \begin{align*} -\frac{3 \, b x^{2} + a}{12 \,{\left (b^{5} x^{6} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.556278, size = 48, normalized size = 1.41 \begin{align*} - \frac{a + 3 b x^{2}}{12 a^{3} b^{2} + 36 a^{2} b^{3} x^{2} + 36 a b^{4} x^{4} + 12 b^{5} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21101, size = 30, normalized size = 0.88 \begin{align*} -\frac{3 \, b x^{2} + a}{12 \,{\left (b x^{2} + a\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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