Optimal. Leaf size=36 \[ \frac{a}{8 b^2 \left (a-b x^2\right )^4}-\frac{1}{6 b^2 \left (a-b x^2\right )^3} \]
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Rubi [A] time = 0.0295651, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {266, 43} \[ \frac{a}{8 b^2 \left (a-b x^2\right )^4}-\frac{1}{6 b^2 \left (a-b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a-b x^2\right )^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a-b x)^5} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a}{b (a-b x)^5}-\frac{1}{b (a-b x)^4}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{8 b^2 \left (a-b x^2\right )^4}-\frac{1}{6 b^2 \left (a-b x^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0096997, size = 25, normalized size = 0.69 \[ -\frac{a-4 b x^2}{24 b^2 \left (a-b x^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 35, normalized size = 1. \begin{align*}{\frac{a}{8\,{b}^{2} \left ( b{x}^{2}-a \right ) ^{4}}}+{\frac{1}{6\,{b}^{2} \left ( b{x}^{2}-a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.9082, size = 81, normalized size = 2.25 \begin{align*} \frac{4 \, b x^{2} - a}{24 \,{\left (b^{6} x^{8} - 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21492, size = 116, normalized size = 3.22 \begin{align*} \frac{4 \, b x^{2} - a}{24 \,{\left (b^{6} x^{8} - 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} - 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.732412, size = 58, normalized size = 1.61 \begin{align*} \frac{- a + 4 b x^{2}}{24 a^{4} b^{2} - 96 a^{3} b^{3} x^{2} + 144 a^{2} b^{4} x^{4} - 96 a b^{5} x^{6} + 24 b^{6} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.33869, size = 53, normalized size = 1.47 \begin{align*} \frac{\frac{4}{{\left (b x^{2} - a\right )}^{3} b} + \frac{3 \, a}{{\left (b x^{2} - a\right )}^{4} b}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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