Optimal. Leaf size=55 \[ -\frac{a^2}{b^3 \sqrt{a+b x^2}}-\frac{2 a \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.0327409, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{a^2}{b^3 \sqrt{a+b x^2}}-\frac{2 a \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/2}}-\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{b^3 \sqrt{a+b x^2}}-\frac{2 a \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{3/2}}{3 b^3}\\ \end{align*}
Mathematica [A] time = 0.0159502, size = 38, normalized size = 0.69 \[ \frac{-8 a^2-4 a b x^2+b^2 x^4}{3 b^3 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.7 \begin{align*} -{\frac{-{b}^{2}{x}^{4}+4\,ab{x}^{2}+8\,{a}^{2}}{3\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29551, size = 93, normalized size = 1.69 \begin{align*} \frac{{\left (b^{2} x^{4} - 4 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.943373, size = 68, normalized size = 1.24 \begin{align*} \begin{cases} - \frac{8 a^{2}}{3 b^{3} \sqrt{a + b x^{2}}} - \frac{4 a x^{2}}{3 b^{2} \sqrt{a + b x^{2}}} + \frac{x^{4}}{3 b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.95599, size = 55, normalized size = 1. \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{b x^{2} + a} a - \frac{3 \, a^{2}}{\sqrt{b x^{2} + a}}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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