Optimal. Leaf size=56 \[ \frac{a^2 \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{5/2}}{5 b^3}-\frac{2 a \left (a+b x^2\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.0307414, antiderivative size = 56, 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} \[ \frac{a^2 \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{5/2}}{5 b^3}-\frac{2 a \left (a+b x^2\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{a+b x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt{a+b x}}-\frac{2 a \sqrt{a+b x}}{b^2}+\frac{(a+b x)^{3/2}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 \sqrt{a+b x^2}}{b^3}-\frac{2 a \left (a+b x^2\right )^{3/2}}{3 b^3}+\frac{\left (a+b x^2\right )^{5/2}}{5 b^3}\\ \end{align*}
Mathematica [A] time = 0.0182842, size = 39, normalized size = 0.7 \[ \frac{\sqrt{a+b x^2} \left (8 a^2-4 a b x^2+3 b^2 x^4\right )}{15 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.6 \begin{align*}{\frac{3\,{b}^{2}{x}^{4}-4\,ab{x}^{2}+8\,{a}^{2}}{15\,{b}^{3}}\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 = 1.27504, size = 78, normalized size = 1.39 \begin{align*} \frac{{\left (3 \, b^{2} x^{4} - 4 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{15 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.781171, size = 68, normalized size = 1.21 \begin{align*} \begin{cases} \frac{8 a^{2} \sqrt{a + b x^{2}}}{15 b^{3}} - \frac{4 a x^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{x^{4} \sqrt{a + b x^{2}}}{5 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 \sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.49782, size = 58, normalized size = 1.04 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x^{2} + a} a^{2}}{15 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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