Optimal. Leaf size=43 \[ \frac{\sqrt{a+b x^2} (A b-a B)}{b^2}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.0335874, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 43} \[ \frac{\sqrt{a+b x^2} (A b-a B)}{b^2}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x \left (A+B x^2\right )}{\sqrt{a+b x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{\sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b \sqrt{a+b x}}+\frac{B \sqrt{a+b x}}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{(A b-a B) \sqrt{a+b x^2}}{b^2}+\frac{B \left (a+b x^2\right )^{3/2}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0217856, size = 33, normalized size = 0.77 \[ \frac{\sqrt{a+b x^2} \left (-2 a B+3 A b+b B x^2\right )}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 30, normalized size = 0.7 \begin{align*}{\frac{bB{x}^{2}+3\,Ab-2\,Ba}{3\,{b}^{2}}\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 = 1.56539, size = 69, normalized size = 1.6 \begin{align*} \frac{{\left (B b x^{2} - 2 \, B a + 3 \, A b\right )} \sqrt{b x^{2} + a}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.506212, size = 70, normalized size = 1.63 \begin{align*} \begin{cases} \frac{A \sqrt{a + b x^{2}}}{b} - \frac{2 B a \sqrt{a + b x^{2}}}{3 b^{2}} + \frac{B x^{2} \sqrt{a + b x^{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{4}}{4}}{\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 = 1.11937, size = 58, normalized size = 1.35 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} B - 3 \, \sqrt{b x^{2} + a} B a + 3 \, \sqrt{b x^{2} + a} A b}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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