Optimal. Leaf size=41 \[ \frac{B \sqrt{a+b x^2}}{b^2}-\frac{A b-a B}{b^2 \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0321411, antiderivative size = 41, 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} \[ \frac{B \sqrt{a+b x^2}}{b^2}-\frac{A b-a B}{b^2 \sqrt{a+b x^2}} \]
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 )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b (a+b x)^{3/2}}+\frac{B}{b \sqrt{a+b x}}\right ) \, dx,x,x^2\right )\\ &=-\frac{A b-a B}{b^2 \sqrt{a+b x^2}}+\frac{B \sqrt{a+b x^2}}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0207583, size = 30, normalized size = 0.73 \[ \frac{2 a B-A b+b B x^2}{b^2 \sqrt{a+b x^2}} \]
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}+Ab-2\,Ba}{{b}^{2}}{\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 = 1.54058, size = 80, normalized size = 1.95 \begin{align*} \frac{{\left (B b x^{2} + 2 \, B a - A b\right )} \sqrt{b x^{2} + a}}{b^{3} x^{2} + a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.588677, size = 66, normalized size = 1.61 \begin{align*} \begin{cases} - \frac{A}{b \sqrt{a + b x^{2}}} + \frac{2 B a}{b^{2} \sqrt{a + b x^{2}}} + \frac{B x^{2}}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{4}}{4}}{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 = 1.10764, size = 46, normalized size = 1.12 \begin{align*} \frac{\sqrt{b x^{2} + a} B + \frac{B a - A b}{\sqrt{b x^{2} + a}}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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