Optimal. Leaf size=46 \[ \frac{\left (a+b x^2\right )^{3/2} (A b-a B)}{3 b^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.0362037, antiderivative size = 46, 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} \[ \frac{\left (a+b x^2\right )^{3/2} (A b-a B)}{3 b^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \sqrt{a+b x^2} \left (A+B x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{a+b x} (A+B x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(A b-a B) \sqrt{a+b x}}{b}+\frac{B (a+b x)^{3/2}}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{(A b-a B) \left (a+b x^2\right )^{3/2}}{3 b^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0225728, size = 34, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{3/2} \left (-2 a B+5 A b+3 b B x^2\right )}{15 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 31, normalized size = 0.7 \begin{align*}{\frac{3\,bB{x}^{2}+5\,Ab-2\,Ba}{15\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \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 = 1.81207, size = 113, normalized size = 2.46 \begin{align*} \frac{{\left (3 \, B b^{2} x^{4} - 2 \, B a^{2} + 5 \, A a b +{\left (B a b + 5 \, A b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.329264, size = 110, normalized size = 2.39 \begin{align*} \begin{cases} \frac{A a \sqrt{a + b x^{2}}}{3 b} + \frac{A x^{2} \sqrt{a + b x^{2}}}{3} - \frac{2 B a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{B a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{B x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\sqrt{a} \left (\frac{A x^{2}}{2} + \frac{B x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11125, size = 63, normalized size = 1.37 \begin{align*} \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A + \frac{{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} B}{b}}{15 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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