Optimal. Leaf size=104 \[ -\frac{a^2 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}-\frac{\left (a+b x^2\right )^{3/2} (8 a B-15 A b x)}{60 b^2}-\frac{a A x \sqrt{a+b x^2}}{8 b}+\frac{B x^2 \left (a+b x^2\right )^{3/2}}{5 b} \]
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Rubi [A] time = 0.0490937, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {833, 780, 195, 217, 206} \[ -\frac{a^2 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}-\frac{\left (a+b x^2\right )^{3/2} (8 a B-15 A b x)}{60 b^2}-\frac{a A x \sqrt{a+b x^2}}{8 b}+\frac{B x^2 \left (a+b x^2\right )^{3/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 833
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^2 (A+B x) \sqrt{a+b x^2} \, dx &=\frac{B x^2 \left (a+b x^2\right )^{3/2}}{5 b}+\frac{\int x (-2 a B+5 A b x) \sqrt{a+b x^2} \, dx}{5 b}\\ &=\frac{B x^2 \left (a+b x^2\right )^{3/2}}{5 b}-\frac{(8 a B-15 A b x) \left (a+b x^2\right )^{3/2}}{60 b^2}-\frac{(a A) \int \sqrt{a+b x^2} \, dx}{4 b}\\ &=-\frac{a A x \sqrt{a+b x^2}}{8 b}+\frac{B x^2 \left (a+b x^2\right )^{3/2}}{5 b}-\frac{(8 a B-15 A b x) \left (a+b x^2\right )^{3/2}}{60 b^2}-\frac{\left (a^2 A\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{8 b}\\ &=-\frac{a A x \sqrt{a+b x^2}}{8 b}+\frac{B x^2 \left (a+b x^2\right )^{3/2}}{5 b}-\frac{(8 a B-15 A b x) \left (a+b x^2\right )^{3/2}}{60 b^2}-\frac{\left (a^2 A\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{8 b}\\ &=-\frac{a A x \sqrt{a+b x^2}}{8 b}+\frac{B x^2 \left (a+b x^2\right )^{3/2}}{5 b}-\frac{(8 a B-15 A b x) \left (a+b x^2\right )^{3/2}}{60 b^2}-\frac{a^2 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.174386, size = 93, normalized size = 0.89 \[ \frac{\sqrt{a+b x^2} \left (-\frac{15 a^{3/2} A \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}-16 a^2 B+a b x (15 A+8 B x)+6 b^2 x^3 (5 A+4 B x)\right )}{120 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 94, normalized size = 0.9 \begin{align*}{\frac{B{x}^{2}}{5\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{2\,Ba}{15\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ax}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{aAx}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\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 = 1.55456, size = 435, normalized size = 4.18 \begin{align*} \left [\frac{15 \, A a^{2} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (24 \, B b^{2} x^{4} + 30 \, A b^{2} x^{3} + 8 \, B a b x^{2} + 15 \, A a b x - 16 \, B a^{2}\right )} \sqrt{b x^{2} + a}}{240 \, b^{2}}, \frac{15 \, A a^{2} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (24 \, B b^{2} x^{4} + 30 \, A b^{2} x^{3} + 8 \, B a b x^{2} + 15 \, A a b x - 16 \, B a^{2}\right )} \sqrt{b x^{2} + a}}{120 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.43068, size = 165, normalized size = 1.59 \begin{align*} \frac{A a^{\frac{3}{2}} x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A \sqrt{a} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{A b x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + B \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18154, size = 109, normalized size = 1.05 \begin{align*} \frac{A a^{2} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{3}{2}}} + \frac{1}{120} \, \sqrt{b x^{2} + a}{\left ({\left (2 \,{\left (3 \,{\left (4 \, B x + 5 \, A\right )} x + \frac{4 \, B a}{b}\right )} x + \frac{15 \, A a}{b}\right )} x - \frac{16 \, B a^{2}}{b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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