Optimal. Leaf size=80 \[ -\frac{a^2 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{\left (a+b x^2\right )^{3/2} (4 A+3 B x)}{12 b}-\frac{a B x \sqrt{a+b x^2}}{8 b} \]
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Rubi [A] time = 0.0256785, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {780, 195, 217, 206} \[ -\frac{a^2 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{\left (a+b x^2\right )^{3/2} (4 A+3 B x)}{12 b}-\frac{a B x \sqrt{a+b x^2}}{8 b} \]
Antiderivative was successfully verified.
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Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x (A+B x) \sqrt{a+b x^2} \, dx &=\frac{(4 A+3 B x) \left (a+b x^2\right )^{3/2}}{12 b}-\frac{(a B) \int \sqrt{a+b x^2} \, dx}{4 b}\\ &=-\frac{a B x \sqrt{a+b x^2}}{8 b}+\frac{(4 A+3 B x) \left (a+b x^2\right )^{3/2}}{12 b}-\frac{\left (a^2 B\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{8 b}\\ &=-\frac{a B x \sqrt{a+b x^2}}{8 b}+\frac{(4 A+3 B x) \left (a+b x^2\right )^{3/2}}{12 b}-\frac{\left (a^2 B\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 B x \sqrt{a+b x^2}}{8 b}+\frac{(4 A+3 B x) \left (a+b x^2\right )^{3/2}}{12 b}-\frac{a^2 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.140984, size = 86, normalized size = 1.08 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} \left (8 a A+3 a B x+8 A b x^2+6 b B x^3\right )-\frac{3 a^{3/2} B \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{24 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 75, normalized size = 0.9 \begin{align*}{\frac{Bx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Bax}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{B{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{A}{3\,b} \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 = 1.58638, size = 381, normalized size = 4.76 \begin{align*} \left [\frac{3 \, B a^{2} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (6 \, B b^{2} x^{3} + 8 \, A b^{2} x^{2} + 3 \, B a b x + 8 \, A a b\right )} \sqrt{b x^{2} + a}}{48 \, b^{2}}, \frac{3 \, B a^{2} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (6 \, B b^{2} x^{3} + 8 \, A b^{2} x^{2} + 3 \, B a b x + 8 \, A a b\right )} \sqrt{b x^{2} + a}}{24 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.37961, size = 124, normalized size = 1.55 \begin{align*} A \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + \frac{B a^{\frac{3}{2}} x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B \sqrt{a} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{B b x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17109, size = 92, normalized size = 1.15 \begin{align*} \frac{B a^{2} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{3}{2}}} + \frac{1}{24} \, \sqrt{b x^{2} + a}{\left ({\left (2 \,{\left (3 \, B x + 4 \, A\right )} x + \frac{3 \, B a}{b}\right )} x + \frac{8 \, A a}{b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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