Optimal. Leaf size=56 \[ \frac {\sqrt {a+b x^2} (2 A+B x)}{2 b}-\frac {a B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {780, 217, 206} \[ \frac {\sqrt {a+b x^2} (2 A+B x)}{2 b}-\frac {a B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 780
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{\sqrt {a+b x^2}} \, dx &=\frac {(2 A+B x) \sqrt {a+b x^2}}{2 b}-\frac {(a B) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b}\\ &=\frac {(2 A+B x) \sqrt {a+b x^2}}{2 b}-\frac {(a B) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b}\\ &=\frac {(2 A+B x) \sqrt {a+b x^2}}{2 b}-\frac {a B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 1.02 \[ \frac {\sqrt {b} \sqrt {a+b x^2} (2 A+B x)-a B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 109, normalized size = 1.95 \[ \left [\frac {B a \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (B b x + 2 \, A b\right )} \sqrt {b x^{2} + a}}{4 \, b^{2}}, \frac {B a \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (B b x + 2 \, A b\right )} \sqrt {b x^{2} + a}}{2 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 50, normalized size = 0.89 \[ \frac {1}{2} \, \sqrt {b x^{2} + a} {\left (\frac {B x}{b} + \frac {2 \, A}{b}\right )} + \frac {B a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 55, normalized size = 0.98 \[ -\frac {B a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}+\frac {\sqrt {b \,x^{2}+a}\, B x}{2 b}+\frac {\sqrt {b \,x^{2}+a}\, A}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 47, normalized size = 0.84 \[ \frac {\sqrt {b x^{2} + a} B x}{2 \, b} - \frac {B a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} + \frac {\sqrt {b x^{2} + a} A}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 82, normalized size = 1.46 \[ \left \{\begin {array}{cl} \frac {2\,B\,x^3+3\,A\,x^2}{6\,\sqrt {a}} & \text {\ if\ \ }b=0\\ \frac {A\,\sqrt {b\,x^2+a}}{b}-\frac {B\,a\,\ln \left (2\,\sqrt {b}\,x+2\,\sqrt {b\,x^2+a}\right )}{2\,b^{3/2}}+\frac {B\,x\,\sqrt {b\,x^2+a}}{2\,b} & \text {\ if\ \ }b\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.26, size = 70, normalized size = 1.25 \[ A \left (\begin {cases} \frac {x^{2}}{2 \sqrt {a}} & \text {for}\: b = 0 \\\frac {\sqrt {a + b x^{2}}}{b} & \text {otherwise} \end {cases}\right ) + \frac {B \sqrt {a} x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {B a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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