Optimal. Leaf size=56 \[ \frac {\sqrt {a+c x^2} (2 A+B x)}{2 c}-\frac {a B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}} \]
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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} \begin {gather*} \frac {\sqrt {a+c x^2} (2 A+B x)}{2 c}-\frac {a B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}} \end {gather*}
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+c x^2}} \, dx &=\frac {(2 A+B x) \sqrt {a+c x^2}}{2 c}-\frac {(a B) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c}\\ &=\frac {(2 A+B x) \sqrt {a+c x^2}}{2 c}-\frac {(a B) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c}\\ &=\frac {(2 A+B x) \sqrt {a+c x^2}}{2 c}-\frac {a B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 57, normalized size = 1.02 \begin {gather*} \frac {\sqrt {c} \sqrt {a+c x^2} (2 A+B x)-a B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 58, normalized size = 1.04 \begin {gather*} \frac {\sqrt {a+c x^2} (2 A+B x)}{2 c}+\frac {a B \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right )}{2 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 109, normalized size = 1.95 \begin {gather*} \left [\frac {B a \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (B c x + 2 \, A c\right )} \sqrt {c x^{2} + a}}{4 \, c^{2}}, \frac {B a \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (B c x + 2 \, A c\right )} \sqrt {c x^{2} + a}}{2 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 50, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} + a} {\left (\frac {B x}{c} + \frac {2 \, A}{c}\right )} + \frac {B a \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 55, normalized size = 0.98 \begin {gather*} -\frac {B a \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}+\frac {\sqrt {c \,x^{2}+a}\, B x}{2 c}+\frac {\sqrt {c \,x^{2}+a}\, A}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 47, normalized size = 0.84 \begin {gather*} \frac {\sqrt {c x^{2} + a} B x}{2 \, c} - \frac {B a \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + a} A}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 82, normalized size = 1.46 \begin {gather*} \left \{\begin {array}{cl} \frac {2\,B\,x^3+3\,A\,x^2}{6\,\sqrt {a}} & \text {\ if\ \ }c=0\\ \frac {A\,\sqrt {c\,x^2+a}}{c}-\frac {B\,a\,\ln \left (2\,\sqrt {c}\,x+2\,\sqrt {c\,x^2+a}\right )}{2\,c^{3/2}}+\frac {B\,x\,\sqrt {c\,x^2+a}}{2\,c} & \text {\ if\ \ }c\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.20, size = 70, normalized size = 1.25 \begin {gather*} A \left (\begin {cases} \frac {x^{2}}{2 \sqrt {a}} & \text {for}\: c = 0 \\\frac {\sqrt {a + c x^{2}}}{c} & \text {otherwise} \end {cases}\right ) + \frac {B \sqrt {a} x \sqrt {1 + \frac {c x^{2}}{a}}}{2 c} - \frac {B a \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{2 c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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