Optimal. Leaf size=43 \[ \frac{A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}}+\frac{B \sqrt{a+c x^2}}{c} \]
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Rubi [A] time = 0.0122925, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {641, 217, 206} \[ \frac{A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}}+\frac{B \sqrt{a+c x^2}}{c} \]
Antiderivative was successfully verified.
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Rule 641
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B x}{\sqrt{a+c x^2}} \, dx &=\frac{B \sqrt{a+c x^2}}{c}+A \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=\frac{B \sqrt{a+c x^2}}{c}+A \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=\frac{B \sqrt{a+c x^2}}{c}+\frac{A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0362145, size = 46, normalized size = 1.07 \[ \frac{A \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{\sqrt{c}}+\frac{B \sqrt{a+c x^2}}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 37, normalized size = 0.9 \begin{align*}{\frac{B}{c}\sqrt{c{x}^{2}+a}}+{A\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \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 = 1.73052, size = 223, normalized size = 5.19 \begin{align*} \left [\frac{A \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \, \sqrt{c x^{2} + a} B}{2 \, c}, -\frac{A \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - \sqrt{c x^{2} + a} B}{c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.27704, size = 102, normalized size = 2.37 \begin{align*} A \left (\begin{cases} \frac{\sqrt{- \frac{a}{c}} \operatorname{asin}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c < 0 \\\frac{\sqrt{\frac{a}{c}} \operatorname{asinh}{\left (x \sqrt{\frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c > 0 \\\frac{\sqrt{- \frac{a}{c}} \operatorname{acosh}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{- a}} & \text{for}\: c > 0 \wedge a < 0 \end{cases}\right ) + B \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15765, size = 53, normalized size = 1.23 \begin{align*} -\frac{A \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2} + a} B}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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