Optimal. Leaf size=79 \[ \frac{1}{2} \sqrt{a+c x^2} (2 A+B x)-\sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{a B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}} \]
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Rubi [A] time = 0.0635644, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {815, 844, 217, 206, 266, 63, 208} \[ \frac{1}{2} \sqrt{a+c x^2} (2 A+B x)-\sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+\frac{a B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 815
Rule 844
Rule 217
Rule 206
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a+c x^2}}{x} \, dx &=\frac{1}{2} (2 A+B x) \sqrt{a+c x^2}+\frac{\int \frac{2 a A c+a B c x}{x \sqrt{a+c x^2}} \, dx}{2 c}\\ &=\frac{1}{2} (2 A+B x) \sqrt{a+c x^2}+(a A) \int \frac{1}{x \sqrt{a+c x^2}} \, dx+\frac{1}{2} (a B) \int \frac{1}{\sqrt{a+c x^2}} \, dx\\ &=\frac{1}{2} (2 A+B x) \sqrt{a+c x^2}+\frac{1}{2} (a A) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )+\frac{1}{2} (a B) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )\\ &=\frac{1}{2} (2 A+B x) \sqrt{a+c x^2}+\frac{a B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}}+\frac{(a A) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{c}\\ &=\frac{1}{2} (2 A+B x) \sqrt{a+c x^2}+\frac{a B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c}}-\sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.216522, size = 100, normalized size = 1.27 \[ \frac{1}{2} \left (\frac{a^{3/2} B \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{c} \sqrt{a+c x^2}}+\sqrt{a+c x^2} (2 A+B x)-2 \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 78, normalized size = 1. \begin{align*}{\frac{Bx}{2}\sqrt{c{x}^{2}+a}}+{\frac{aB}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-A\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) +A\sqrt{c{x}^{2}+a} \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 = 1.76741, size = 859, normalized size = 10.87 \begin{align*} \left [\frac{B a \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \, A \sqrt{a} c \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (B c x + 2 \, A c\right )} \sqrt{c x^{2} + a}}{4 \, c}, -\frac{B a \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - A \sqrt{a} c \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) -{\left (B c x + 2 \, A c\right )} \sqrt{c x^{2} + a}}{2 \, c}, \frac{4 \, A \sqrt{-a} c \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) + 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}, -\frac{B a \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) - 2 \, A \sqrt{-a} c \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) -{\left (B c x + 2 \, A c\right )} \sqrt{c x^{2} + a}}{2 \, c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.96154, size = 107, normalized size = 1.35 \begin{align*} - A \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )} + \frac{A a}{\sqrt{c} x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{A \sqrt{c} x}{\sqrt{\frac{a}{c x^{2}} + 1}} + \frac{B \sqrt{a} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17688, size = 105, normalized size = 1.33 \begin{align*} \frac{2 \, A a \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{B a \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, \sqrt{c}} + \frac{1}{2} \, \sqrt{c x^{2} + a}{\left (B x + 2 \, A\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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