Optimal. Leaf size=48 \[ -a \tanh ^{-1}\left (\sqrt{1-d^2 x^2}\right )+\frac{b \sin ^{-1}(d x)}{d}-\frac{c \sqrt{1-d^2 x^2}}{d^2} \]
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Rubi [A] time = 0.183471, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {1609, 1809, 844, 216, 266, 63, 208} \[ -a \tanh ^{-1}\left (\sqrt{1-d^2 x^2}\right )+\frac{b \sin ^{-1}(d x)}{d}-\frac{c \sqrt{1-d^2 x^2}}{d^2} \]
Antiderivative was successfully verified.
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Rule 1609
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{x \sqrt{1-d x} \sqrt{1+d x}} \, dx &=\int \frac{a+b x+c x^2}{x \sqrt{1-d^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-d^2 x^2}}{d^2}-\frac{\int \frac{-a d^2-b d^2 x}{x \sqrt{1-d^2 x^2}} \, dx}{d^2}\\ &=-\frac{c \sqrt{1-d^2 x^2}}{d^2}+a \int \frac{1}{x \sqrt{1-d^2 x^2}} \, dx+b \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx\\ &=-\frac{c \sqrt{1-d^2 x^2}}{d^2}+\frac{b \sin ^{-1}(d x)}{d}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-d^2 x}} \, dx,x,x^2\right )\\ &=-\frac{c \sqrt{1-d^2 x^2}}{d^2}+\frac{b \sin ^{-1}(d x)}{d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\frac{1}{d^2}-\frac{x^2}{d^2}} \, dx,x,\sqrt{1-d^2 x^2}\right )}{d^2}\\ &=-\frac{c \sqrt{1-d^2 x^2}}{d^2}+\frac{b \sin ^{-1}(d x)}{d}-a \tanh ^{-1}\left (\sqrt{1-d^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0516867, size = 48, normalized size = 1. \[ -a \tanh ^{-1}\left (\sqrt{1-d^2 x^2}\right )+\frac{b \sin ^{-1}(d x)}{d}-\frac{c \sqrt{1-d^2 x^2}}{d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0., size = 96, normalized size = 2. \begin{align*}{\frac{{\it csgn} \left ( d \right ) }{{d}^{2}} \left ( -{\it csgn} \left ( d \right ){\it Artanh} \left ({\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) a{d}^{2}-{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}c+\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{- \left ( dx+1 \right ) \left ( dx-1 \right ) }}}} \right ) bd \right ) \sqrt{-dx+1}\sqrt{dx+1}{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 4.22597, size = 89, normalized size = 1.85 \begin{align*} -a \log \left (\frac{2 \, \sqrt{-d^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{b \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{\sqrt{-d^{2} x^{2} + 1} c}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18311, size = 196, normalized size = 4.08 \begin{align*} \frac{a d^{2} \log \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{x}\right ) - 2 \, b d \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right ) - \sqrt{d x + 1} \sqrt{-d x + 1} c}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 28.2392, size = 245, normalized size = 5.1 \begin{align*} \frac{i a{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{a{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i b{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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