Optimal. Leaf size=63 \[ \frac{\left (2 a d^2+c\right ) \sin ^{-1}(d x)}{2 d^3}-\frac{b \sqrt{1-d^2 x^2}}{d^2}-\frac{c x \sqrt{1-d^2 x^2}}{2 d^2} \]
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Rubi [A] time = 0.060904, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {899, 1815, 641, 216} \[ \frac{\left (2 a d^2+c\right ) \sin ^{-1}(d x)}{2 d^3}-\frac{b \sqrt{1-d^2 x^2}}{d^2}-\frac{c x \sqrt{1-d^2 x^2}}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 899
Rule 1815
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{a+b x+c x^2}{\sqrt{1-d x} \sqrt{1+d x}} \, dx &=\int \frac{a+b x+c x^2}{\sqrt{1-d^2 x^2}} \, dx\\ &=-\frac{c x \sqrt{1-d^2 x^2}}{2 d^2}-\frac{\int \frac{-c-2 a d^2-2 b d^2 x}{\sqrt{1-d^2 x^2}} \, dx}{2 d^2}\\ &=-\frac{b \sqrt{1-d^2 x^2}}{d^2}-\frac{c x \sqrt{1-d^2 x^2}}{2 d^2}-\frac{\left (-c-2 a d^2\right ) \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx}{2 d^2}\\ &=-\frac{b \sqrt{1-d^2 x^2}}{d^2}-\frac{c x \sqrt{1-d^2 x^2}}{2 d^2}+\frac{\left (c+2 a d^2\right ) \sin ^{-1}(d x)}{2 d^3}\\ \end{align*}
Mathematica [A] time = 0.0322668, size = 45, normalized size = 0.71 \[ \frac{\left (2 a d^2+c\right ) \sin ^{-1}(d x)-d \sqrt{1-d^2 x^2} (2 b+c x)}{2 d^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0., size = 117, normalized size = 1.9 \begin{align*} -{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{3}}\sqrt{-dx+1}\sqrt{dx+1} \left ({\it csgn} \left ( d \right ) d\sqrt{-{d}^{2}{x}^{2}+1}xc-2\,\arctan \left ({\frac{{\it csgn} \left ( d \right ) dx}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) a{d}^{2}+2\,{\it csgn} \left ( d \right ) d\sqrt{-{d}^{2}{x}^{2}+1}b-\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \right ) c \right ){\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 = 2.39676, size = 105, normalized size = 1.67 \begin{align*} \frac{a \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{\sqrt{-d^{2} x^{2} + 1} c x}{2 \, d^{2}} - \frac{\sqrt{-d^{2} x^{2} + 1} b}{d^{2}} + \frac{c \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04285, size = 167, normalized size = 2.65 \begin{align*} -\frac{{\left (c d x + 2 \, b d\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 2 \,{\left (2 \, a d^{2} + c\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{2 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 20.8624, size = 282, normalized size = 4.48 \begin{align*} - \frac{i a{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{a{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 b{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{b{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}} - \frac{i c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} + \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.86588, size = 97, normalized size = 1.54 \begin{align*} -\frac{{\left ({\left (d x + 1\right )} c d^{4} + 2 \, b d^{5} - c d^{4}\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 2 \,{\left (2 \, a d^{6} + c d^{4}\right )} \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{d x + 1}\right )}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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