Optimal. Leaf size=87 \[ \frac{\left (2 A e^2+C d^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3}-\frac{B \sqrt{d^2-e^2 x^2}}{e^2}-\frac{C x \sqrt{d^2-e^2 x^2}}{2 e^2} \]
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Rubi [A] time = 0.0510074, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1815, 641, 217, 203} \[ \frac{\left (2 A e^2+C d^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3}-\frac{B \sqrt{d^2-e^2 x^2}}{e^2}-\frac{C x \sqrt{d^2-e^2 x^2}}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 1815
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{C x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{\int \frac{-C d^2-2 A e^2-2 B e^2 x}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^2}\\ &=-\frac{B \sqrt{d^2-e^2 x^2}}{e^2}-\frac{C x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{\left (-C d^2-2 A e^2\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^2}\\ &=-\frac{B \sqrt{d^2-e^2 x^2}}{e^2}-\frac{C x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{\left (-C d^2-2 A e^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^2}\\ &=-\frac{B \sqrt{d^2-e^2 x^2}}{e^2}-\frac{C x \sqrt{d^2-e^2 x^2}}{2 e^2}+\frac{\left (C d^2+2 A e^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3}\\ \end{align*}
Mathematica [A] time = 0.0417679, size = 67, normalized size = 0.77 \[ \frac{\left (2 A e^2+C d^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-e (2 B+C x) \sqrt{d^2-e^2 x^2}}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.105, size = 108, normalized size = 1.2 \begin{align*} -{\frac{Cx}{2\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{C{d}^{2}}{2\,{e}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{B}{{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{A\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52561, size = 126, normalized size = 1.45 \begin{align*} \frac{A \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} + \frac{C d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{2}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} C x}{2 \, e^{2}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} B}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77747, size = 153, normalized size = 1.76 \begin{align*} -\frac{2 \,{\left (C d^{2} + 2 \, A e^{2}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + \sqrt{-e^{2} x^{2} + d^{2}}{\left (C e x + 2 \, B e\right )}}{2 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.22441, size = 264, normalized size = 3.03 \begin{align*} A \left (\begin{cases} \frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{asin}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge e^{2} > 0 \\\frac{\sqrt{- \frac{d^{2}}{e^{2}}} \operatorname{asinh}{\left (x \sqrt{- \frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge e^{2} < 0 \\\frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{acosh}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{- d^{2}}} & \text{for}\: d^{2} < 0 \wedge e^{2} < 0 \end{cases}\right ) + B \left (\begin{cases} \frac{x^{2}}{2 \sqrt{d^{2}}} & \text{for}\: e^{2} = 0 \\- \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{2}} & \text{otherwise} \end{cases}\right ) + C \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{i d x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}}{2 e^{2}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{d x}{2 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{3}}{2 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21307, size = 70, normalized size = 0.8 \begin{align*} \frac{1}{2} \,{\left (C d^{2} + 2 \, A e^{2}\right )} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{2} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (C x e^{\left (-2\right )} + 2 \, B e^{\left (-2\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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