Optimal. Leaf size=143 \[ -\frac{\sqrt{d^2-e^2 x^2} \left (3 e (A e+B d)+2 C d^2\right )}{3 e^3}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (e (2 A e+B d)+C d^2\right )}{2 e^3}-\frac{x \sqrt{d^2-e^2 x^2} (B e+C d)}{2 e^2}-\frac{C x^2 \sqrt{d^2-e^2 x^2}}{3 e} \]
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Rubi [A] time = 0.198975, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1815, 641, 217, 203} \[ -\frac{\sqrt{d^2-e^2 x^2} \left (3 e (A e+B d)+2 C d^2\right )}{3 e^3}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (e (2 A e+B d)+C d^2\right )}{2 e^3}-\frac{x \sqrt{d^2-e^2 x^2} (B e+C d)}{2 e^2}-\frac{C x^2 \sqrt{d^2-e^2 x^2}}{3 e} \]
Antiderivative was successfully verified.
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Rule 1815
Rule 641
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{(d+e x) \left (A+B x+C x^2\right )}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{C x^2 \sqrt{d^2-e^2 x^2}}{3 e}-\frac{\int \frac{-3 A d e^2-e \left (2 C d^2+3 e (B d+A e)\right ) x-3 e^2 (C d+B e) x^2}{\sqrt{d^2-e^2 x^2}} \, dx}{3 e^2}\\ &=-\frac{(C d+B e) x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{C x^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{\int \frac{3 d e^2 \left (C d^2+e (B d+2 A e)\right )+2 e^3 \left (2 C d^2+3 e (B d+A e)\right ) x}{\sqrt{d^2-e^2 x^2}} \, dx}{6 e^4}\\ &=-\frac{\left (2 C d^2+3 e (B d+A e)\right ) \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{(C d+B e) x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{C x^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{\left (d \left (C d^2+e (B d+2 A e)\right )\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 e^2}\\ &=-\frac{\left (2 C d^2+3 e (B d+A e)\right ) \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{(C d+B e) x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{C x^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{\left (d \left (C d^2+e (B d+2 A e)\right )\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{\left (2 C d^2+3 e (B d+A e)\right ) \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{(C d+B e) x \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{C x^2 \sqrt{d^2-e^2 x^2}}{3 e}+\frac{d \left (C d^2+e (B d+2 A e)\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3}\\ \end{align*}
Mathematica [A] time = 0.115612, size = 103, normalized size = 0.72 \[ \frac{3 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (e (2 A e+B d)+C d^2\right )-\sqrt{d^2-e^2 x^2} \left (3 e (2 A e+2 B d+B e x)+C \left (4 d^2+3 d e x+2 e^2 x^2\right )\right )}{6 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 234, normalized size = 1.6 \begin{align*} -{\frac{C{x}^{2}}{3\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{2\,C{d}^{2}}{3\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{Bx}{2\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{Cdx}{2\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{B{d}^{2}}{2\,e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{C{d}^{3}}{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{A}{e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{Bd}{{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{Ad\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 = 1.55194, size = 234, normalized size = 1.64 \begin{align*} -\frac{\sqrt{-e^{2} x^{2} + d^{2}} C x^{2}}{3 \, e} + \frac{A d \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} + \frac{{\left (C d + B e\right )} d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{2}} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} C d^{2}}{3 \, e^{3}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} B d}{e^{2}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} A}{e} - \frac{\sqrt{-e^{2} x^{2} + d^{2}}{\left (C d + B e\right )} x}{2 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76141, size = 236, normalized size = 1.65 \begin{align*} -\frac{6 \,{\left (C d^{3} + B d^{2} e + 2 \, A d e^{2}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (2 \, C e^{2} x^{2} + 4 \, C d^{2} + 6 \, B d e + 6 \, A e^{2} + 3 \,{\left (C d e + B e^{2}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.5767, size = 488, normalized size = 3.41 \begin{align*} A d \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 ) + A e \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 ) + B d \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 ) + B e \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 ) + C d \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 ) + C e \left (\begin{cases} - \frac{2 d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{4}} - \frac{x^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 e^{2}} & \text{for}\: e \neq 0 \\\frac{x^{4}}{4 \sqrt{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 = 1.19084, size = 131, normalized size = 0.92 \begin{align*} \frac{1}{2} \,{\left (C d^{3} + B d^{2} e + 2 \, A d e^{2}\right )} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{6} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (2 \, C x e^{\left (-1\right )} + 3 \,{\left (C d e^{3} + B e^{4}\right )} e^{\left (-5\right )}\right )} x + 2 \,{\left (2 \, C d^{2} e^{2} + 3 \, B d e^{3} + 3 \, A e^{4}\right )} e^{\left (-5\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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