Optimal. Leaf size=125 \[ \frac{1}{8} x \sqrt{d^2-e^2 x^2} \left (4 A+\frac{C d^2}{e^2}\right )+\frac{d^2 \left (4 A e^2+C d^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2} \]
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Rubi [A] time = 0.068973, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1815, 641, 195, 217, 203} \[ \frac{1}{8} x \sqrt{d^2-e^2 x^2} \left (4 A+\frac{C d^2}{e^2}\right )+\frac{d^2 \left (4 A e^2+C d^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2} \]
Antiderivative was successfully verified.
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Rule 1815
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \left (A+B x+C x^2\right ) \sqrt{d^2-e^2 x^2} \, dx &=-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{\int \left (-C d^2-4 A e^2-4 B e^2 x\right ) \sqrt{d^2-e^2 x^2} \, dx}{4 e^2}\\ &=-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{\left (-C d^2-4 A e^2\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{4 e^2}\\ &=\frac{\left (C d^2+4 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{\left (d^2 \left (-C d^2-4 A e^2\right )\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=\frac{\left (C d^2+4 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{\left (d^2 \left (-C d^2-4 A e^2\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 )}{8 e^2}\\ &=\frac{\left (C d^2+4 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{B \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}-\frac{C x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}+\frac{d^2 \left (C d^2+4 A e^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}\\ \end{align*}
Mathematica [A] time = 0.143962, size = 121, normalized size = 0.97 \[ \frac{\sqrt{d^2-e^2 x^2} \left (e \sqrt{1-\frac{e^2 x^2}{d^2}} \left (12 A e^2 x-8 B d^2+8 B e^2 x^2-3 C d^2 x+6 C e^2 x^3\right )+3 \left (4 A d e^2+C d^3\right ) \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{24 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 154, normalized size = 1.2 \begin{align*} -{\frac{Cx}{4\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{C{d}^{2}x}{8\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{C{d}^{4}}{8\,{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}{3\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{Ax}{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{A{d}^{2}}{2}\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 = 1.52992, size = 188, normalized size = 1.5 \begin{align*} \frac{A d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} + \frac{C d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{2}} + \frac{1}{2} \, \sqrt{-e^{2} x^{2} + d^{2}} A x + \frac{\sqrt{-e^{2} x^{2} + d^{2}} C d^{2} x}{8 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} C x}{4 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} B}{3 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0362, size = 227, normalized size = 1.82 \begin{align*} -\frac{6 \,{\left (C d^{4} + 4 \, A d^{2} e^{2}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (6 \, C e^{3} x^{3} + 8 \, B e^{3} x^{2} - 8 \, B d^{2} e - 3 \,{\left (C d^{2} e - 4 \, A e^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.47671, size = 347, normalized size = 2.78 \begin{align*} A \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{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} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) + B \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) + C \left (\begin{cases} - \frac{i d^{4} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{8 e^{3}} + \frac{i d^{3} x}{8 e^{2} \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} - \frac{3 i d x^{3}}{8 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{5}}{4 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \frac{\left |{e^{2} x^{2}}\right |}{\left |{d^{2}}\right |} > 1 \\\frac{d^{4} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{8 e^{3}} - \frac{d^{3} x}{8 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{3 d x^{3}}{8 \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - \frac{e^{2} x^{5}}{4 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 = 1.14138, size = 115, normalized size = 0.92 \begin{align*} \frac{1}{8} \,{\left (C d^{4} + 4 \, A d^{2} e^{2}\right )} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )} \mathrm{sgn}\left (d\right ) - \frac{1}{24} \,{\left (8 \, B d^{2} e^{\left (-2\right )} -{\left (2 \,{\left (3 \, C x + 4 \, B\right )} x - 3 \,{\left (C d^{2} e^{2} - 4 \, A e^{4}\right )} e^{\left (-4\right )}\right )} x\right )} \sqrt{-x^{2} e^{2} + d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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