Optimal. Leaf size=191 \[ -\frac{x \sqrt{d^2-e^2 x^2} \left (4 e (A e+2 B d)+7 C d^2\right )}{8 e^2}-\frac{d \sqrt{d^2-e^2 x^2} \left (e (6 A e+5 B d)+4 C d^2\right )}{3 e^3}+\frac{d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (12 A e^2+8 B d e+7 C d^2\right )}{8 e^3}-\frac{x^2 \sqrt{d^2-e^2 x^2} (B e+2 C d)}{3 e}-\frac{1}{4} C x^3 \sqrt{d^2-e^2 x^2} \]
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Rubi [A] time = 0.378195, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1815, 641, 217, 203} \[ -\frac{x \sqrt{d^2-e^2 x^2} \left (4 e (A e+2 B d)+7 C d^2\right )}{8 e^2}-\frac{d \sqrt{d^2-e^2 x^2} \left (e (6 A e+5 B d)+4 C d^2\right )}{3 e^3}+\frac{d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (12 A e^2+8 B d e+7 C d^2\right )}{8 e^3}-\frac{x^2 \sqrt{d^2-e^2 x^2} (B e+2 C d)}{3 e}-\frac{1}{4} C x^3 \sqrt{d^2-e^2 x^2} \]
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)^2 \left (A+B x+C x^2\right )}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{1}{4} C x^3 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{-4 A d^2 e^2-4 d e^2 (B d+2 A e) x-e^2 \left (7 C d^2+4 e (2 B d+A e)\right ) x^2-4 e^3 (2 C d+B e) x^3}{\sqrt{d^2-e^2 x^2}} \, dx}{4 e^2}\\ &=-\frac{(2 C d+B e) x^2 \sqrt{d^2-e^2 x^2}}{3 e}-\frac{1}{4} C x^3 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{12 A d^2 e^4+4 d e^3 \left (4 C d^2+e (5 B d+6 A e)\right ) x+3 e^4 \left (7 C d^2+4 e (2 B d+A e)\right ) x^2}{\sqrt{d^2-e^2 x^2}} \, dx}{12 e^4}\\ &=-\frac{\left (7 C d^2+4 e (2 B d+A e)\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{(2 C d+B e) x^2 \sqrt{d^2-e^2 x^2}}{3 e}-\frac{1}{4} C x^3 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{-3 d^2 e^4 \left (7 C d^2+8 B d e+12 A e^2\right )-8 d e^5 \left (4 C d^2+e (5 B d+6 A e)\right ) x}{\sqrt{d^2-e^2 x^2}} \, dx}{24 e^6}\\ &=-\frac{d \left (4 C d^2+e (5 B d+6 A e)\right ) \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{\left (7 C d^2+4 e (2 B d+A e)\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{(2 C d+B e) x^2 \sqrt{d^2-e^2 x^2}}{3 e}-\frac{1}{4} C x^3 \sqrt{d^2-e^2 x^2}+\frac{\left (d^2 \left (7 C d^2+8 B d e+12 A e^2\right )\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=-\frac{d \left (4 C d^2+e (5 B d+6 A e)\right ) \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{\left (7 C d^2+4 e (2 B d+A e)\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{(2 C d+B e) x^2 \sqrt{d^2-e^2 x^2}}{3 e}-\frac{1}{4} C x^3 \sqrt{d^2-e^2 x^2}+\frac{\left (d^2 \left (7 C d^2+8 B d e+12 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{d \left (4 C d^2+e (5 B d+6 A e)\right ) \sqrt{d^2-e^2 x^2}}{3 e^3}-\frac{\left (7 C d^2+4 e (2 B d+A e)\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{(2 C d+B e) x^2 \sqrt{d^2-e^2 x^2}}{3 e}-\frac{1}{4} C x^3 \sqrt{d^2-e^2 x^2}+\frac{d^2 \left (7 C d^2+8 B d e+12 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 = 0.183379, size = 139, normalized size = 0.73 \[ \frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (4 e (3 A e+2 B d)+7 C d^2\right )-\sqrt{d^2-e^2 x^2} \left (4 e \left (3 A e (4 d+e x)+2 B \left (5 d^2+3 d e x+e^2 x^2\right )\right )+C \left (21 d^2 e x+32 d^3+16 d e^2 x^2+6 e^3 x^3\right )\right )}{24 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 301, normalized size = 1.6 \begin{align*} -{\frac{C{x}^{3}}{4}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{7\,C{d}^{2}x}{8\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{7\,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{{x}^{2}B}{3}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{2\,Cd{x}^{2}}{3\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{5\,B{d}^{2}}{3\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{4\,{d}^{3}C}{3\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{Ax}{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{Bdx}{e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{3\,{d}^{2}A}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{B{d}^{3}}{e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-2\,{\frac{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}Ad}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51843, size = 390, normalized size = 2.04 \begin{align*} -\frac{1}{4} \, \sqrt{-e^{2} x^{2} + d^{2}} C x^{3} + \frac{A d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} + \frac{3 \, C d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{2}} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} C d^{2} x}{8 \, e^{2}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} B d^{2}}{e^{2}} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} A d}{e} - \frac{\sqrt{-e^{2} x^{2} + d^{2}}{\left (2 \, C d e + B e^{2}\right )} x^{2}}{3 \, e^{2}} + \frac{{\left (C d^{2} + 2 \, B d e + A e^{2}\right )} 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}}{\left (C d^{2} + 2 \, B d e + A e^{2}\right )} x}{2 \, e^{2}} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}}{\left (2 \, C d e + B e^{2}\right )} d^{2}}{3 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38753, size = 319, normalized size = 1.67 \begin{align*} -\frac{6 \,{\left (7 \, C d^{4} + 8 \, B d^{3} e + 12 \, 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} + 32 \, C d^{3} + 40 \, B d^{2} e + 48 \, A d e^{2} + 8 \,{\left (2 \, C d e^{2} + B e^{3}\right )} x^{2} + 3 \,{\left (7 \, C d^{2} e + 8 \, B d e^{2} + 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 [A] time = 14.1762, size = 898, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17706, size = 177, normalized size = 0.93 \begin{align*} \frac{1}{8} \,{\left (7 \, C d^{4} + 8 \, B d^{3} e + 12 \, 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} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (2 \,{\left (3 \, C x + 4 \,{\left (2 \, C d e^{4} + B e^{5}\right )} e^{\left (-5\right )}\right )} x + 3 \,{\left (7 \, C d^{2} e^{3} + 8 \, B d e^{4} + 4 \, A e^{5}\right )} e^{\left (-5\right )}\right )} x + 8 \,{\left (4 \, C d^{3} e^{2} + 5 \, B d^{2} e^{3} + 6 \, A d e^{4}\right )} e^{\left (-5\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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