Optimal. Leaf size=186 \[ \frac{d x \sqrt{d^2-e^2 x^2} \left (e (4 A e+B d)+C d^2\right )}{8 e^2}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (5 e (A e+B d)+2 C d^2\right )}{15 e^3}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (e (4 A e+B d)+C d^2\right )}{8 e^3}-\frac{x \left (d^2-e^2 x^2\right )^{3/2} (B e+C d)}{4 e^2}-\frac{C x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e} \]
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Rubi [A] time = 0.227334, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1815, 641, 195, 217, 203} \[ \frac{d x \sqrt{d^2-e^2 x^2} \left (e (4 A e+B d)+C d^2\right )}{8 e^2}-\frac{\left (d^2-e^2 x^2\right )^{3/2} \left (5 e (A e+B d)+2 C d^2\right )}{15 e^3}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (e (4 A e+B d)+C d^2\right )}{8 e^3}-\frac{x \left (d^2-e^2 x^2\right )^{3/2} (B e+C d)}{4 e^2}-\frac{C x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e} \]
Antiderivative was successfully verified.
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Rule 1815
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int (d+e x) \left (A+B x+C x^2\right ) \sqrt{d^2-e^2 x^2} \, dx &=-\frac{C x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac{\int \sqrt{d^2-e^2 x^2} \left (-5 A d e^2-e \left (2 C d^2+5 e (B d+A e)\right ) x-5 e^2 (C d+B e) x^2\right ) \, dx}{5 e^2}\\ &=-\frac{(C d+B e) x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{C x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{\int \left (5 d e^2 \left (C d^2+e (B d+4 A e)\right )+4 e^3 \left (2 C d^2+5 e (B d+A e)\right ) x\right ) \sqrt{d^2-e^2 x^2} \, dx}{20 e^4}\\ &=-\frac{\left (2 C d^2+5 e (B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 e^3}-\frac{(C d+B e) x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{C x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{\left (d \left (C d^2+e (B d+4 A e)\right )\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{4 e^2}\\ &=\frac{d \left (C d^2+e (B d+4 A e)\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{\left (2 C d^2+5 e (B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 e^3}-\frac{(C d+B e) x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{C x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{\left (d^3 \left (C d^2+e (B d+4 A e)\right )\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=\frac{d \left (C d^2+e (B d+4 A e)\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{\left (2 C d^2+5 e (B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 e^3}-\frac{(C d+B e) x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{C x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{\left (d^3 \left (C d^2+e (B d+4 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 )}{8 e^2}\\ &=\frac{d \left (C d^2+e (B d+4 A e)\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{\left (2 C d^2+5 e (B d+A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 e^3}-\frac{(C d+B e) x \left (d^2-e^2 x^2\right )^{3/2}}{4 e^2}-\frac{C x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}+\frac{d^3 \left (C d^2+e (B d+4 A e)\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{8 e^3}\\ \end{align*}
Mathematica [A] time = 0.308719, size = 190, normalized size = 1.02 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (5 e \left (4 A e \left (-2 d^2+3 d e x+2 e^2 x^2\right )+B \left (-3 d^2 e x-8 d^3+8 d e^2 x^2+6 e^3 x^3\right )\right )+C \left (-8 d^2 e^2 x^2-15 d^3 e x-16 d^4+30 d e^3 x^3+24 e^4 x^4\right )\right )+15 \sin ^{-1}\left (\frac{e x}{d}\right ) \left (d^2 e (4 A e+B d)+C d^4\right )\right )}{120 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 304, normalized size = 1.6 \begin{align*} -{\frac{C{x}^{2}}{5\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{2\,C{d}^{2}}{15\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{Bx}{4\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{Cdx}{4\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{2}xB}{8\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{C{d}^{3}x}{8\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{{d}^{4}B}{8\,e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{{d}^{5}C}{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{A}{3\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{Bd}{3\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{Adx}{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{A{d}^{3}}{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 = 1.78755, size = 304, normalized size = 1.63 \begin{align*} \frac{A d^{3} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} + \frac{1}{2} \, \sqrt{-e^{2} x^{2} + d^{2}} A d x - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} C x^{2}}{5 \, e} + \frac{{\left (C d + B e\right )} d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{2}} + \frac{\sqrt{-e^{2} x^{2} + d^{2}}{\left (C d + B e\right )} d^{2} x}{8 \, e^{2}} - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} C d^{2}}{15 \, e^{3}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} B d}{3 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} A}{3 \, e} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}{\left (C d + B e\right )} x}{4 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51104, size = 375, normalized size = 2.02 \begin{align*} -\frac{30 \,{\left (C d^{5} + B d^{4} e + 4 \, A d^{3} e^{2}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (24 \, C e^{4} x^{4} - 16 \, C d^{4} - 40 \, B d^{3} e - 40 \, A d^{2} e^{2} + 30 \,{\left (C d e^{3} + B e^{4}\right )} x^{3} - 8 \,{\left (C d^{2} e^{2} - 5 \, B d e^{3} - 5 \, A e^{4}\right )} x^{2} - 15 \,{\left (C d^{3} e + B d^{2} e^{2} - 4 \, A d e^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{120 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 11.4995, size = 675, normalized size = 3.63 \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 = 1.18175, size = 216, normalized size = 1.16 \begin{align*} \frac{1}{8} \,{\left (C d^{5} + B d^{4} e + 4 \, A d^{3} e^{2}\right )} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )} \mathrm{sgn}\left (d\right ) + \frac{1}{120} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (2 \,{\left (3 \,{\left (4 \, C x e + 5 \,{\left (C d e^{6} + B e^{7}\right )} e^{\left (-6\right )}\right )} x - 4 \,{\left (C d^{2} e^{5} - 5 \, B d e^{6} - 5 \, A e^{7}\right )} e^{\left (-6\right )}\right )} x - 15 \,{\left (C d^{3} e^{4} + B d^{2} e^{5} - 4 \, A d e^{6}\right )} e^{\left (-6\right )}\right )} x - 8 \,{\left (2 \, C d^{4} e^{3} + 5 \, B d^{3} e^{4} + 5 \, A d^{2} e^{5}\right )} e^{\left (-6\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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