Optimal. Leaf size=236 \[ -\frac{x^2 \sqrt{d^2-e^2 x^2} \left (5 e (A e+3 B d)+19 C d^2\right )}{15 e}-\frac{d x \sqrt{d^2-e^2 x^2} \left (12 A e^2+15 B d e+13 C d^2\right )}{8 e^2}-\frac{d^2 \sqrt{d^2-e^2 x^2} \left (55 A e^2+45 B d e+38 C d^2\right )}{15 e^3}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (20 A e^2+15 B d e+13 C d^2\right )}{8 e^3}-\frac{1}{4} x^3 \sqrt{d^2-e^2 x^2} (B e+3 C d)-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2} \]
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Rubi [A] time = 0.656543, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1815, 641, 217, 203} \[ -\frac{x^2 \sqrt{d^2-e^2 x^2} \left (5 e (A e+3 B d)+19 C d^2\right )}{15 e}-\frac{d x \sqrt{d^2-e^2 x^2} \left (12 A e^2+15 B d e+13 C d^2\right )}{8 e^2}-\frac{d^2 \sqrt{d^2-e^2 x^2} \left (55 A e^2+45 B d e+38 C d^2\right )}{15 e^3}+\frac{d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (20 A e^2+15 B d e+13 C d^2\right )}{8 e^3}-\frac{1}{4} x^3 \sqrt{d^2-e^2 x^2} (B e+3 C d)-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2} \]
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)^3 \left (A+B x+C x^2\right )}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{-5 A d^3 e^2-5 d^2 e^2 (B d+3 A e) x-5 d e^2 \left (C d^2+3 e (B d+A e)\right ) x^2-e^3 \left (19 C d^2+5 e (3 B d+A e)\right ) x^3-5 e^4 (3 C d+B e) x^4}{\sqrt{d^2-e^2 x^2}} \, dx}{5 e^2}\\ &=-\frac{1}{4} (3 C d+B e) x^3 \sqrt{d^2-e^2 x^2}-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{20 A d^3 e^4+20 d^2 e^4 (B d+3 A e) x+5 d e^4 \left (13 C d^2+15 B d e+12 A e^2\right ) x^2+4 e^5 \left (19 C d^2+5 e (3 B d+A e)\right ) x^3}{\sqrt{d^2-e^2 x^2}} \, dx}{20 e^4}\\ &=-\frac{\left (19 C d^2+5 e (3 B d+A e)\right ) x^2 \sqrt{d^2-e^2 x^2}}{15 e}-\frac{1}{4} (3 C d+B e) x^3 \sqrt{d^2-e^2 x^2}-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2}-\frac{\int \frac{-60 A d^3 e^6-4 d^2 e^5 \left (38 C d^2+45 B d e+55 A e^2\right ) x-15 d e^6 \left (13 C d^2+15 B d e+12 A e^2\right ) x^2}{\sqrt{d^2-e^2 x^2}} \, dx}{60 e^6}\\ &=-\frac{d \left (13 C d^2+15 B d e+12 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{\left (19 C d^2+5 e (3 B d+A e)\right ) x^2 \sqrt{d^2-e^2 x^2}}{15 e}-\frac{1}{4} (3 C d+B e) x^3 \sqrt{d^2-e^2 x^2}-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2}+\frac{\int \frac{15 d^3 e^6 \left (13 C d^2+15 B d e+20 A e^2\right )+8 d^2 e^7 \left (38 C d^2+45 B d e+55 A e^2\right ) x}{\sqrt{d^2-e^2 x^2}} \, dx}{120 e^8}\\ &=-\frac{d^2 \left (38 C d^2+45 B d e+55 A e^2\right ) \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{d \left (13 C d^2+15 B d e+12 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{\left (19 C d^2+5 e (3 B d+A e)\right ) x^2 \sqrt{d^2-e^2 x^2}}{15 e}-\frac{1}{4} (3 C d+B e) x^3 \sqrt{d^2-e^2 x^2}-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2}+\frac{\left (d^3 \left (13 C d^2+15 B d e+20 A e^2\right )\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^2}\\ &=-\frac{d^2 \left (38 C d^2+45 B d e+55 A e^2\right ) \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{d \left (13 C d^2+15 B d e+12 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{\left (19 C d^2+5 e (3 B d+A e)\right ) x^2 \sqrt{d^2-e^2 x^2}}{15 e}-\frac{1}{4} (3 C d+B e) x^3 \sqrt{d^2-e^2 x^2}-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2}+\frac{\left (d^3 \left (13 C d^2+15 B d e+20 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^2 \left (38 C d^2+45 B d e+55 A e^2\right ) \sqrt{d^2-e^2 x^2}}{15 e^3}-\frac{d \left (13 C d^2+15 B d e+12 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{8 e^2}-\frac{\left (19 C d^2+5 e (3 B d+A e)\right ) x^2 \sqrt{d^2-e^2 x^2}}{15 e}-\frac{1}{4} (3 C d+B e) x^3 \sqrt{d^2-e^2 x^2}-\frac{1}{5} C e x^4 \sqrt{d^2-e^2 x^2}+\frac{d^3 \left (13 C d^2+15 B d e+20 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 = 0.453851, size = 174, normalized size = 0.74 \[ \frac{15 d^3 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (5 e (4 A e+3 B d)+13 C d^2\right )-\sqrt{d^2-e^2 x^2} \left (5 e \left (4 A e \left (22 d^2+9 d e x+2 e^2 x^2\right )+3 B \left (15 d^2 e x+24 d^3+8 d e^2 x^2+2 e^3 x^3\right )\right )+C \left (152 d^2 e^2 x^2+195 d^3 e x+304 d^4+90 d e^3 x^3+24 e^4 x^4\right )\right )}{120 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 374, normalized size = 1.6 \begin{align*} -{\frac{Ce{x}^{4}}{5}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{19\,C{d}^{2}{x}^{2}}{15\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{38\,C{d}^{4}}{15\,{e}^{3}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{{x}^{3}eB}{4}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{3\,d{x}^{3}C}{4}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{15\,{d}^{2}xB}{8\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{13\,C{d}^{3}x}{8\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{15\,{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{13\,{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{e{x}^{2}A}{3}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{x}^{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}dB-{\frac{11\,{d}^{2}A}{3\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-3\,{\frac{{d}^{3}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}B}{{e}^{2}}}-{\frac{3\,Adx}{2}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{5\,{d}^{3}A}{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 [B] time = 1.5055, size = 575, normalized size = 2.44 \begin{align*} -\frac{1}{5} \, \sqrt{-e^{2} x^{2} + d^{2}} C e x^{4} - \frac{4 \, \sqrt{-e^{2} x^{2} + d^{2}} C d^{2} x^{2}}{15 \, e} + \frac{A d^{3} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} C d^{4}}{15 \, e^{3}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} B d^{3}}{e^{2}} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} A d^{2}}{e} - \frac{{\left (3 \, C d e^{2} + B e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}} x^{3}}{4 \, e^{2}} - \frac{{\left (3 \, C d^{2} e + 3 \, B d e^{2} + A e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}} x^{2}}{3 \, e^{2}} + \frac{3 \,{\left (3 \, C d e^{2} + B e^{3}\right )} d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{4}} + \frac{{\left (C d^{3} + 3 \, B d^{2} e + 3 \, A d e^{2}\right )} d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{2}} - \frac{3 \,{\left (3 \, C d e^{2} + B e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}} d^{2} x}{8 \, e^{4}} - \frac{{\left (C d^{3} + 3 \, B d^{2} e + 3 \, A d e^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}} x}{2 \, e^{2}} - \frac{2 \,{\left (3 \, C d^{2} e + 3 \, B d e^{2} + A e^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}} d^{2}}{3 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35253, size = 406, normalized size = 1.72 \begin{align*} -\frac{30 \,{\left (13 \, C d^{5} + 15 \, B d^{4} e + 20 \, 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} + 304 \, C d^{4} + 360 \, B d^{3} e + 440 \, A d^{2} e^{2} + 30 \,{\left (3 \, C d e^{3} + B e^{4}\right )} x^{3} + 8 \,{\left (19 \, C d^{2} e^{2} + 15 \, B d e^{3} + 5 \, A e^{4}\right )} x^{2} + 15 \,{\left (13 \, C d^{3} e + 15 \, B d^{2} e^{2} + 12 \, 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 [A] time = 20.234, size = 1277, normalized size = 5.41 \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 = 1.23252, size = 224, normalized size = 0.95 \begin{align*} \frac{1}{8} \,{\left (13 \, C d^{5} + 15 \, B d^{4} e + 20 \, 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 (3 \, C d e^{6} + B e^{7}\right )} e^{\left (-6\right )}\right )} x + 4 \,{\left (19 \, C d^{2} e^{5} + 15 \, B d e^{6} + 5 \, A e^{7}\right )} e^{\left (-6\right )}\right )} x + 15 \,{\left (13 \, C d^{3} e^{4} + 15 \, B d^{2} e^{5} + 12 \, A d e^{6}\right )} e^{\left (-6\right )}\right )} x + 8 \,{\left (38 \, C d^{4} e^{3} + 45 \, B d^{3} e^{4} + 55 \, 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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