Optimal. Leaf size=236 \[ \frac{d^2 x \sqrt{d^2-e^2 x^2} \left (10 A e^2+4 B d e+3 C d^2\right )}{16 e^2}-\frac{d \left (d^2-e^2 x^2\right )^{3/2} \left (e (10 A e+7 B d)+4 C d^2\right )}{15 e^3}-\frac{x \left (d^2-e^2 x^2\right )^{3/2} \left (2 e (A e+2 B d)+3 C d^2\right )}{8 e^2}+\frac{d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (10 A e^2+4 B d e+3 C d^2\right )}{16 e^3}-\frac{x^2 \left (d^2-e^2 x^2\right )^{3/2} (B e+2 C d)}{5 e}-\frac{1}{6} C x^3 \left (d^2-e^2 x^2\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.469868, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {1815, 641, 195, 217, 203} \[ \frac{d^2 x \sqrt{d^2-e^2 x^2} \left (10 A e^2+4 B d e+3 C d^2\right )}{16 e^2}-\frac{d \left (d^2-e^2 x^2\right )^{3/2} \left (e (10 A e+7 B d)+4 C d^2\right )}{15 e^3}-\frac{x \left (d^2-e^2 x^2\right )^{3/2} \left (2 e (A e+2 B d)+3 C d^2\right )}{8 e^2}+\frac{d^4 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (10 A e^2+4 B d e+3 C d^2\right )}{16 e^3}-\frac{x^2 \left (d^2-e^2 x^2\right )^{3/2} (B e+2 C d)}{5 e}-\frac{1}{6} C x^3 \left (d^2-e^2 x^2\right )^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1815
Rule 641
Rule 195
Rule 217
Rule 203
Rubi steps
\begin{align*} \int (d+e x)^2 \left (A+B x+C x^2\right ) \sqrt{d^2-e^2 x^2} \, dx &=-\frac{1}{6} C x^3 \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int \sqrt{d^2-e^2 x^2} \left (-6 A d^2 e^2-6 d e^2 (B d+2 A e) x-3 e^2 \left (3 C d^2+2 e (2 B d+A e)\right ) x^2-6 e^3 (2 C d+B e) x^3\right ) \, dx}{6 e^2}\\ &=-\frac{(2 C d+B e) x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac{1}{6} C x^3 \left (d^2-e^2 x^2\right )^{3/2}+\frac{\int \sqrt{d^2-e^2 x^2} \left (30 A d^2 e^4+6 d e^3 \left (4 C d^2+e (7 B d+10 A e)\right ) x+15 e^4 \left (3 C d^2+2 e (2 B d+A e)\right ) x^2\right ) \, dx}{30 e^4}\\ &=-\frac{\left (3 C d^2+2 e (2 B d+A e)\right ) x \left (d^2-e^2 x^2\right )^{3/2}}{8 e^2}-\frac{(2 C d+B e) x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac{1}{6} C x^3 \left (d^2-e^2 x^2\right )^{3/2}-\frac{\int \left (-15 d^2 e^4 \left (3 C d^2+4 B d e+10 A e^2\right )-24 d e^5 \left (4 C d^2+e (7 B d+10 A e)\right ) x\right ) \sqrt{d^2-e^2 x^2} \, dx}{120 e^6}\\ &=-\frac{d \left (4 C d^2+e (7 B d+10 A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 e^3}-\frac{\left (3 C d^2+2 e (2 B d+A e)\right ) x \left (d^2-e^2 x^2\right )^{3/2}}{8 e^2}-\frac{(2 C d+B e) x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac{1}{6} C x^3 \left (d^2-e^2 x^2\right )^{3/2}+\frac{\left (d^2 \left (3 C d^2+4 B d e+10 A e^2\right )\right ) \int \sqrt{d^2-e^2 x^2} \, dx}{8 e^2}\\ &=\frac{d^2 \left (3 C d^2+4 B d e+10 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{16 e^2}-\frac{d \left (4 C d^2+e (7 B d+10 A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 e^3}-\frac{\left (3 C d^2+2 e (2 B d+A e)\right ) x \left (d^2-e^2 x^2\right )^{3/2}}{8 e^2}-\frac{(2 C d+B e) x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac{1}{6} C x^3 \left (d^2-e^2 x^2\right )^{3/2}+\frac{\left (d^4 \left (3 C d^2+4 B d e+10 A e^2\right )\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{16 e^2}\\ &=\frac{d^2 \left (3 C d^2+4 B d e+10 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{16 e^2}-\frac{d \left (4 C d^2+e (7 B d+10 A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 e^3}-\frac{\left (3 C d^2+2 e (2 B d+A e)\right ) x \left (d^2-e^2 x^2\right )^{3/2}}{8 e^2}-\frac{(2 C d+B e) x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac{1}{6} C x^3 \left (d^2-e^2 x^2\right )^{3/2}+\frac{\left (d^4 \left (3 C d^2+4 B d e+10 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 )}{16 e^2}\\ &=\frac{d^2 \left (3 C d^2+4 B d e+10 A e^2\right ) x \sqrt{d^2-e^2 x^2}}{16 e^2}-\frac{d \left (4 C d^2+e (7 B d+10 A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{15 e^3}-\frac{\left (3 C d^2+2 e (2 B d+A e)\right ) x \left (d^2-e^2 x^2\right )^{3/2}}{8 e^2}-\frac{(2 C d+B e) x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac{1}{6} C x^3 \left (d^2-e^2 x^2\right )^{3/2}+\frac{d^4 \left (3 C d^2+4 B d e+10 A e^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^3}\\ \end{align*}
Mathematica [A] time = 0.51806, size = 226, normalized size = 0.96 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\sqrt{1-\frac{e^2 x^2}{d^2}} \left (2 e \left (5 A e \left (9 d^2 e x-16 d^3+16 d e^2 x^2+6 e^3 x^3\right )+B \left (32 d^2 e^2 x^2-30 d^3 e x-56 d^4+60 d e^3 x^3+24 e^4 x^4\right )\right )+C \left (-32 d^3 e^2 x^2+50 d^2 e^3 x^3-45 d^4 e x-64 d^5+96 d e^4 x^4+40 e^5 x^5\right )\right )+15 \sin ^{-1}\left (\frac{e x}{d}\right ) \left (2 d^3 e (5 A e+2 B d)+3 C d^5\right )\right )}{240 e^3 \sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.087, size = 371, normalized size = 1.6 \begin{align*} -{\frac{C{x}^{3}}{6} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,C{d}^{2}x}{8\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{3\,C{d}^{4}x}{16\,{e}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{3\,C{d}^{6}}{16\,{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}{5} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{2\,Cd{x}^{2}}{5\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{7\,B{d}^{2}}{15\,{e}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{4\,{d}^{3}C}{15\,{e}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{xA}{4} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{Bdx}{2\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{d}^{2}xA}{8}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{{d}^{3}xB}{4\,e}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{5\,{d}^{4}A}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{{d}^{5}B}{4\,e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{2\,Ad}{3\,e} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51519, size = 505, normalized size = 2.14 \begin{align*} -\frac{1}{6} \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} C x^{3} + \frac{A d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} + \frac{C d^{6} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{16 \, \sqrt{e^{2}} e^{2}} + \frac{1}{2} \, \sqrt{-e^{2} x^{2} + d^{2}} A d^{2} x + \frac{\sqrt{-e^{2} x^{2} + d^{2}} C d^{4} x}{16 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} C d^{2} x}{8 \, e^{2}} + \frac{{\left (C d^{2} + 2 \, B d e + A e^{2}\right )} d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} B d^{2}}{3 \, e^{2}} - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}} A d}{3 \, e} + \frac{\sqrt{-e^{2} x^{2} + d^{2}}{\left (C d^{2} + 2 \, B d e + A e^{2}\right )} d^{2} x}{8 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}{\left (2 \, C d e + B e^{2}\right )} x^{2}}{5 \, e^{2}} - \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}{\left (C d^{2} + 2 \, B d e + A e^{2}\right )} x}{4 \, e^{2}} - \frac{2 \,{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{3}{2}}{\left (2 \, C d e + B e^{2}\right )} d^{2}}{15 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.39827, size = 464, normalized size = 1.97 \begin{align*} -\frac{30 \,{\left (3 \, C d^{6} + 4 \, B d^{5} e + 10 \, A d^{4} e^{2}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (40 \, C e^{5} x^{5} - 64 \, C d^{5} - 112 \, B d^{4} e - 160 \, A d^{3} e^{2} + 48 \,{\left (2 \, C d e^{4} + B e^{5}\right )} x^{4} + 10 \,{\left (5 \, C d^{2} e^{3} + 12 \, B d e^{4} + 6 \, A e^{5}\right )} x^{3} - 32 \,{\left (C d^{3} e^{2} - 2 \, B d^{2} e^{3} - 5 \, A d e^{4}\right )} x^{2} - 15 \,{\left (3 \, C d^{4} e + 4 \, B d^{3} e^{2} - 6 \, A d^{2} e^{3}\right )} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 21.467, size = 1239, normalized size = 5.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15015, size = 266, normalized size = 1.13 \begin{align*} \frac{1}{16} \,{\left (3 \, C d^{6} + 4 \, B d^{5} e + 10 \, A d^{4} e^{2}\right )} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-3\right )} \mathrm{sgn}\left (d\right ) + \frac{1}{240} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, C x e^{2} + 6 \,{\left (2 \, C d e^{9} + B e^{10}\right )} e^{\left (-8\right )}\right )} x + 5 \,{\left (5 \, C d^{2} e^{8} + 12 \, B d e^{9} + 6 \, A e^{10}\right )} e^{\left (-8\right )}\right )} x - 16 \,{\left (C d^{3} e^{7} - 2 \, B d^{2} e^{8} - 5 \, A d e^{9}\right )} e^{\left (-8\right )}\right )} x - 15 \,{\left (3 \, C d^{4} e^{6} + 4 \, B d^{3} e^{7} - 6 \, A d^{2} e^{8}\right )} e^{\left (-8\right )}\right )} x - 16 \,{\left (4 \, C d^{5} e^{5} + 7 \, B d^{4} e^{6} + 10 \, A d^{3} e^{7}\right )} e^{\left (-8\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]