Optimal. Leaf size=216 \[ \frac{d^2 \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \sqrt{d-e x} \sqrt{d+e x} \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^6}-\frac{x^3 \sqrt{d-e x} \sqrt{d+e x} \left (6 b e^2+5 c d^2\right )}{24 e^4}+\frac{c x^5 (e x-d) \sqrt{d+e x}}{6 e^2 \sqrt{d-e x}} \]
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Rubi [A] time = 0.205081, antiderivative size = 245, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {520, 1267, 459, 321, 217, 203} \[ -\frac{x \left (d^2-e^2 x^2\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^2 \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (8 a e^4+6 b d^2 e^2+5 c d^4\right )}{16 e^7 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x^3 \left (d^2-e^2 x^2\right ) \left (6 b e^2+5 c d^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 520
Rule 1267
Rule 459
Rule 321
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b x^2+c x^4\right )}{\sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{d^2-e^2 x^2} \int \frac{x^2 \left (a+b x^2+c x^4\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d^2-e^2 x^2} \int \frac{x^2 \left (-6 a e^2-\left (5 c d^2+6 b e^2\right ) x^2\right )}{\sqrt{d^2-e^2 x^2}} \, dx}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{x^2}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \left (d^2-e^2 x^2\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \left (d^2-e^2 x^2\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt{d^2-e^2 x^2}\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^6 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) x \left (d^2-e^2 x^2\right )}{16 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (5 c d^2+6 b e^2\right ) x^3 \left (d^2-e^2 x^2\right )}{24 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^5 \left (d^2-e^2 x^2\right )}{6 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^2 \left (5 c d^4+6 b d^2 e^2+8 a e^4\right ) \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{16 e^7 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.808645, size = 202, normalized size = 0.94 \[ -\frac{e x \sqrt{d-e x} \sqrt{d+e x} \left (6 \left (4 a e^4+3 b d^2 e^2+2 b e^4 x^2\right )+c \left (10 d^2 e^2 x^2+15 d^4+8 e^4 x^4\right )\right )-\frac{6 d^{3/2} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right ) \left (8 a e^4+10 b d^2 e^2+11 c d^4\right )}{\sqrt{\frac{e x}{d}+1}}+96 d^2 \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{48 e^7} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.035, size = 273, normalized size = 1.3 \begin{align*} -{\frac{{\it csgn} \left ( e \right ) }{48\,{e}^{7}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 8\,{\it csgn} \left ( e \right ){x}^{5}c{e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+12\,{\it csgn} \left ( e \right ){x}^{3}b{e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+10\,{\it csgn} \left ( e \right ){x}^{3}c{d}^{2}{e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+24\,{\it csgn} \left ( e \right ){e}^{5}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}xa+18\,{\it csgn} \left ( e \right ){e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}xb{d}^{2}+15\,{\it csgn} \left ( e \right ) e\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}xc{d}^{4}-24\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) a{d}^{2}{e}^{4}-18\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) b{d}^{4}{e}^{2}-15\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) c{d}^{6} \right ){\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56008, size = 309, normalized size = 1.43 \begin{align*} -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{5}}{6 \, e^{2}} - \frac{5 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{3}}{24 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{3}}{4 \, e^{2}} + \frac{5 \, c d^{6} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{16 \, \sqrt{e^{2}} e^{6}} + \frac{3 \, b d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{4}} + \frac{a d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{2}} - \frac{5 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4} x}{16 \, e^{6}} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2} x}{8 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a x}{2 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43201, size = 298, normalized size = 1.38 \begin{align*} -\frac{{\left (8 \, c e^{5} x^{5} + 2 \,{\left (5 \, c d^{2} e^{3} + 6 \, b e^{5}\right )} x^{3} + 3 \,{\left (5 \, c d^{4} e + 6 \, b d^{2} e^{3} + 8 \, a e^{5}\right )} x\right )} \sqrt{e x + d} \sqrt{-e x + d} + 6 \,{\left (5 \, c d^{6} + 6 \, b d^{4} e^{2} + 8 \, a d^{2} e^{4}\right )} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e x}\right )}{48 \, e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 125.927, size = 362, normalized size = 1.68 \begin{align*} - \frac{i a d^{2}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{3}} + \frac{a d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{3}} - \frac{i b d^{4}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{7}{4}, - \frac{5}{4} & - \frac{3}{2}, - \frac{3}{2}, -1, 1 \\-2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{5}} + \frac{b d^{4}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 1 & \\- \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, -2, -2, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{5}} - \frac{i c d^{6}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{11}{4}, - \frac{9}{4} & - \frac{5}{2}, - \frac{5}{2}, -2, 1 \\-3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, -2, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{7}} + \frac{c d^{6}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{7}{2}, - \frac{13}{4}, -3, - \frac{11}{4}, - \frac{5}{2}, 1 & \\- \frac{13}{4}, - \frac{11}{4} & - \frac{7}{2}, -3, -3, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1521, size = 257, normalized size = 1.19 \begin{align*} \frac{1}{34603008} \,{\left ({\left (33 \, c d^{5} e^{36} + 30 \, b d^{3} e^{38} + 24 \, a d e^{40} -{\left (85 \, c d^{4} e^{36} + 54 \, b d^{2} e^{38} - 2 \,{\left (55 \, c d^{3} e^{36} + 18 \, b d e^{38} -{\left (45 \, c d^{2} e^{36} + 4 \,{\left ({\left (x e + d\right )} c e^{36} - 5 \, c d e^{36}\right )}{\left (x e + d\right )} + 6 \, b e^{38}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 24 \, a e^{40}\right )}{\left (x e + d\right )}\right )} \sqrt{x e + d} \sqrt{-x e + d} + 6 \,{\left (5 \, c d^{6} e^{36} + 6 \, b d^{4} e^{38} + 8 \, a d^{2} e^{40}\right )} \arcsin \left (\frac{\sqrt{2} \sqrt{x e + d}}{2 \, \sqrt{d}}\right )\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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