Optimal. Leaf size=109 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (a e^4+b d^2 e^2+c d^4\right )}{e^6}+\frac{(d-e x)^{3/2} (d+e x)^{3/2} \left (b e^2+2 c d^2\right )}{3 e^6}-\frac{c (d-e x)^{5/2} (d+e x)^{5/2}}{5 e^6} \]
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Rubi [A] time = 0.1221, antiderivative size = 149, normalized size of antiderivative = 1.37, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {520, 1247, 698} \[ -\frac{\left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^6 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^2 \left (b e^2+2 c d^2\right )}{3 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )^3}{5 e^6 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 520
Rule 1247
Rule 698
Rubi steps
\begin{align*} \int \frac{x \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 \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{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \frac{a+b x+c x^2}{\sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \left (\frac{c d^4+b d^2 e^2+a e^4}{e^4 \sqrt{d^2-e^2 x}}+\frac{\left (-2 c d^2-b e^2\right ) \sqrt{d^2-e^2 x}}{e^4}+\frac{c \left (d^2-e^2 x\right )^{3/2}}{e^4}\right ) \, dx,x,x^2\right )}{2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (c d^4+b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )}{e^6 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (2 c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )^2}{3 e^6 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )^3}{5 e^6 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 0.703123, size = 194, normalized size = 1.78 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (5 \left (3 a e^4+2 b d^2 e^2+b e^4 x^2\right )+c \left (4 d^2 e^2 x^2+8 d^4+3 e^4 x^4\right )\right )+\frac{30 \sqrt{d} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{\sqrt{\frac{e x}{d}+1}}-30 d \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{15 e^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 73, normalized size = 0.7 \begin{align*} -{\frac{3\,c{x}^{4}{e}^{4}+5\,b{e}^{4}{x}^{2}+4\,c{d}^{2}{e}^{2}{x}^{2}+15\,a{e}^{4}+10\,b{d}^{2}{e}^{2}+8\,c{d}^{4}}{15\,{e}^{6}}\sqrt{-ex+d}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.63594, size = 188, normalized size = 1.72 \begin{align*} -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{4}}{5 \, e^{2}} - \frac{4 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{2}}{15 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{2}}{3 \, e^{2}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4}}{15 \, e^{6}} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2}}{3 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81681, size = 162, normalized size = 1.49 \begin{align*} -\frac{{\left (3 \, c e^{4} x^{4} + 8 \, c d^{4} + 10 \, b d^{2} e^{2} + 15 \, a e^{4} +{\left (4 \, c d^{2} e^{2} + 5 \, b e^{4}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{15 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 77.8217, size = 350, normalized size = 3.21 \begin{align*} - \frac{i a d{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{2}} - \frac{a d{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{2}} - \frac{i b d^{3}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{4}} - \frac{b d^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{4}} - \frac{i c d^{5}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{9}{4}, - \frac{7}{4} & -2, -2, - \frac{3}{2}, 1 \\- \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{6}} - \frac{c d^{5}{G_{6, 6}^{2, 6}\left (\begin{matrix} -3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, -2, 1 & \\- \frac{11}{4}, - \frac{9}{4} & -3, - \frac{5}{2}, - \frac{5}{2}, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15936, size = 153, normalized size = 1.4 \begin{align*} -\frac{1}{276480} \,{\left (15 \, c d^{4} e^{25} + 15 \, b d^{2} e^{27} -{\left (20 \, c d^{3} e^{25} + 10 \, b d e^{27} -{\left (22 \, c d^{2} e^{25} + 3 \,{\left ({\left (x e + d\right )} c e^{25} - 4 \, c d e^{25}\right )}{\left (x e + d\right )} + 5 \, b e^{27}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 15 \, a e^{29}\right )} \sqrt{x e + d} \sqrt{-x e + d} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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