Optimal. Leaf size=210 \[ -\frac{(d-e x)^{5/2} (d+e x)^{5/2} \left (a e^4+3 b d^2 e^2+6 c d^4\right )}{5 e^{10}}+\frac{d^2 (d-e x)^{3/2} (d+e x)^{3/2} \left (2 a e^4+3 b d^2 e^2+4 c d^4\right )}{3 e^{10}}-\frac{d^4 \sqrt{d-e x} \sqrt{d+e x} \left (a e^4+b d^2 e^2+c d^4\right )}{e^{10}}+\frac{(d-e x)^{7/2} (d+e x)^{7/2} \left (b e^2+4 c d^2\right )}{7 e^{10}}-\frac{c (d-e x)^{9/2} (d+e x)^{9/2}}{9 e^{10}} \]
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Rubi [A] time = 0.314532, antiderivative size = 278, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114, Rules used = {520, 1251, 897, 1153} \[ -\frac{\left (d^2-e^2 x^2\right )^3 \left (a e^4+3 b d^2 e^2+6 c d^4\right )}{5 e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^2 \left (d^2-e^2 x^2\right )^2 \left (2 a e^4+3 b d^2 e^2+4 c d^4\right )}{3 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^4 \left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (d^2-e^2 x^2\right )^4 \left (b e^2+4 c d^2\right )}{7 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )^5}{9 e^{10} \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 520
Rule 1251
Rule 897
Rule 1153
Rubi steps
\begin{align*} \int \frac{x^5 \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^5 \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{x^2 \left (a+b x+c x^2\right )}{\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{d^2}{e^2}-\frac{x^2}{e^2}\right )^2 \left (\frac{c d^4+b d^2 e^2+a e^4}{e^4}-\frac{\left (2 c d^2+b e^2\right ) x^2}{e^4}+\frac{c x^4}{e^4}\right ) \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{e^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^8+b d^6 e^2+a d^4 e^4}{e^8}-\frac{d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) x^2}{e^8}+\frac{\left (6 c d^4+3 b d^2 e^2+a e^4\right ) x^4}{e^8}-\frac{\left (4 c d^2+b e^2\right ) x^6}{e^8}+\frac{c x^8}{e^8}\right ) \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{d^4 \left (c d^4+b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )}{e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^2 \left (4 c d^4+3 b d^2 e^2+2 a e^4\right ) \left (d^2-e^2 x^2\right )^2}{3 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (6 c d^4+3 b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )^3}{5 e^{10} \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (4 c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )^4}{7 e^{10} \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )^5}{9 e^{10} \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 1.43718, size = 265, normalized size = 1.26 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (21 a e^4 \left (4 d^2 e^2 x^2+8 d^4+3 e^4 x^4\right )+9 b \left (6 d^2 e^6 x^4+8 d^4 e^4 x^2+16 d^6 e^2+5 e^8 x^6\right )+c \left (64 d^6 e^2 x^2+48 d^4 e^4 x^4+40 d^2 e^6 x^6+128 d^8+35 e^8 x^8\right )\right )+\frac{630 d^{9/2} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{\sqrt{\frac{e x}{d}+1}}-630 d^5 \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{315 e^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 145, normalized size = 0.7 \begin{align*} -{\frac{35\,c{x}^{8}{e}^{8}+45\,b{e}^{8}{x}^{6}+40\,c{d}^{2}{e}^{6}{x}^{6}+63\,a{e}^{8}{x}^{4}+54\,b{d}^{2}{e}^{6}{x}^{4}+48\,c{d}^{4}{e}^{4}{x}^{4}+84\,a{d}^{2}{e}^{6}{x}^{2}+72\,b{d}^{4}{e}^{4}{x}^{2}+64\,c{d}^{6}{e}^{2}{x}^{2}+168\,a{d}^{4}{e}^{4}+144\,b{d}^{6}{e}^{2}+128\,c{d}^{8}}{315\,{e}^{10}}\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 = 1.64729, size = 398, normalized size = 1.9 \begin{align*} -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{8}}{9 \, e^{2}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{6}}{63 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{6}}{7 \, e^{2}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4} x^{4}}{105 \, e^{6}} - \frac{6 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2} x^{4}}{35 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a x^{4}}{5 \, e^{2}} - \frac{64 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{6} x^{2}}{315 \, e^{8}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{4} x^{2}}{35 \, e^{6}} - \frac{4 \, \sqrt{-e^{2} x^{2} + d^{2}} a d^{2} x^{2}}{15 \, e^{4}} - \frac{128 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{8}}{315 \, e^{10}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{6}}{35 \, e^{8}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} a d^{4}}{15 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77388, size = 317, normalized size = 1.51 \begin{align*} -\frac{{\left (35 \, c e^{8} x^{8} + 128 \, c d^{8} + 144 \, b d^{6} e^{2} + 168 \, a d^{4} e^{4} + 5 \,{\left (8 \, c d^{2} e^{6} + 9 \, b e^{8}\right )} x^{6} + 3 \,{\left (16 \, c d^{4} e^{4} + 18 \, b d^{2} e^{6} + 21 \, a e^{8}\right )} x^{4} + 4 \,{\left (16 \, c d^{6} e^{2} + 18 \, b d^{4} e^{4} + 21 \, a d^{2} e^{6}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{315 \, e^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25458, size = 328, normalized size = 1.56 \begin{align*} -\frac{1}{2807562240} \,{\left (315 \, c d^{8} e^{81} + 315 \, b d^{6} e^{83} + 315 \, a d^{4} e^{85} -{\left (840 \, c d^{7} e^{81} + 630 \, b d^{5} e^{83} + 420 \, a d^{3} e^{85} -{\left (1932 \, c d^{6} e^{81} + 1071 \, b d^{4} e^{83} + 462 \, a d^{2} e^{85} -{\left (2952 \, c d^{5} e^{81} + 1116 \, b d^{3} e^{83} + 252 \, a d e^{85} -{\left (3098 \, c d^{4} e^{81} + 729 \, b d^{2} e^{83} - 5 \,{\left (440 \, c d^{3} e^{81} + 54 \, b d e^{83} -{\left (204 \, c d^{2} e^{81} + 7 \,{\left ({\left (x e + d\right )} c e^{81} - 8 \, c d e^{81}\right )}{\left (x e + d\right )} + 9 \, b e^{83}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 63 \, a e^{85}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )}\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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