Optimal. Leaf size=159 \[ \frac{(d-e x)^{3/2} (d+e x)^{3/2} \left (a e^4+2 b d^2 e^2+3 c d^4\right )}{3 e^8}-\frac{d^2 \sqrt{d-e x} \sqrt{d+e x} \left (a e^4+b d^2 e^2+c d^4\right )}{e^8}-\frac{(d-e x)^{5/2} (d+e x)^{5/2} \left (b e^2+3 c d^2\right )}{5 e^8}+\frac{c (d-e x)^{7/2} (d+e x)^{7/2}}{7 e^8} \]
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Rubi [A] time = 0.189442, antiderivative size = 213, normalized size of antiderivative = 1.34, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {520, 1251, 771} \[ \frac{\left (d^2-e^2 x^2\right )^2 \left (a e^4+2 b d^2 e^2+3 c d^4\right )}{3 e^8 \sqrt{d-e x} \sqrt{d+e x}}-\frac{d^2 \left (d^2-e^2 x^2\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{e^8 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right )^3 \left (b e^2+3 c d^2\right )}{5 e^8 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \left (d^2-e^2 x^2\right )^4}{7 e^8 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 520
Rule 1251
Rule 771
Rubi steps
\begin{align*} \int \frac{x^3 \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^3 \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 \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{c d^6+b d^4 e^2+a d^2 e^4}{e^6 \sqrt{d^2-e^2 x}}+\frac{\left (-3 c d^4-2 b d^2 e^2-a e^4\right ) \sqrt{d^2-e^2 x}}{e^6}+\frac{\left (3 c d^2+b e^2\right ) \left (d^2-e^2 x\right )^{3/2}}{e^6}-\frac{c \left (d^2-e^2 x\right )^{5/2}}{e^6}\right ) \, dx,x,x^2\right )}{2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{d^2 \left (c d^4+b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )}{e^8 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (3 c d^4+2 b d^2 e^2+a e^4\right ) \left (d^2-e^2 x^2\right )^2}{3 e^8 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (3 c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )^3}{5 e^8 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \left (d^2-e^2 x^2\right )^4}{7 e^8 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 1.0886, size = 232, normalized size = 1.46 \[ -\frac{\sqrt{d-e x} \sqrt{d+e x} \left (35 a e^4 \left (2 d^2+e^2 x^2\right )+7 b \left (4 d^2 e^4 x^2+8 d^4 e^2+3 e^6 x^4\right )+3 c \left (8 d^4 e^2 x^2+6 d^2 e^4 x^4+16 d^6+5 e^6 x^6\right )\right )+\frac{210 d^{5/2} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{\sqrt{\frac{e x}{d}+1}}-210 d^3 \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )}{105 e^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 109, normalized size = 0.7 \begin{align*} -{\frac{15\,c{x}^{6}{e}^{6}+21\,b{e}^{6}{x}^{4}+18\,c{d}^{2}{e}^{4}{x}^{4}+35\,a{e}^{6}{x}^{2}+28\,b{d}^{2}{e}^{4}{x}^{2}+24\,c{d}^{4}{e}^{2}{x}^{2}+70\,a{d}^{2}{e}^{4}+56\,b{d}^{4}{e}^{2}+48\,c{d}^{6}}{105\,{e}^{8}}\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 = 1.49211, size = 293, normalized size = 1.84 \begin{align*} -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{6}}{7 \, e^{2}} - \frac{6 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x^{4}}{35 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x^{4}}{5 \, e^{2}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{4} x^{2}}{35 \, e^{6}} - \frac{4 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{2} x^{2}}{15 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} a x^{2}}{3 \, e^{2}} - \frac{16 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{6}}{35 \, e^{8}} - \frac{8 \, \sqrt{-e^{2} x^{2} + d^{2}} b d^{4}}{15 \, e^{6}} - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} a d^{2}}{3 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91068, size = 238, normalized size = 1.5 \begin{align*} -\frac{{\left (15 \, c e^{6} x^{6} + 48 \, c d^{6} + 56 \, b d^{4} e^{2} + 70 \, a d^{2} e^{4} + 3 \,{\left (6 \, c d^{2} e^{4} + 7 \, b e^{6}\right )} x^{4} +{\left (24 \, c d^{4} e^{2} + 28 \, b d^{2} e^{4} + 35 \, a e^{6}\right )} x^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{105 \, e^{8}} \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 = 1.14837, size = 239, normalized size = 1.5 \begin{align*} -\frac{1}{44728320} \,{\left (105 \, c d^{6} e^{49} + 105 \, b d^{4} e^{51} + 105 \, a d^{2} e^{53} -{\left (210 \, c d^{5} e^{49} + 140 \, b d^{3} e^{51} + 70 \, a d e^{53} -{\left (357 \, c d^{4} e^{49} + 154 \, b d^{2} e^{51} - 3 \,{\left (124 \, c d^{3} e^{49} + 28 \, b d e^{51} -{\left (81 \, c d^{2} e^{49} + 5 \,{\left ({\left (x e + d\right )} c e^{49} - 6 \, c d e^{49}\right )}{\left (x e + d\right )} + 7 \, b e^{51}\right )}{\left (x e + d\right )}\right )}{\left (x e + d\right )} + 35 \, a e^{53}\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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