Optimal. Leaf size=128 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (8 a e^4+4 b d^2 e^2+3 c d^4\right )}{4 e^5}-\frac{x \sqrt{d-e x} \sqrt{d+e x} \left (4 b e^2+3 c d^2\right )}{8 e^4}+\frac{c x^3 (e x-d) \sqrt{d+e x}}{4 e^2 \sqrt{d-e x}} \]
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Rubi [A] time = 0.0911319, antiderivative size = 179, normalized size of antiderivative = 1.4, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {520, 1159, 388, 217, 203} \[ \frac{\sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right ) \left (8 a e^4+4 b d^2 e^2+3 c d^4\right )}{8 e^5 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (4 b e^2+3 c d^2\right )}{8 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 520
Rule 1159
Rule 388
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{\sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{d^2-e^2 x^2} \int \frac{a+b x^2+c x^4}{\sqrt{d^2-e^2 x^2}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d^2-e^2 x^2} \int \frac{-4 a e^2-\left (3 c d^2+4 b e^2\right ) x^2}{\sqrt{d^2-e^2 x^2}} \, dx}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (3 c d^2+4 b e^2\right ) x \left (d^2-e^2 x^2\right )}{8 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (\left (-8 a e^4+d^2 \left (-3 c d^2-4 b e^2\right )\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{8 e^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (3 c d^2+4 b e^2\right ) x \left (d^2-e^2 x^2\right )}{8 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (\left (-8 a e^4+d^2 \left (-3 c d^2-4 b e^2\right )\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 )}{8 e^4 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (3 c d^2+4 b e^2\right ) x \left (d^2-e^2 x^2\right )}{8 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (3 c d^4+4 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 )}{8 e^5 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.581851, size = 157, normalized size = 1.23 \[ -\frac{16 \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )+e x \sqrt{d-e x} \sqrt{d+e x} \left (4 b e^2+3 c d^2+2 c e^2 x^2\right )-\frac{2 d^{5/2} \sqrt{\frac{e x}{d}+1} \left (4 b e^2+5 c d^2\right ) \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right )}{\sqrt{d+e x}}}{8 e^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 191, normalized size = 1.5 \begin{align*} -{\frac{{\it csgn} \left ( e \right ) }{8\,{e}^{5}}\sqrt{-ex+d}\sqrt{ex+d} \left ( 2\,{\it csgn} \left ( e \right ){x}^{3}c{e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+4\,{\it csgn} \left ( e \right ){e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}xb+3\,{\it csgn} \left ( e \right ) e\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}xc{d}^{2}-8\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) a{e}^{4}-4\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) b{d}^{2}{e}^{2}-3\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) c{d}^{4} \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 = 1.50604, size = 201, normalized size = 1.57 \begin{align*} -\frac{\sqrt{-e^{2} x^{2} + d^{2}} c x^{3}}{4 \, e^{2}} + \frac{a \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} + \frac{3 \, c d^{4} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{8 \, \sqrt{e^{2}} e^{4}} + \frac{b d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}} e^{2}} - \frac{3 \, \sqrt{-e^{2} x^{2} + d^{2}} c d^{2} x}{8 \, e^{4}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}} b x}{2 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58024, size = 227, normalized size = 1.77 \begin{align*} -\frac{{\left (2 \, c e^{3} x^{3} +{\left (3 \, c d^{2} e + 4 \, b e^{3}\right )} x\right )} \sqrt{e x + d} \sqrt{-e x + d} + 2 \,{\left (3 \, c d^{4} + 4 \, b d^{2} e^{2} + 8 \, a e^{4}\right )} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e x}\right )}{8 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 37.9723, size = 325, normalized size = 2.54 \begin{align*} - \frac{i a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e} + \frac{a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e} - \frac{i b 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{b 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 c 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{c 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13785, size = 170, normalized size = 1.33 \begin{align*} \frac{1}{114688} \,{\left ({\left (5 \, c d^{3} e^{16} + 4 \, b d e^{18} -{\left (9 \, c d^{2} e^{16} + 2 \,{\left ({\left (x e + d\right )} c e^{16} - 3 \, c d e^{16}\right )}{\left (x e + d\right )} + 4 \, b e^{18}\right )}{\left (x e + d\right )}\right )} \sqrt{x e + d} \sqrt{-x e + d} + 2 \,{\left (3 \, c d^{4} e^{16} + 4 \, b d^{2} e^{18} + 8 \, a e^{20}\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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