Optimal. Leaf size=102 \[ -\frac{a \sqrt{d-e x} \sqrt{d+e x}}{d^2 x}-\frac{\left (2 b e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right )}{e^3}+\frac{c x (e x-d) \sqrt{d+e x}}{2 e^2 \sqrt{d-e x}} \]
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Rubi [A] time = 0.121831, antiderivative size = 155, normalized size of antiderivative = 1.52, number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {520, 1265, 388, 217, 203} \[ -\frac{a \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}+\frac{\sqrt{d^2-e^2 x^2} \left (2 b e^2+c d^2\right ) \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x \left (d^2-e^2 x^2\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 520
Rule 1265
Rule 388
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{x^2 \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}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{\sqrt{d^2-e^2 x^2} \int \frac{-b d^2-c d^2 x^2}{\sqrt{d^2-e^2 x^2}} \, dx}{d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x \left (d^2-e^2 x^2\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (\left (2 b+\frac{c d^2}{e^2}\right ) \sqrt{d^2-e^2 x^2}\right ) \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx}{2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x \left (d^2-e^2 x^2\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (\left (2 b+\frac{c d^2}{e^2}\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 )}{2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{d^2 x \sqrt{d-e x} \sqrt{d+e x}}-\frac{c x \left (d^2-e^2 x^2\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (c d^2+2 b e^2\right ) \sqrt{d^2-e^2 x^2} \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.564506, size = 135, normalized size = 1.32 \[ -\frac{\frac{e \sqrt{d-e x} \sqrt{d+e x} \left (2 a e^2+c d^2 x^2\right )}{d^2 x}+4 \left (b e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right )-\frac{2 c d^{5/2} \sqrt{\frac{e x}{d}+1} \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right )}{\sqrt{d+e x}}}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 148, normalized size = 1.5 \begin{align*} -{\frac{{\it csgn} \left ( e \right ) }{2\,{d}^{2}{e}^{3}x}\sqrt{-ex+d}\sqrt{ex+d} \left ({\it csgn} \left ( e \right ){x}^{2}c{d}^{2}e\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}-2\,\arctan \left ({\frac{{\it csgn} \left ( e \right ) ex}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) xb{d}^{2}{e}^{2}-\arctan \left ({{\it csgn} \left ( e \right ) ex{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ) xc{d}^{4}+2\,{\it csgn} \left ( e \right ){e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}a \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52415, size = 203, normalized size = 1.99 \begin{align*} -\frac{2 \,{\left (c d^{4} + 2 \, b d^{2} e^{2}\right )} x \arctan \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{e x}\right ) +{\left (c d^{2} e x^{2} + 2 \, a e^{3}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{2 \, d^{2} e^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 54.2156, size = 287, normalized size = 2.81 \begin{align*} \frac{i a e{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{a e{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{i b{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{b{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 c 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{c 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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