Optimal. Leaf size=99 \[ -\frac{\left (a e^2+2 b d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d-e x} \sqrt{d+e x}}{d}\right )}{2 d^3}-\frac{a \sqrt{d-e x} \sqrt{d+e x}}{2 d^2 x^2}-\frac{c \sqrt{d-e x} \sqrt{d+e x}}{e^2} \]
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Rubi [A] time = 0.251905, antiderivative size = 155, normalized size of antiderivative = 1.57, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {520, 1251, 897, 1157, 388, 208} \[ -\frac{\sqrt{d^2-e^2 x^2} \left (a e^2+2 b d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{c \left (d^2-e^2 x^2\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 520
Rule 1251
Rule 897
Rule 1157
Rule 388
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{x^3 \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^3 \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}{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 \frac{\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}}{\left (\frac{d^2}{e^2}-\frac{x^2}{e^2}\right )^2} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\sqrt{d^2-e^2 x^2} \operatorname{Subst}\left (\int \frac{-a-\frac{2 \left (c d^4+b d^2 e^2\right )}{e^4}+\frac{2 c d^2 x^2}{e^4}}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{c \left (d^2-e^2 x^2\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (e^2 \left (\frac{2 c d^4}{e^6}+\frac{-a-\frac{2 \left (c d^4+b d^2 e^2\right )}{e^4}}{e^2}\right ) \sqrt{d^2-e^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{2 d^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{c \left (d^2-e^2 x^2\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \left (d^2-e^2 x^2\right )}{2 d^2 x^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (2 b d^2+a e^2\right ) \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^3 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [B] time = 0.216946, size = 233, normalized size = 2.35 \[ \frac{-e^2 x^2 \sqrt{d^2-e^2 x^2} \left (a e^2+2 b d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )-a d^3 e^2+a d e^4 x^2+2 c d^3 e^2 x^4-4 c d^{9/2} x^2 \sqrt{d-e x} \sqrt{\frac{e x}{d}+1} \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right )+4 c d^4 x^2 \sqrt{d-e x} \sqrt{d+e x} \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right )-2 c d^5 x^2}{2 d^3 e^2 x^2 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.025, size = 163, normalized size = 1.7 \begin{align*} -{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{3}{e}^{2}{x}^{2}}\sqrt{-ex+d}\sqrt{ex+d} \left ( \ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{2}a{e}^{4}+2\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ){x}^{2}b{d}^{2}{e}^{2}+2\,{\it csgn} \left ( d \right ){x}^{2}c{d}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+{\it csgn} \left ( d \right ) ad{e}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \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 = 1.29746, size = 215, normalized size = 2.17 \begin{align*} -\frac{2 \, c d^{4} x^{2} -{\left (2 \, b d^{2} e^{2} + a e^{4}\right )} x^{2} \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right ) +{\left (2 \, c d^{3} x^{2} + a d e^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{2 \, d^{3} e^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 64.1964, size = 270, normalized size = 2.73 \begin{align*} \frac{i a e^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{a e^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \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}} d^{3}} + \frac{i b{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{b{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \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}} d} - \frac{i c 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{c 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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