Optimal. Leaf size=93 \[ -\frac{a \tanh ^{-1}\left (\frac{\sqrt{d-e x} \sqrt{d+e x}}{d}\right )}{d}-\frac{\sqrt{d-e x} \sqrt{d+e x} \left (b e^2+c d^2\right )}{e^4}+\frac{c (d-e x)^{3/2} (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A] time = 0.164607, antiderivative size = 151, normalized size of antiderivative = 1.62, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {520, 1251, 897, 1153, 208} \[ -\frac{a \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (d^2-e^2 x^2\right ) \left (b e^2+c d^2\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 520
Rule 1251
Rule 897
Rule 1153
Rule 208
Rubi steps
\begin{align*} \int \frac{a+b x^2+c x^4}{x \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 \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 \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}}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, 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 (b+\frac{c d^2}{e^2}-\frac{c x^2}{e^2}+\frac{a}{\frac{d^2}{e^2}-\frac{x^2}{e^2}}\right ) \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (a \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 )}{e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{\left (c d^2+b e^2\right ) \left (d^2-e^2 x^2\right )}{e^4 \sqrt{d-e x} \sqrt{d+e x}}+\frac{c \left (d^2-e^2 x^2\right )^2}{3 e^4 \sqrt{d-e x} \sqrt{d+e x}}-\frac{a \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}
Mathematica [B] time = 0.891118, size = 217, normalized size = 2.33 \[ \frac{-\frac{3 a \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d \sqrt{d-e x}}+\frac{\left (e^2 x^2-d^2\right ) \left (3 b e^2+2 c d^2+c e^2 x^2\right )}{e^4 \sqrt{d-e x}}-\frac{6 d^{3/2} \sqrt{\frac{e x}{d}+1} \left (b e^2+c d^2\right ) \sin ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{2} \sqrt{d}}\right )}{e^4}+\frac{6 d \sqrt{d+e x} \left (b e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d-e x}}{\sqrt{d+e x}}\right )}{e^4}}{3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.044, size = 143, normalized size = 1.5 \begin{align*} -{\frac{{\it csgn} \left ( d \right ) }{3\,d{e}^{4}}\sqrt{-ex+d}\sqrt{ex+d} \left ({\it csgn} \left ( d \right ){x}^{2}cd{e}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+3\,\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) bd{e}^{2}+2\,\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) c{d}^{3}+3\,\ln \left ( 2\,{\frac{d \left ( \sqrt{-{e}^{2}{x}^{2}+{d}^{2}}{\it csgn} \left ( d \right ) +d \right ) }{x}} \right ) a{e}^{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 [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 = 1.59832, size = 178, normalized size = 1.91 \begin{align*} \frac{3 \, a e^{4} \log \left (\frac{\sqrt{e x + d} \sqrt{-e x + d} - d}{x}\right ) -{\left (c d e^{2} x^{2} + 2 \, c d^{3} + 3 \, b d e^{2}\right )} \sqrt{e x + d} \sqrt{-e x + d}}{3 \, d e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 45.0443, size = 304, normalized size = 3.27 \begin{align*} \frac{i a{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{a{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 b 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{b 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}} - \frac{i c d^{3}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} e^{4}} - \frac{c d^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \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}} e^{4}} \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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