Optimal. Leaf size=77 \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c d-b e}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b \sqrt{d}} \]
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Rubi [A] time = 0.0576517, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {707, 1093, 208} \[ \frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c d-b e}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 707
Rule 1093
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx &=(2 e) \operatorname{Subst}\left (\int \frac{1}{c d^2-b d e-(2 c d-b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )\\ &=\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b}-\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{b}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b \sqrt{d}}+\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b \sqrt{c d-b e}}\\ \end{align*}
Mathematica [A] time = 0.0841091, size = 75, normalized size = 0.97 \[ \frac{\frac{2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c d-b e}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{\sqrt{d}}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.25, size = 62, normalized size = 0.8 \begin{align*} -2\,{\frac{c}{b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{1}{b\sqrt{d}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \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 = 2.33209, size = 880, normalized size = 11.43 \begin{align*} \left [\frac{d \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) + \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right )}{b d}, \frac{2 \, d \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{-\frac{c}{c d - b e}}}{c e x + c d}\right ) + \sqrt{d} \log \left (\frac{e x - 2 \, \sqrt{e x + d} \sqrt{d} + 2 \, d}{x}\right )}{b d}, \frac{d \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e + 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) + 2 \, \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right )}{b d}, \frac{2 \,{\left (d \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{-\frac{c}{c d - b e}}}{c e x + c d}\right ) + \sqrt{-d} \arctan \left (\frac{\sqrt{e x + d} \sqrt{-d}}{d}\right )\right )}}{b d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.2026, size = 80, normalized size = 1.04 \begin{align*} \frac{2 c \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{c}{b e - c d}} \sqrt{d + e x}} \right )}}{b \sqrt{\frac{c}{b e - c d}} \left (b e - c d\right )} + \frac{2 \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{d}} \sqrt{d + e x}} \right )}}{b d \sqrt{- \frac{1}{d}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17955, size = 96, normalized size = 1.25 \begin{align*} -\frac{2 \, c \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b} + \frac{2 \, \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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