Optimal. Leaf size=62 \[ \frac{2 \sqrt{d+e x}}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.0292775, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {27, 50, 63, 208} \[ \frac{2 \sqrt{d+e x}}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 27
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) \sqrt{d+e x}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{\sqrt{d+e x}}{a+b x} \, dx\\ &=\frac{2 \sqrt{d+e x}}{b}+\frac{(b d-a e) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{b}\\ &=\frac{2 \sqrt{d+e x}}{b}+\frac{(2 (b d-a e)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b e}\\ &=\frac{2 \sqrt{d+e x}}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0249448, size = 62, normalized size = 1. \[ \frac{2 \sqrt{d+e x}}{b}-\frac{2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 92, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{ex+d}}{b}}-2\,{\frac{ae}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{d}{\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \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 = 1.0368, size = 306, normalized size = 4.94 \begin{align*} \left [\frac{\sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \, \sqrt{e x + d}}{b}, -\frac{2 \,{\left (\sqrt{-\frac{b d - a e}{b}} \arctan \left (-\frac{\sqrt{e x + d} b \sqrt{-\frac{b d - a e}{b}}}{b d - a e}\right ) - \sqrt{e x + d}\right )}}{b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.0554, size = 61, normalized size = 0.98 \begin{align*} \frac{2 \left (\frac{e \sqrt{d + e x}}{b} - \frac{e \left (a e - b d\right ) \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b^{2} \sqrt{\frac{a e - b d}{b}}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15411, size = 90, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (b d - a e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b} + \frac{2 \, \sqrt{x e + d}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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