Optimal. Leaf size=80 \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^2 (a+b x)} \]
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Rubi [A] time = 0.0373813, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 43} \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x)}{e^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{d+e x} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{a b+b^2 x}{d+e x} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{b^2}{e}-\frac{b (b d-a e)}{e (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac{b x \sqrt{a^2+2 a b x+b^2 x^2}}{e (a+b x)}-\frac{(b d-a e) \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0156397, size = 42, normalized size = 0.52 \[ \frac{\sqrt{(a+b x)^2} ((a e-b d) \log (d+e x)+b e x)}{e^2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.201, size = 44, normalized size = 0.6 \begin{align*}{\frac{{\it csgn} \left ( bx+a \right ) \left ( \ln \left ( bxe+bd \right ) ae-\ln \left ( bxe+bd \right ) bd+bxe+ae \right ) }{{e}^{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 = 1.54714, size = 54, normalized size = 0.68 \begin{align*} \frac{b e x -{\left (b d - a e\right )} \log \left (e x + d\right )}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.585905, size = 20, normalized size = 0.25 \begin{align*} \frac{b x}{e} + \frac{\left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20215, size = 61, normalized size = 0.76 \begin{align*} b x e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) -{\left (b d \mathrm{sgn}\left (b x + a\right ) - a e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-2\right )} \log \left ({\left | x e + d \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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