Optimal. Leaf size=85 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) (d+e x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)} \]
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Rubi [A] time = 0.0405717, antiderivative size = 85, 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} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) (d+e x)}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)} \]
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)^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{a b+b^2 x}{(d+e x)^2} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b (b d-a e)}{e (d+e x)^2}+\frac{b^2}{e (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) (d+e x)}+\frac{b \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 = 0.0194631, size = 50, normalized size = 0.59 \[ \frac{\sqrt{(a+b x)^2} (-a e+b (d+e x) \log (d+e x)+b d)}{e^2 (a+b x) (d+e x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.227, size = 51, normalized size = 0.6 \begin{align*}{\frac{{\it csgn} \left ( bx+a \right ) \left ( \ln \left ( bxe+bd \right ) xbe+\ln \left ( bxe+bd \right ) bd-ae+bd \right ) }{{e}^{2} \left ( ex+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 = 1.57975, size = 78, normalized size = 0.92 \begin{align*} \frac{b d - a e +{\left (b e x + b d\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.379246, size = 27, normalized size = 0.32 \begin{align*} \frac{b \log{\left (d + e x \right )}}{e^{2}} - \frac{a e - b d}{d e^{2} + e^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22229, size = 69, normalized size = 0.81 \begin{align*} b e^{\left (-2\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (b d \mathrm{sgn}\left (b x + a\right ) - a e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{x e + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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