Optimal. Leaf size=46 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 (d+e x)^2 (b d-a e)} \]
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Rubi [A] time = 0.0198825, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 37} \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 (d+e x)^2 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^3} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{a b+b^2 x}{(d+e x)^3} \, dx}{a b+b^2 x}\\ &=\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 (b d-a e) (d+e x)^2}\\ \end{align*}
Mathematica [A] time = 0.0163794, size = 44, normalized size = 0.96 \[ -\frac{\sqrt{(a+b x)^2} (a e+b (d+2 e x))}{2 e^2 (a+b x) (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 41, normalized size = 0.9 \begin{align*} -{\frac{2\,bxe+ae+bd}{2\, \left ( ex+d \right ) ^{2}{e}^{2} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{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 = 1.55283, size = 81, normalized size = 1.76 \begin{align*} -\frac{2 \, b e x + b d + a e}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.451762, size = 39, normalized size = 0.85 \begin{align*} - \frac{a e + b d + 2 b e x}{2 d^{2} e^{2} + 4 d e^{3} x + 2 e^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13743, size = 59, normalized size = 1.28 \begin{align*} -\frac{{\left (2 \, b x e \mathrm{sgn}\left (b x + a\right ) + b d \mathrm{sgn}\left (b x + a\right ) + a e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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