Optimal. Leaf size=39 \[ -\frac{a+b x}{2 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2} \]
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Rubi [A] time = 0.028474, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 32} \[ -\frac{a+b x}{2 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{a+b x}{(d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{a+b x}{\left (a b+b^2 x\right ) (d+e x)^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{(d+e x)^3} \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{a+b x}{2 e (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0113091, size = 30, normalized size = 0.77 \[ -\frac{a+b x}{2 e \sqrt{(a+b x)^2} (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 27, normalized size = 0.7 \begin{align*} -{\frac{bx+a}{2\,e \left ( ex+d \right ) ^{2}}{\frac{1}{\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 = 1.4958, size = 49, normalized size = 1.26 \begin{align*} -\frac{1}{2 \,{\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.371408, size = 26, normalized size = 0.67 \begin{align*} - \frac{1}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12804, size = 24, normalized size = 0.62 \begin{align*} -\frac{e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right )}{2 \,{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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