Optimal. Leaf size=39 \[ -\frac{2 (a+b x)}{e \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}} \]
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Rubi [A] time = 0.0294353, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {770, 21, 32} \[ -\frac{2 (a+b x)}{e \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}} \]
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)^{3/2} \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)^{3/2}} \, 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)^{3/2}} \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 (a+b x)}{e \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0141988, size = 30, normalized size = 0.77 \[ -\frac{2 (a+b x)}{e \sqrt{(a+b x)^2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 27, normalized size = 0.7 \begin{align*} -2\,{\frac{bx+a}{e\sqrt{ex+d}\sqrt{ \left ( bx+a \right ) ^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19779, size = 27, normalized size = 0.69 \begin{align*} -\frac{2 \, \sqrt{e x + d}}{e^{2} x + d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.09469, size = 43, normalized size = 1.1 \begin{align*} -\frac{2 \, \sqrt{e x + d}}{e^{2} x + d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b x}{\left (d + e x\right )^{\frac{3}{2}} \sqrt{\left (a + b x\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14671, size = 24, normalized size = 0.62 \begin{align*} -\frac{2 \, e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right )}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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