Optimal. Leaf size=34 \[ -\frac{1}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0043055, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {607} \[ -\frac{1}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 607
Rubi steps
\begin{align*} \int \frac{1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac{1}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0085033, size = 23, normalized size = 0.68 \[ -\frac{a+b x}{2 b \left ((a+b x)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 20, normalized size = 0.6 \begin{align*} -{\frac{bx+a}{2\,b} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46967, size = 22, normalized size = 0.65 \begin{align*} -\frac{1}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61633, size = 49, normalized size = 1.44 \begin{align*} -\frac{1}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \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{1}{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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