Optimal. Leaf size=35 \[ -\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 a x^2} \]
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Rubi [A] time = 0.0130166, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 37} \[ -\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 a x^2} \]
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}}{x^3} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{a b+b^2 x}{x^3} \, dx}{a b+b^2 x}\\ &=-\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 a x^2}\\ \end{align*}
Mathematica [A] time = 0.0069257, size = 31, normalized size = 0.89 \[ -\frac{\sqrt{(a+b x)^2} (a+2 b x)}{2 x^2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 28, normalized size = 0.8 \begin{align*} -{\frac{2\,bx+a}{2\,{x}^{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 = 2.02235, size = 30, normalized size = 0.86 \begin{align*} -\frac{2 \, b x + a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.582048, size = 12, normalized size = 0.34 \begin{align*} - \frac{a + 2 b x}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23143, size = 53, normalized size = 1.51 \begin{align*} -\frac{b^{2} \mathrm{sgn}\left (b x + a\right )}{2 \, a} - \frac{2 \, b x \mathrm{sgn}\left (b x + a\right ) + a \mathrm{sgn}\left (b x + a\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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