Optimal. Leaf size=108 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}+\frac{b B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.043048, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {770, 76} \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}+\frac{b B \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
Antiderivative was successfully verified.
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Rule 770
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \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{\left (a b+b^2 x\right ) (A+B x)}{x^3} \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a A b}{x^3}+\frac{b (A b+a B)}{x^2}+\frac{b^2 B}{x}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{(A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0184731, size = 48, normalized size = 0.44 \[ -\frac{\sqrt{(a+b x)^2} \left (a (A+2 B x)+2 A b x-2 b B x^2 \log (x)\right )}{2 x^2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 37, normalized size = 0.3 \begin{align*} -{\frac{{\it csgn} \left ( bx+a \right ) \left ( -2\,B\ln \left ( bx \right ) b{x}^{2}+2\,Abx+2\,aBx+aA \right ) }{2\,{x}^{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 = 1.27811, size = 70, normalized size = 0.65 \begin{align*} \frac{2 \, B b x^{2} \log \left (x\right ) - A a - 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.450945, size = 26, normalized size = 0.24 \begin{align*} B b \log{\left (x \right )} - \frac{A a + x \left (2 A b + 2 B a\right )}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18057, size = 68, normalized size = 0.63 \begin{align*} B b \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) - \frac{A a \mathrm{sgn}\left (b x + a\right ) + 2 \,{\left (B a \mathrm{sgn}\left (b x + a\right ) + A b \mathrm{sgn}\left (b x + a\right )\right )} x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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