Optimal. Leaf size=80 \[ \frac{A \log (x) (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) (A b-a B) \log (a+b x)}{a b \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0500902, antiderivative size = 80, 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, 72} \[ \frac{A \log (x) (a+b x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) (A b-a B) \log (a+b x)}{a b \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 72
Rubi steps
\begin{align*} \int \frac{A+B x}{x \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}{x \left (a b+b^2 x\right )} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{A}{a b x}+\frac{-A b+a B}{a b (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{A (a+b x) \log (x)}{a \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (a+b x) \log (a+b x)}{a b \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0208284, size = 44, normalized size = 0.55 \[ \frac{(a+b x) ((a B-A b) \log (a+b x)+A b \log (x))}{a b \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 47, normalized size = 0.6 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( Ab\ln \left ( x \right ) -A\ln \left ( bx+a \right ) b+B\ln \left ( bx+a \right ) a \right ) }{ba}{\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 = 1.54205, size = 63, normalized size = 0.79 \begin{align*} \frac{A b \log \left (x\right ) +{\left (B a - A b\right )} \log \left (b x + a\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.499915, size = 41, normalized size = 0.51 \begin{align*} \frac{A \log{\left (x \right )}}{a} + \frac{\left (- A b + B a\right ) \log{\left (x + \frac{- A a + \frac{a \left (- A b + B a\right )}{b}}{- 2 A b + B a} \right )}}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13453, size = 66, normalized size = 0.82 \begin{align*} \frac{A \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right )}{a} + \frac{{\left (B a \mathrm{sgn}\left (b x + a\right ) - A b \mathrm{sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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