Optimal. Leaf size=65 \[ -\frac{A b-a B}{a^2 (a+b x)}-\frac{\log (x) (2 A b-a B)}{a^3}+\frac{(2 A b-a B) \log (a+b x)}{a^3}-\frac{A}{a^2 x} \]
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Rubi [A] time = 0.0505052, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 77} \[ -\frac{A b-a B}{a^2 (a+b x)}-\frac{\log (x) (2 A b-a B)}{a^3}+\frac{(2 A b-a B) \log (a+b x)}{a^3}-\frac{A}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{A+B x}{x^2 (a+b x)^2} \, dx\\ &=\int \left (\frac{A}{a^2 x^2}+\frac{-2 A b+a B}{a^3 x}-\frac{b (-A b+a B)}{a^2 (a+b x)^2}-\frac{b (-2 A b+a B)}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac{A}{a^2 x}-\frac{A b-a B}{a^2 (a+b x)}-\frac{(2 A b-a B) \log (x)}{a^3}+\frac{(2 A b-a B) \log (a+b x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0388051, size = 56, normalized size = 0.86 \[ \frac{\frac{a (a B-A b)}{a+b x}+\log (x) (a B-2 A b)+(2 A b-a B) \log (a+b x)-\frac{a A}{x}}{a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 78, normalized size = 1.2 \begin{align*} -{\frac{A}{{a}^{2}x}}-2\,{\frac{Ab\ln \left ( x \right ) }{{a}^{3}}}+{\frac{\ln \left ( x \right ) B}{{a}^{2}}}+2\,{\frac{\ln \left ( bx+a \right ) Ab}{{a}^{3}}}-{\frac{\ln \left ( bx+a \right ) B}{{a}^{2}}}-{\frac{Ab}{{a}^{2} \left ( bx+a \right ) }}+{\frac{B}{a \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06314, size = 90, normalized size = 1.38 \begin{align*} -\frac{A a -{\left (B a - 2 \, A b\right )} x}{a^{2} b x^{2} + a^{3} x} - \frac{{\left (B a - 2 \, A b\right )} \log \left (b x + a\right )}{a^{3}} + \frac{{\left (B a - 2 \, A b\right )} \log \left (x\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34274, size = 227, normalized size = 3.49 \begin{align*} -\frac{A a^{2} -{\left (B a^{2} - 2 \, A a b\right )} x +{\left ({\left (B a b - 2 \, A b^{2}\right )} x^{2} +{\left (B a^{2} - 2 \, A a b\right )} x\right )} \log \left (b x + a\right ) -{\left ({\left (B a b - 2 \, A b^{2}\right )} x^{2} +{\left (B a^{2} - 2 \, A a b\right )} x\right )} \log \left (x\right )}{a^{3} b x^{2} + a^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.655717, size = 128, normalized size = 1.97 \begin{align*} \frac{- A a + x \left (- 2 A b + B a\right )}{a^{3} x + a^{2} b x^{2}} + \frac{\left (- 2 A b + B a\right ) \log{\left (x + \frac{- 2 A a b + B a^{2} - a \left (- 2 A b + B a\right )}{- 4 A b^{2} + 2 B a b} \right )}}{a^{3}} - \frac{\left (- 2 A b + B a\right ) \log{\left (x + \frac{- 2 A a b + B a^{2} + a \left (- 2 A b + B a\right )}{- 4 A b^{2} + 2 B a b} \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09354, size = 96, normalized size = 1.48 \begin{align*} \frac{{\left (B a - 2 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{B a x - 2 \, A b x - A a}{{\left (b x^{2} + a x\right )} a^{2}} - \frac{{\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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