Optimal. Leaf size=45 \[ \frac{a (A b-a B)}{b^3 (a+b x)}+\frac{(A b-2 a B) \log (a+b x)}{b^3}+\frac{B x}{b^2} \]
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Rubi [A] time = 0.0362801, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {27, 77} \[ \frac{a (A b-a B)}{b^3 (a+b x)}+\frac{(A b-2 a B) \log (a+b x)}{b^3}+\frac{B x}{b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{x (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{x (A+B x)}{(a+b x)^2} \, dx\\ &=\int \left (\frac{B}{b^2}+\frac{a (-A b+a B)}{b^2 (a+b x)^2}+\frac{A b-2 a B}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{B x}{b^2}+\frac{a (A b-a B)}{b^3 (a+b x)}+\frac{(A b-2 a B) \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0241257, size = 41, normalized size = 0.91 \[ \frac{\frac{a (A b-a B)}{a+b x}+(A b-2 a B) \log (a+b x)+b B x}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 61, normalized size = 1.4 \begin{align*}{\frac{Bx}{{b}^{2}}}+{\frac{aA}{{b}^{2} \left ( bx+a \right ) }}-{\frac{B{a}^{2}}{{b}^{3} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) A}{{b}^{2}}}-2\,{\frac{\ln \left ( bx+a \right ) aB}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00007, size = 72, normalized size = 1.6 \begin{align*} -\frac{B a^{2} - A a b}{b^{4} x + a b^{3}} + \frac{B x}{b^{2}} - \frac{{\left (2 \, B a - A b\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2461, size = 149, normalized size = 3.31 \begin{align*} \frac{B b^{2} x^{2} + B a b x - B a^{2} + A a b -{\left (2 \, B a^{2} - A a b +{\left (2 \, B a b - A b^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.464968, size = 44, normalized size = 0.98 \begin{align*} \frac{B x}{b^{2}} - \frac{- A a b + B a^{2}}{a b^{3} + b^{4} x} - \frac{\left (- A b + 2 B a\right ) \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17415, size = 69, normalized size = 1.53 \begin{align*} \frac{B x}{b^{2}} - \frac{{\left (2 \, B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac{B a^{2} - A a b}{{\left (b x + a\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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