Optimal. Leaf size=72 \[ \frac{A}{a^3 (a+b x)}+\frac{A}{2 a^2 (a+b x)^2}-\frac{A \log (a+b x)}{a^4}+\frac{A \log (x)}{a^4}+\frac{A b-a B}{3 a b (a+b x)^3} \]
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Rubi [A] time = 0.0492757, antiderivative size = 72, 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} \[ \frac{A}{a^3 (a+b x)}+\frac{A}{2 a^2 (a+b x)^2}-\frac{A \log (a+b x)}{a^4}+\frac{A \log (x)}{a^4}+\frac{A b-a B}{3 a b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{A+B x}{x (a+b x)^4} \, dx\\ &=\int \left (\frac{A}{a^4 x}+\frac{-A b+a B}{a (a+b x)^4}-\frac{A b}{a^2 (a+b x)^3}-\frac{A b}{a^3 (a+b x)^2}-\frac{A b}{a^4 (a+b x)}\right ) \, dx\\ &=\frac{A b-a B}{3 a b (a+b x)^3}+\frac{A}{2 a^2 (a+b x)^2}+\frac{A}{a^3 (a+b x)}+\frac{A \log (x)}{a^4}-\frac{A \log (a+b x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0462261, size = 65, normalized size = 0.9 \[ \frac{\frac{a \left (11 a^2 A b-2 a^3 B+15 a A b^2 x+6 A b^3 x^2\right )}{b (a+b x)^3}-6 A \log (a+b x)+6 A \log (x)}{6 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 72, normalized size = 1. \begin{align*}{\frac{A\ln \left ( x \right ) }{{a}^{4}}}+{\frac{A}{3\,a \left ( bx+a \right ) ^{3}}}-{\frac{B}{3\,b \left ( bx+a \right ) ^{3}}}-{\frac{A\ln \left ( bx+a \right ) }{{a}^{4}}}+{\frac{A}{{a}^{3} \left ( bx+a \right ) }}+{\frac{A}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0186, size = 123, normalized size = 1.71 \begin{align*} \frac{6 \, A b^{3} x^{2} + 15 \, A a b^{2} x - 2 \, B a^{3} + 11 \, A a^{2} b}{6 \,{\left (a^{3} b^{4} x^{3} + 3 \, a^{4} b^{3} x^{2} + 3 \, a^{5} b^{2} x + a^{6} b\right )}} - \frac{A \log \left (b x + a\right )}{a^{4}} + \frac{A \log \left (x\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.3773, size = 336, normalized size = 4.67 \begin{align*} \frac{6 \, A a b^{3} x^{2} + 15 \, A a^{2} b^{2} x - 2 \, B a^{4} + 11 \, A a^{3} b - 6 \,{\left (A b^{4} x^{3} + 3 \, A a b^{3} x^{2} + 3 \, A a^{2} b^{2} x + A a^{3} b\right )} \log \left (b x + a\right ) + 6 \,{\left (A b^{4} x^{3} + 3 \, A a b^{3} x^{2} + 3 \, A a^{2} b^{2} x + A a^{3} b\right )} \log \left (x\right )}{6 \,{\left (a^{4} b^{4} x^{3} + 3 \, a^{5} b^{3} x^{2} + 3 \, a^{6} b^{2} x + a^{7} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.784203, size = 90, normalized size = 1.25 \begin{align*} \frac{A \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{4}} + \frac{11 A a^{2} b + 15 A a b^{2} x + 6 A b^{3} x^{2} - 2 B a^{3}}{6 a^{6} b + 18 a^{5} b^{2} x + 18 a^{4} b^{3} x^{2} + 6 a^{3} b^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1663, size = 96, normalized size = 1.33 \begin{align*} -\frac{A \log \left ({\left | b x + a \right |}\right )}{a^{4}} + \frac{A \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{6 \, A a b^{3} x^{2} + 15 \, A a^{2} b^{2} x - 2 \, B a^{4} + 11 \, A a^{3} b}{6 \,{\left (b x + a\right )}^{3} a^{4} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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