Optimal. Leaf size=111 \[ -\frac{3 A b-a B}{a^4 (a+b x)}-\frac{2 A b-a B}{2 a^3 (a+b x)^2}-\frac{A b-a B}{3 a^2 (a+b x)^3}-\frac{\log (x) (4 A b-a B)}{a^5}+\frac{(4 A b-a B) \log (a+b x)}{a^5}-\frac{A}{a^4 x} \]
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Rubi [A] time = 0.1038, antiderivative size = 111, 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} \[ -\frac{3 A b-a B}{a^4 (a+b x)}-\frac{2 A b-a B}{2 a^3 (a+b x)^2}-\frac{A b-a B}{3 a^2 (a+b x)^3}-\frac{\log (x) (4 A b-a B)}{a^5}+\frac{(4 A b-a B) \log (a+b x)}{a^5}-\frac{A}{a^4 x} \]
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 )^2} \, dx &=\int \frac{A+B x}{x^2 (a+b x)^4} \, dx\\ &=\int \left (\frac{A}{a^4 x^2}+\frac{-4 A b+a B}{a^5 x}-\frac{b (-A b+a B)}{a^2 (a+b x)^4}-\frac{b (-2 A b+a B)}{a^3 (a+b x)^3}-\frac{b (-3 A b+a B)}{a^4 (a+b x)^2}-\frac{b (-4 A b+a B)}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{A}{a^4 x}-\frac{A b-a B}{3 a^2 (a+b x)^3}-\frac{2 A b-a B}{2 a^3 (a+b x)^2}-\frac{3 A b-a B}{a^4 (a+b x)}-\frac{(4 A b-a B) \log (x)}{a^5}+\frac{(4 A b-a B) \log (a+b x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0653748, size = 102, normalized size = 0.92 \[ \frac{\frac{2 a^3 (a B-A b)}{(a+b x)^3}+\frac{3 a^2 (a B-2 A b)}{(a+b x)^2}+\frac{6 a (a B-3 A b)}{a+b x}+6 \log (x) (a B-4 A b)+6 (4 A b-a B) \log (a+b x)-\frac{6 a A}{x}}{6 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 132, normalized size = 1.2 \begin{align*} -{\frac{A}{{a}^{4}x}}-4\,{\frac{Ab\ln \left ( x \right ) }{{a}^{5}}}+{\frac{\ln \left ( x \right ) B}{{a}^{4}}}-{\frac{Ab}{{a}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{B}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}}-3\,{\frac{Ab}{{a}^{4} \left ( bx+a \right ) }}+{\frac{B}{{a}^{3} \left ( bx+a \right ) }}+4\,{\frac{\ln \left ( bx+a \right ) Ab}{{a}^{5}}}-{\frac{\ln \left ( bx+a \right ) B}{{a}^{4}}}-{\frac{Ab}{3\,{a}^{2} \left ( bx+a \right ) ^{3}}}+{\frac{B}{3\,a \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07159, size = 181, normalized size = 1.63 \begin{align*} -\frac{6 \, A a^{3} - 6 \,{\left (B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 15 \,{\left (B a^{2} b - 4 \, A a b^{2}\right )} x^{2} - 11 \,{\left (B a^{3} - 4 \, A a^{2} b\right )} x}{6 \,{\left (a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{3} + 3 \, a^{6} b x^{2} + a^{7} x\right )}} - \frac{{\left (B a - 4 \, A b\right )} \log \left (b x + a\right )}{a^{5}} + \frac{{\left (B a - 4 \, A b\right )} \log \left (x\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.3564, size = 562, normalized size = 5.06 \begin{align*} -\frac{6 \, A a^{4} - 6 \,{\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \,{\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} - 11 \,{\left (B a^{4} - 4 \, A a^{3} b\right )} x + 6 \,{\left ({\left (B a b^{3} - 4 \, A b^{4}\right )} x^{4} + 3 \,{\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \,{\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} - 4 \, A a^{3} b\right )} x\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (B a b^{3} - 4 \, A b^{4}\right )} x^{4} + 3 \,{\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \,{\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} +{\left (B a^{4} - 4 \, A a^{3} b\right )} x\right )} \log \left (x\right )}{6 \,{\left (a^{5} b^{3} x^{4} + 3 \, a^{6} b^{2} x^{3} + 3 \, a^{7} b x^{2} + a^{8} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.11369, size = 204, normalized size = 1.84 \begin{align*} \frac{- 6 A a^{3} + x^{3} \left (- 24 A b^{3} + 6 B a b^{2}\right ) + x^{2} \left (- 60 A a b^{2} + 15 B a^{2} b\right ) + x \left (- 44 A a^{2} b + 11 B a^{3}\right )}{6 a^{7} x + 18 a^{6} b x^{2} + 18 a^{5} b^{2} x^{3} + 6 a^{4} b^{3} x^{4}} + \frac{\left (- 4 A b + B a\right ) \log{\left (x + \frac{- 4 A a b + B a^{2} - a \left (- 4 A b + B a\right )}{- 8 A b^{2} + 2 B a b} \right )}}{a^{5}} - \frac{\left (- 4 A b + B a\right ) \log{\left (x + \frac{- 4 A a b + B a^{2} + a \left (- 4 A b + B a\right )}{- 8 A b^{2} + 2 B a b} \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18083, size = 165, normalized size = 1.49 \begin{align*} \frac{{\left (B a - 4 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (B a b - 4 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{6 \, A a^{4} - 6 \,{\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 15 \,{\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} - 11 \,{\left (B a^{4} - 4 \, A a^{3} b\right )} x}{6 \,{\left (b x + a\right )}^{3} a^{5} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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