Optimal. Leaf size=113 \[ -\frac{b^2 (A b-a B)}{a^4 (a+b x)}-\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac{b^2 (4 A b-3 a B) \log (a+b x)}{a^5}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{A}{3 a^2 x^3} \]
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Rubi [A] time = 0.0920331, antiderivative size = 113, 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} \[ -\frac{b^2 (A b-a B)}{a^4 (a+b x)}-\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac{b^2 (4 A b-3 a B) \log (a+b x)}{a^5}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{A}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{A+B x}{x^4 (a+b x)^2} \, dx\\ &=\int \left (\frac{A}{a^2 x^4}+\frac{-2 A b+a B}{a^3 x^3}-\frac{b (-3 A b+2 a B)}{a^4 x^2}+\frac{b^2 (-4 A b+3 a B)}{a^5 x}-\frac{b^3 (-A b+a B)}{a^4 (a+b x)^2}-\frac{b^3 (-4 A b+3 a B)}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{A}{3 a^2 x^3}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{b^2 (A b-a B)}{a^4 (a+b x)}-\frac{b^2 (4 A b-3 a B) \log (x)}{a^5}+\frac{b^2 (4 A b-3 a B) \log (a+b x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.086037, size = 106, normalized size = 0.94 \[ \frac{-\frac{3 a^2 (a B-2 A b)}{x^2}-\frac{2 a^3 A}{x^3}+\frac{6 a b^2 (a B-A b)}{a+b x}+6 b^2 \log (x) (3 a B-4 A b)+6 b^2 (4 A b-3 a B) \log (a+b x)+\frac{6 a b (2 a B-3 A b)}{x}}{6 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 134, normalized size = 1.2 \begin{align*} -{\frac{A}{3\,{a}^{2}{x}^{3}}}+{\frac{Ab}{{a}^{3}{x}^{2}}}-{\frac{B}{2\,{a}^{2}{x}^{2}}}-3\,{\frac{A{b}^{2}}{{a}^{4}x}}+2\,{\frac{bB}{{a}^{3}x}}-4\,{\frac{A{b}^{3}\ln \left ( x \right ) }{{a}^{5}}}+3\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{4}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{5}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{4}}}-{\frac{A{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}+{\frac{{b}^{2}B}{{a}^{3} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00453, size = 173, normalized size = 1.53 \begin{align*} -\frac{2 \, A a^{3} - 6 \,{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} +{\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} x}{6 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} - \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{5}} + \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (x\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3164, size = 378, normalized size = 3.35 \begin{align*} -\frac{2 \, A a^{4} - 6 \,{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} +{\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x + 6 \,{\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} +{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} +{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.885723, size = 219, normalized size = 1.94 \begin{align*} \frac{- 2 A a^{3} + x^{3} \left (- 24 A b^{3} + 18 B a b^{2}\right ) + x^{2} \left (- 12 A a b^{2} + 9 B a^{2} b\right ) + x \left (4 A a^{2} b - 3 B a^{3}\right )}{6 a^{5} x^{3} + 6 a^{4} b x^{4}} + \frac{b^{2} \left (- 4 A b + 3 B a\right ) \log{\left (x + \frac{- 4 A a b^{3} + 3 B a^{2} b^{2} - a b^{2} \left (- 4 A b + 3 B a\right )}{- 8 A b^{4} + 6 B a b^{3}} \right )}}{a^{5}} - \frac{b^{2} \left (- 4 A b + 3 B a\right ) \log{\left (x + \frac{- 4 A a b^{3} + 3 B a^{2} b^{2} + a b^{2} \left (- 4 A b + 3 B a\right )}{- 8 A b^{4} + 6 B a b^{3}} \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13514, size = 180, normalized size = 1.59 \begin{align*} \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac{{\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{2 \, A a^{4} - 6 \,{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} +{\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x}{6 \,{\left (b x + a\right )} a^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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