Optimal. Leaf size=106 \[ \frac{b^2 (A b-a B)}{a^4 x}+\frac{b^3 \log (x) (A b-a B)}{a^5}-\frac{b^3 (A b-a B) \log (a+b x)}{a^5}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{3 a^2 x^3}-\frac{A}{4 a x^4} \]
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Rubi [A] time = 0.0595751, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ \frac{b^2 (A b-a B)}{a^4 x}+\frac{b^3 \log (x) (A b-a B)}{a^5}-\frac{b^3 (A b-a B) \log (a+b x)}{a^5}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{3 a^2 x^3}-\frac{A}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^5 (a+b x)} \, dx &=\int \left (\frac{A}{a x^5}+\frac{-A b+a B}{a^2 x^4}-\frac{b (-A b+a B)}{a^3 x^3}+\frac{b^2 (-A b+a B)}{a^4 x^2}-\frac{b^3 (-A b+a B)}{a^5 x}+\frac{b^4 (-A b+a B)}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{A}{4 a x^4}+\frac{A b-a B}{3 a^2 x^3}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{b^2 (A b-a B)}{a^4 x}+\frac{b^3 (A b-a B) \log (x)}{a^5}-\frac{b^3 (A b-a B) \log (a+b x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.062978, size = 100, normalized size = 0.94 \[ \frac{\frac{a \left (2 a^2 b x (2 A+3 B x)+a^3 (-(3 A+4 B x))-6 a b^2 x^2 (A+2 B x)+12 A b^3 x^3\right )}{x^4}+12 b^3 \log (x) (A b-a B)-12 b^3 (A b-a B) \log (a+b x)}{12 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 125, normalized size = 1.2 \begin{align*} -{\frac{A}{4\,a{x}^{4}}}+{\frac{Ab}{3\,{a}^{2}{x}^{3}}}-{\frac{B}{3\,a{x}^{3}}}+{\frac{{b}^{4}\ln \left ( x \right ) A}{{a}^{5}}}-{\frac{{b}^{3}B\ln \left ( x \right ) }{{a}^{4}}}-{\frac{{b}^{2}A}{2\,{a}^{3}{x}^{2}}}+{\frac{bB}{2\,{a}^{2}{x}^{2}}}+{\frac{{b}^{3}A}{{a}^{4}x}}-{\frac{{b}^{2}B}{{a}^{3}x}}-{\frac{{b}^{4}\ln \left ( bx+a \right ) A}{{a}^{5}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) B}{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00312, size = 151, normalized size = 1.42 \begin{align*} \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (b x + a\right )}{a^{5}} - \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (x\right )}{a^{5}} - \frac{3 \, A a^{3} + 12 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 4 \,{\left (B a^{3} - A a^{2} b\right )} x}{12 \, a^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95608, size = 250, normalized size = 2.36 \begin{align*} \frac{12 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (b x + a\right ) - 12 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (x\right ) - 3 \, A a^{4} - 12 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 6 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} - 4 \,{\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.939351, size = 189, normalized size = 1.78 \begin{align*} - \frac{3 A a^{3} + x^{3} \left (- 12 A b^{3} + 12 B a b^{2}\right ) + x^{2} \left (6 A a b^{2} - 6 B a^{2} b\right ) + x \left (- 4 A a^{2} b + 4 B a^{3}\right )}{12 a^{4} x^{4}} - \frac{b^{3} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{4} + B a^{2} b^{3} - a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} + \frac{b^{3} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{4} + B a^{2} b^{3} + a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.184, size = 165, normalized size = 1.56 \begin{align*} -\frac{{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{{\left (B a b^{4} - A b^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{3 \, A a^{4} + 12 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 6 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 4 \,{\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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