Optimal. Leaf size=86 \[ -\frac{b^2 \log (x) (A b-a B)}{a^4}+\frac{b^2 (A b-a B) \log (a+b x)}{a^4}+\frac{A b-a B}{2 a^2 x^2}-\frac{b (A b-a B)}{a^3 x}-\frac{A}{3 a x^3} \]
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Rubi [A] time = 0.0506325, antiderivative size = 86, 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} \[ -\frac{b^2 \log (x) (A b-a B)}{a^4}+\frac{b^2 (A b-a B) \log (a+b x)}{a^4}+\frac{A b-a B}{2 a^2 x^2}-\frac{b (A b-a B)}{a^3 x}-\frac{A}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x^4 (a+b x)} \, dx &=\int \left (\frac{A}{a x^4}+\frac{-A b+a B}{a^2 x^3}-\frac{b (-A b+a B)}{a^3 x^2}+\frac{b^2 (-A b+a B)}{a^4 x}-\frac{b^3 (-A b+a B)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac{A}{3 a x^3}+\frac{A b-a B}{2 a^2 x^2}-\frac{b (A b-a B)}{a^3 x}-\frac{b^2 (A b-a B) \log (x)}{a^4}+\frac{b^2 (A b-a B) \log (a+b x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0499314, size = 81, normalized size = 0.94 \[ \frac{\frac{a \left (a^2 (-(2 A+3 B x))+3 a b x (A+2 B x)-6 A b^2 x^2\right )}{x^3}+6 b^2 \log (x) (a B-A b)+6 b^2 (A b-a B) \log (a+b x)}{6 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 101, normalized size = 1.2 \begin{align*} -{\frac{A}{3\,a{x}^{3}}}+{\frac{Ab}{2\,{a}^{2}{x}^{2}}}-{\frac{B}{2\,a{x}^{2}}}-{\frac{{b}^{2}A}{{a}^{3}x}}+{\frac{bB}{{a}^{2}x}}-{\frac{A\ln \left ( x \right ){b}^{3}}{{a}^{4}}}+{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{3}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{4}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03134, size = 120, normalized size = 1.4 \begin{align*} -\frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (b x + a\right )}{a^{4}} + \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (x\right )}{a^{4}} - \frac{2 \, A a^{2} - 6 \,{\left (B a b - A b^{2}\right )} x^{2} + 3 \,{\left (B a^{2} - A a b\right )} x}{6 \, a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94261, size = 203, normalized size = 2.36 \begin{align*} -\frac{6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (b x + a\right ) - 6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (x\right ) + 2 \, A a^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.93392, size = 165, normalized size = 1.92 \begin{align*} \frac{- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 6 B a b\right ) + x \left (3 A a b - 3 B a^{2}\right )}{6 a^{3} x^{3}} + \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} - a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} - \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} + a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31399, size = 134, normalized size = 1.56 \begin{align*} \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, A a^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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