Optimal. Leaf size=57 \[ \frac{A}{a^2 (a+b x)}-\frac{A \log (a+b x)}{a^3}+\frac{A \log (x)}{a^3}+\frac{A b-a B}{2 a b (a+b x)^2} \]
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Rubi [A] time = 0.0357317, antiderivative size = 57, 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} \[ \frac{A}{a^2 (a+b x)}-\frac{A \log (a+b x)}{a^3}+\frac{A \log (x)}{a^3}+\frac{A b-a B}{2 a b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{x (a+b x)^3} \, dx &=\int \left (\frac{A}{a^3 x}+\frac{-A b+a B}{a (a+b x)^3}-\frac{A b}{a^2 (a+b x)^2}-\frac{A b}{a^3 (a+b x)}\right ) \, dx\\ &=\frac{A b-a B}{2 a b (a+b x)^2}+\frac{A}{a^2 (a+b x)}+\frac{A \log (x)}{a^3}-\frac{A \log (a+b x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0410038, size = 53, normalized size = 0.93 \[ \frac{\frac{a \left (a^2 (-B)+3 a A b+2 A b^2 x\right )}{b (a+b x)^2}-2 A \log (a+b x)+2 A \log (x)}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 59, normalized size = 1. \begin{align*}{\frac{A\ln \left ( x \right ) }{{a}^{3}}}+{\frac{A}{2\,a \left ( bx+a \right ) ^{2}}}-{\frac{B}{2\,b \left ( bx+a \right ) ^{2}}}-{\frac{A\ln \left ( bx+a \right ) }{{a}^{3}}}+{\frac{A}{{a}^{2} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43088, size = 92, normalized size = 1.61 \begin{align*} \frac{2 \, A b^{2} x - B a^{2} + 3 \, A a b}{2 \,{\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac{A \log \left (b x + a\right )}{a^{3}} + \frac{A \log \left (x\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98025, size = 236, normalized size = 4.14 \begin{align*} \frac{2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b - 2 \,{\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (b x + a\right ) + 2 \,{\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.67209, size = 63, normalized size = 1.11 \begin{align*} \frac{A \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{3}} + \frac{3 A a b + 2 A b^{2} x - B a^{2}}{2 a^{4} b + 4 a^{3} b^{2} x + 2 a^{2} b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20935, size = 80, normalized size = 1.4 \begin{align*} -\frac{A \log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac{A \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b}{2 \,{\left (b x + a\right )}^{2} a^{3} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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